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
      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.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 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 : x = 0
⊢ oangle o w x = oangle o y z

Next tactic:
rcases h0yz with (h0yz | h0yz)
/-- 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) <;> rcases h0yz with (h0yz | h0yz) <;> 

Proof state:
case pos.intro.intro.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)
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
h0yz : y = 0
⊢ oangle o w x = oangle o y z

Next tactic:
simp [h0wx, h0yz]
/-- 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) <;> rcases h0yz with (h0yz | h0yz) <;> 

Proof state:
case pos.intro.intro.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 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
h0yz : z = 0
⊢ oangle o w x = oangle o y z

Next tactic:
simp [h0wx, h0yz]
/-- 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) <;> rcases h0yz with (h0yz | h0yz) <;> 

Proof state:
case pos.intro.intro.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 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 : x = 0
h0yz : y = 0
⊢ oangle o w x = oangle o y z

Next tactic:
simp [h0wx, h0yz]
/-- 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) <;> rcases h0yz with (h0yz | h0yz) <;> 

Proof state:
case pos.intro.intro.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 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 : x = 0
h0yz : z = 0
⊢ oangle o w x = oangle o y z

Next tactic:
simp [h0wx, h0yz]
/-- 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) <;> rcases h0yz with (h0yz | h0yz) <;> simp [h0wx, h0yz]
  · 

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)
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:
push_neg at 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'
    rcases h0wx with (h0wx | h0wx) <;> rcases h0yz with (h0yz | h0yz) <;> simp [h0wx, h0yz]
  · push_neg at h0
    

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)
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:
rw [Real.Angle.eq_iff_abs_toReal_eq_of_sign_eq 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
    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) <;> rcases h0yz with (h0yz | h0yz) <;> simp [h0wx, h0yz]
  · push_neg at h0
    rw [Real.Angle.eq_iff_abs_toReal_eq_of_sign_eq hs]
    

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)
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.toReal (oangle o w x)| = |Real.Angle.toReal (oangle o y z)|

Next tactic:
rwa [o.angle_eq_abs_oangle_toReal h0.1.1 h0.1.2,
      o.angle_eq_abs_oangle_toReal h0.2.1 h0.2.2] at h
/-- If the signs of two oriented angles between nonzero vectors are equal, the oriented angles are
equal if and only if the unoriented angles are equal. -/
theorem angle_eq_iff_oangle_eq_of_sign_eq {w x y z : V} (hw : w ≠ 0) (hx : x ≠ 0) (hy : y ≠ 0)
    (hz : z ≠ 0) (hs : (o.oangle w x).sign = (o.oangle y z).sign) :
    InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ↔
    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
hw : w ≠ 0
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z)
⊢ InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ↔ oangle o w x = oangle o y z

Next tactic:
refine' ⟨fun h => o.oangle_eq_of_angle_eq_of_sign_eq h hs, fun h => _⟩
/-- If the signs of two oriented angles between nonzero vectors are equal, the oriented angles are
equal if and only if the unoriented angles are equal. -/
theorem angle_eq_iff_oangle_eq_of_sign_eq {w x y z : V} (hw : w ≠ 0) (hx : x ≠ 0) (hy : y ≠ 0)
    (hz : z ≠ 0) (hs : (o.oangle w x).sign = (o.oangle y z).sign) :
    InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ↔
    o.oangle w x = o.oangle y z := by
  refine' ⟨fun h => o.oangle_eq_of_angle_eq_of_sign_eq h hs, fun h => _⟩
  

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
hw : w ≠ 0
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z)
h : oangle o w x = oangle o y z
⊢ InnerProductGeometry.angle w x = InnerProductGeometry.angle y z

Next tactic:
rw [o.angle_eq_abs_oangle_toReal hw hx, o.angle_eq_abs_oangle_toReal hy hz, h]
/-- The oriented angle between two vectors equals the unoriented angle if the sign is positive. -/
theorem oangle_eq_angle_of_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) :
    o.oangle x y = InnerProductGeometry.angle x y := 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)
x y : V
h : Real.Angle.sign (oangle o x y) = 1
⊢ oangle o x y = ↑(InnerProductGeometry.angle x y)

Next tactic:
by_cases hx : x = 0
/-- The oriented angle between two vectors equals the unoriented angle if the sign is positive. -/
theorem oangle_eq_angle_of_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) :
    o.oangle x y = InnerProductGeometry.angle x y := by
  by_cases hx : x = 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)
x y : V
h : Real.Angle.sign (oangle o x y) = 1
hx : x = 0
⊢ oangle o x y = ↑(InnerProductGeometry.angle x y)

Next tactic:
exfalso
/-- The oriented angle between two vectors equals the unoriented angle if the sign is positive. -/
theorem oangle_eq_angle_of_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) :
    o.oangle x y = InnerProductGeometry.angle x y := by
  by_cases hx : x = 0; · exfalso; 

Proof state:
case pos.h
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 : Real.Angle.sign (oangle o x y) = 1
hx : x = 0
⊢ False

Next tactic:
simp [hx] at h
/-- The oriented angle between two vectors equals the unoriented angle if the sign is positive. -/
theorem oangle_eq_angle_of_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) :
    o.oangle x y = InnerProductGeometry.angle x y := by
  by_cases hx : x = 0; · exfalso; simp [hx] 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 : Real.Angle.sign (oangle o x y) = 1
hx : ¬x = 0
⊢ oangle o x y = ↑(InnerProductGeometry.angle x y)

Next tactic:
by_cases hy : y = 0
/-- The oriented angle between two vectors equals the unoriented angle if the sign is positive. -/
theorem oangle_eq_angle_of_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) :
    o.oangle x y = InnerProductGeometry.angle x y := by
  by_cases hx : x = 0; · exfalso; simp [hx] at h
  by_cases hy : y = 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)
x y : V
h : Real.Angle.sign (oangle o x y) = 1
hx : ¬x = 0
hy : y = 0
⊢ oangle o x y = ↑(InnerProductGeometry.angle x y)

Next tactic:
exfalso
/-- The oriented angle between two vectors equals the unoriented angle if the sign is positive. -/
theorem oangle_eq_angle_of_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) :
    o.oangle x y = InnerProductGeometry.angle x y := by
  by_cases hx : x = 0; · exfalso; simp [hx] at h
  by_cases hy : y = 0; · exfalso; 

Proof state:
case pos.h
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 : Real.Angle.sign (oangle o x y) = 1
hx : ¬x = 0
hy : y = 0
⊢ False

Next tactic:
simp [hy] at h