l3lab/ntp-mathlib huggingface dataset:
/-- If the sign of the oriented angle between two vectors is zero, either one of the vectors is
zero or the unoriented angle is 0 or π. -/
theorem eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero {x y : V}
    (h : (o.oangle x y).sign = 0) :
    x = 0 ∨ y = 0 ∨ InnerProductGeometry.angle x y = 0 ∨ InnerProductGeometry.angle x y = π := by
  by_cases hx : x = 0; · simp [hx]
  by_cases hy : y = 0; · simp [hy]
  rw [o.angle_eq_abs_oangle_toReal hx hy]
  rw [Real.Angle.sign_eq_zero_iff] at h
  

Proof state:
case neg
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : oangle o x y = 0 ∨ oangle o x y = ↑π
hx : ¬x = 0
hy : ¬y = 0
⊢ x = 0 ∨ y = 0 ∨ |Real.Angle.toReal (oangle o x y)| = 0 ∨ |Real.Angle.toReal (oangle o x y)| = π

Next tactic:
rcases h with (h | h)
/-- If the sign of the oriented angle between two vectors is zero, either one of the vectors is
zero or the unoriented angle is 0 or π. -/
theorem eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero {x y : V}
    (h : (o.oangle x y).sign = 0) :
    x = 0 ∨ y = 0 ∨ InnerProductGeometry.angle x y = 0 ∨ InnerProductGeometry.angle x y = π := by
  by_cases hx : x = 0; · simp [hx]
  by_cases hy : y = 0; · simp [hy]
  rw [o.angle_eq_abs_oangle_toReal hx hy]
  rw [Real.Angle.sign_eq_zero_iff] at h
  rcases h with (h | h) <;> 

Proof state:
case neg.inl
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : ¬x = 0
hy : ¬y = 0
h : oangle o x y = 0
⊢ x = 0 ∨ y = 0 ∨ |Real.Angle.toReal (oangle o x y)| = 0 ∨ |Real.Angle.toReal (oangle o x y)| = π

Next tactic:
simp [h, Real.pi_pos.le]
/-- If the sign of the oriented angle between two vectors is zero, either one of the vectors is
zero or the unoriented angle is 0 or π. -/
theorem eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero {x y : V}
    (h : (o.oangle x y).sign = 0) :
    x = 0 ∨ y = 0 ∨ InnerProductGeometry.angle x y = 0 ∨ InnerProductGeometry.angle x y = π := by
  by_cases hx : x = 0; · simp [hx]
  by_cases hy : y = 0; · simp [hy]
  rw [o.angle_eq_abs_oangle_toReal hx hy]
  rw [Real.Angle.sign_eq_zero_iff] at h
  rcases h with (h | h) <;> 

Proof state:
case neg.inr
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : ¬x = 0
hy : ¬y = 0
h : oangle o x y = ↑π
⊢ x = 0 ∨ y = 0 ∨ |Real.Angle.toReal (oangle o x y)| = 0 ∨ |Real.Angle.toReal (oangle o x y)| = π

Next tactic:
simp [h, Real.pi_pos.le]
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z)
⊢ oangle o w x = oangle o y z

Next tactic:
by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · 

Proof state:
case pos
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z)
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
⊢ oangle o w x = oangle o y z

Next tactic:
have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · simpa using hs
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z)
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
⊢ Real.Angle.sign (oangle o w x) = 0 ∧ Real.Angle.sign (oangle o y z) = 0

Next tactic:
rcases h0 with ((rfl | rfl) | rfl | rfl)
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · 

Proof state:
case inl.inl
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
h : InnerProductGeometry.angle 0 x = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o 0 x) = Real.Angle.sign (oangle o y z)
⊢ Real.Angle.sign (oangle o 0 x) = 0 ∧ Real.Angle.sign (oangle o y z) = 0

Next tactic:
simpa using hs.symm
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · 

Proof state:
case inl.inr
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w y z : V
h : InnerProductGeometry.angle w 0 = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o w 0) = Real.Angle.sign (oangle o y z)
⊢ Real.Angle.sign (oangle o w 0) = 0 ∧ Real.Angle.sign (oangle o y z) = 0

Next tactic:
simpa using hs.symm
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · 

Proof state:
case inr.inl
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle 0 z
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o 0 z)
⊢ Real.Angle.sign (oangle o w x) = 0 ∧ Real.Angle.sign (oangle o 0 z) = 0

Next tactic:
simpa using hs
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · 

Proof state:
case inr.inr
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x y : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y 0)
⊢ Real.Angle.sign (oangle o w x) = 0 ∧ Real.Angle.sign (oangle o y 0) = 0

Next tactic:
simpa using hs
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · simpa using hs
    

