l3lab/ntp-mathlib huggingface dataset:
/-- 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⟩
    have hpi : π / 2 ≠ π := 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
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
⊢ π / 2 ≠ π

Next tactic:
intro hpi
/-- 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⟩
    have hpi : π / 2 ≠ π := by
      intro hpi
      

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
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π / 2 = π
⊢ False

Next tactic:
rw [div_eq_iff, eq_comm, ← sub_eq_zero, mul_two, add_sub_cancel] at hpi
/-- 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⟩
    have hpi : π / 2 ≠ π := by
      intro hpi
      rw [div_eq_iff, eq_comm, ← sub_eq_zero, mul_two, add_sub_cancel] at hpi
      · 

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
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π = 0
⊢ False

Next tactic:
exact Real.pi_pos.ne.symm hpi
/-- 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⟩
    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
      · 

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
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π / 2 = π
⊢ 2 ≠ 0

Next tactic:
exact two_ne_zero
/-- 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⟩
    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
    

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
hpi : π / 2 ≠ π
⊢ oangle o w x = oangle o y z

Next tactic:
have h0wx : w = 0 ∨ x = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hswx
      simpa [hwx, Real.pi_pos.ne.symm, hpi] using h0'
/-- 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⟩
    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
    have h0wx : w = 0 ∨ x = 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
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
hpi : π / 2 ≠ π
⊢ w = 0 ∨ x = 0

Next tactic:
have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hswx
/-- 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⟩
    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
    have h0wx : w = 0 ∨ x = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hswx
      

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
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π / 2 ≠ π
h0' : w = 0 ∨ x = 0 ∨ InnerProductGeometry.angle w x = 0 ∨ InnerProductGeometry.angle w x = π
⊢ w = 0 ∨ x = 0

Next tactic:
simpa [hwx, Real.pi_pos.ne.symm, hpi] using h0'
/-- 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⟩
    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
    have h0wx : w = 0 ∨ x = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hswx
      simpa [hwx, Real.pi_pos.ne.symm, hpi] using h0'
    

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
hpi : π / 2 ≠ π
h0wx : w = 0 ∨ x = 0
⊢ oangle o w x = oangle o y z

Next tactic:
have h0yz : y = 0 ∨ z = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hsyz
      simpa [hyz, Real.pi_pos.ne.symm, hpi] using h0'
/-- 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⟩
    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
    have h0wx : w = 0 ∨ x = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hswx
      simpa [hwx, Real.pi_pos.ne.symm, hpi] using h0'
    have h0yz : y = 0 ∨ z = 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
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
hpi : π / 2 ≠ π
h0wx : w = 0 ∨ x = 0
⊢ y = 0 ∨ z = 0

Next tactic:
have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero 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⟩
    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⟩
    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
    have h0wx : w = 0 ∨ x = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hswx
      simpa [hwx, Real.pi_pos.ne.symm, hpi] using h0'
    have h0yz : y = 0 ∨ z = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hsyz
      

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
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π / 2 ≠ π
h0wx : w = 0 ∨ x = 0
h0' : y = 0 ∨ z = 0 ∨ InnerProductGeometry.angle y z = 0 ∨ InnerProductGeometry.angle y z = π
⊢ y = 0 ∨ z = 0

Next tactic:
simpa [hyz, Real.pi_pos.ne.symm, hpi] using h0'
/-- 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⟩
    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
    have h0wx : w = 0 ∨ x = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hswx
      simpa [hwx, Real.pi_pos.ne.symm, hpi] using h0'
    have h0yz : y = 0 ∨ z = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hsyz
      simpa [hyz, Real.pi_pos.ne.symm, hpi] using h0'
    

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
hpi : π / 2 ≠ π
h0wx : w = 0 ∨ x = 0
h0yz : y = 0 ∨ z = 0
⊢ oangle o w x = oangle o y z

Next tactic:
rcases h0wx with (h0wx | h0wx)
/-- 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⟩
    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
    have h0wx : w = 0 ∨ x = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hswx
      simpa [hwx, Real.pi_pos.ne.symm, hpi] using h0'
    have h0yz : y = 0 ∨ z = 0 := by
      have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hsyz
      simpa [hyz, Real.pi_pos.ne.symm, hpi] using h0'
    rcases h0wx with (h0wx | h0wx) <;> 

Proof state:
case pos.intro.intro.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 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
hpi : π / 2 ≠ π
h0yz : y = 0 ∨ z = 0
h0wx : w = 0
⊢ oangle o w x = oangle o y z

Next tactic:
rcases h0yz with (h0yz | h0yz)