Proof state:
case pos
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z)
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hs' : Real.Angle.sign (oangle o w x) = 0 ∧ Real.Angle.sign (oangle o y z) = 0
⊢ oangle o w x = oangle o y z

Next tactic:
rcases hs' with ⟨hswx, hsyz⟩
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · simpa using hs
    rcases hs' with ⟨hswx, hsyz⟩
    

Proof state:
case pos.intro
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z)
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : Real.Angle.sign (oangle o w x) = 0
hsyz : Real.Angle.sign (oangle o y z) = 0
⊢ oangle o w x = oangle o y z

Next tactic:
have h' : InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using h.symm
      · simpa using h.symm
      · simpa using h
      · simpa using h
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · simpa using hs
    rcases hs' with ⟨hswx, hsyz⟩
    have h' : InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2 := by
      

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z)
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : Real.Angle.sign (oangle o w x) = 0
hsyz : Real.Angle.sign (oangle o y z) = 0
⊢ InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2

Next tactic:
rcases h0 with ((rfl | rfl) | rfl | rfl)
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · simpa using hs
    rcases hs' with ⟨hswx, hsyz⟩
    have h' : InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · 

Proof state:
case inl.inl
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hsyz : Real.Angle.sign (oangle o y z) = 0
h : InnerProductGeometry.angle 0 x = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o 0 x) = Real.Angle.sign (oangle o y z)
hswx : Real.Angle.sign (oangle o 0 x) = 0
⊢ InnerProductGeometry.angle 0 x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2

Next tactic:
simpa using h.symm
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · simpa using hs
    rcases hs' with ⟨hswx, hsyz⟩
    have h' : InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using h.symm
      · 

Proof state:
case inl.inr
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w y z : V
hsyz : Real.Angle.sign (oangle o y z) = 0
h : InnerProductGeometry.angle w 0 = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o w 0) = Real.Angle.sign (oangle o y z)
hswx : Real.Angle.sign (oangle o w 0) = 0
⊢ InnerProductGeometry.angle w 0 = π / 2 ∧ InnerProductGeometry.angle y z = π / 2

Next tactic:
simpa using h.symm
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · simpa using hs
    rcases hs' with ⟨hswx, hsyz⟩
    have h' : InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using h.symm
      · simpa using h.symm
      · 

Proof state:
case inr.inl
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x z : V
hswx : Real.Angle.sign (oangle o w x) = 0
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle 0 z
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o 0 z)
hsyz : Real.Angle.sign (oangle o 0 z) = 0
⊢ InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle 0 z = π / 2

Next tactic:
simpa using h
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · simpa using hs
    rcases hs' with ⟨hswx, hsyz⟩
    have h' : InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using h.symm
      · simpa using h.symm
      · simpa using h
      · 

Proof state:
case inr.inr
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x y : V
hswx : Real.Angle.sign (oangle o w x) = 0
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y 0)
hsyz : Real.Angle.sign (oangle o y 0) = 0
⊢ InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y 0 = π / 2

Next tactic:
simpa using h
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · simpa using hs
    rcases hs' with ⟨hswx, hsyz⟩
    have h' : InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using h.symm
      · simpa using h.symm
      · simpa using h
      · simpa using h
    

Proof state:
case pos.intro
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z)
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : Real.Angle.sign (oangle o w x) = 0
hsyz : Real.Angle.sign (oangle o y z) = 0
h' : InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2
⊢ oangle o w x = oangle o y z

Next tactic:
rcases h' with ⟨hwx, hyz⟩
/-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z := by
  by_cases h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
  · have hs' : (o.oangle w x).sign = 0 ∧ (o.oangle y z).sign = 0 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using hs.symm
      · simpa using hs.symm
      · simpa using hs
      · simpa using hs
    rcases hs' with ⟨hswx, hsyz⟩
    have h' : InnerProductGeometry.angle w x = π / 2 ∧ InnerProductGeometry.angle y z = π / 2 := by
      rcases h0 with ((rfl | rfl) | rfl | rfl)
      · simpa using h.symm
      · simpa using h.symm
      · simpa using h
      · simpa using h
    rcases h' with ⟨hwx, hyz⟩
    

Proof state:
case pos.intro.intro
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z)
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : Real.Angle.sign (oangle o w x) = 0
hsyz : Real.Angle.sign (oangle o y z) = 0
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
⊢ oangle o w x = oangle o y z

Next tactic:
have hpi : π / 2 ≠ π := by
      intro hpi
      rw [div_eq_iff, eq_comm, ← sub_eq_zero, mul_two, add_sub_cancel] at hpi
      · exact Real.pi_pos.ne.symm hpi
      · exact two_ne_zero