l3lab/ntp-mathlib huggingface dataset:
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    have H1 :
      Real.sin (angle x (x - y)) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖y‖ := by
      ring
    have H2 :
      ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
        ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖ := by
      ring
    have H3 :
      ⟪x, x⟫ * (⟪x, x⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪y, y⟫)) - (⟪x, x⟫ - ⟪x, y⟫) * (⟪x, x⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
H2 :
  inner x (x - y) / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
    inner x (x - y) / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖
⊢ inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =
    inner x x * inner y y - inner x y * inner x y

Next tactic:
ring
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    have H1 :
      Real.sin (angle x (x - y)) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖y‖ := by
      ring
    have H2 :
      ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
        ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖ := by
      ring
    have H3 :
      ⟪x, x⟫ * (⟪x, x⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪y, y⟫)) - (⟪x, x⟫ - ⟪x, y⟫) * (⟪x, x⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      ring
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
H2 :
  inner x (x - y) / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
    inner x (x - y) / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖
H3 :
  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =
    inner x x * inner y y - inner x y * inner x y
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
              inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Next tactic:
have H4 :
      ⟪y, y⟫ * (⟪y, y⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪x, x⟫)) - (⟪y, y⟫ - ⟪x, y⟫) * (⟪y, y⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      ring
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    have H1 :
      Real.sin (angle x (x - y)) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖y‖ := by
      ring
    have H2 :
      ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
        ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖ := by
      ring
    have H3 :
      ⟪x, x⟫ * (⟪x, x⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪y, y⟫)) - (⟪x, x⟫ - ⟪x, y⟫) * (⟪x, x⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      ring
    have H4 :
      ⟪y, y⟫ * (⟪y, y⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪x, x⟫)) - (⟪y, y⟫ - ⟪x, y⟫) * (⟪y, y⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
H2 :
  inner x (x - y) / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
    inner x (x - y) / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖
H3 :
  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =
    inner x x * inner y y - inner x y * inner x y
⊢ inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =
    inner x x * inner y y - inner x y * inner x y

Next tactic:
ring
/-- The cosine of the sum of the angles of a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_add_angle_sub_add_angle_sub_eq_neg_one {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1 := by
  

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
⊢ Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1

Next tactic:
rw [add_assoc, Real.cos_add, cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle hx hy,
    sin_angle_sub_add_angle_sub_rev_eq_sin_angle hx hy, mul_neg, ← neg_add', add_comm, ← sq, ← sq,
    Real.sin_sq_add_cos_sq]
/-- The sine of the sum of the angles of a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_add_angle_sub_add_angle_sub_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x y + angle x (x - y) + angle y (y - x)) = 0 := by
  

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
⊢ Real.sin (angle x y + angle x (x - y) + angle y (y - x)) = 0

Next tactic:
rw [add_assoc, Real.sin_add, cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle hx hy,
    sin_angle_sub_add_angle_sub_rev_eq_sin_angle hx hy]
/-- The sine of the sum of the angles of a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_add_angle_sub_add_angle_sub_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x y + angle x (x - y) + angle y (y - x)) = 0 := by
  rw [add_assoc, Real.sin_add, cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle hx hy,
    sin_angle_sub_add_angle_sub_rev_eq_sin_angle hx hy]
  

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
⊢ Real.sin (angle x y) * -Real.cos (angle x y) + Real.cos (angle x y) * Real.sin (angle x y) = 0

Next tactic:
ring
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
⊢ angle x y + angle x (x - y) + angle y (y - x) = π

Next tactic:
have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
⊢ angle x y + angle x (x - y) + angle y (y - x) = π

Next tactic:
have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
hsin : Real.sin (angle x y + angle x (x - y) + angle y (y - x)) = 0
⊢ angle x y + angle x (x - y) + angle y (y - x) = π

Next tactic:
rw [Real.sin_eq_zero_iff] at hsin
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
hsin : ∃ n, ↑n * π = angle x y + angle x (x - y) + angle y (y - x)
⊢ angle x y + angle x (x - y) + angle y (y - x) = π

Next tactic:
cases' hsin with n hn
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  cases' hsin with n hn
  

Proof state:
case intro
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
n : ℤ
hn : ↑n * π = angle x y + angle x (x - y) + angle y (y - x)
⊢ angle x y + angle x (x - y) + angle y (y - x) = π

Next tactic:
symm at hn
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  cases' hsin with n hn
  symm at hn
  

Proof state:
case intro
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
n : ℤ
hn : angle x y + angle x (x - y) + angle y (y - x) = ↑n * π
⊢ angle x y + angle x (x - y) + angle y (y - x) = π

Next tactic:
have h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x) :=
    add_nonneg (add_nonneg (angle_nonneg _ _) (angle_nonneg _ _)) (angle_nonneg _ _)
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  cases' hsin with n hn
  symm at hn
  have h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x) :=
    add_nonneg (add_nonneg (angle_nonneg _ _) (angle_nonneg _ _)) (angle_nonneg _ _)
  

Proof state:
case intro
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
n : ℤ
hn : angle x y + angle x (x - y) + angle y (y - x) = ↑n * π
h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x)
⊢ angle x y + angle x (x - y) + angle y (y - x) = π

Next tactic:
have h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π := by
    by_contra hnlt
    have hxy : angle x y = π := by
      by_contra hxy
      exact hnlt (add_lt_add_of_lt_of_le (add_lt_add_of_lt_of_le (lt_of_le_of_ne
        (angle_le_pi _ _) hxy) (angle_le_pi _ _)) (angle_le_pi _ _))
    rw [hxy] at hnlt
    rw [angle_eq_pi_iff] at hxy
    rcases hxy with ⟨hx, ⟨r, ⟨hr, hxr⟩⟩⟩
    rw [hxr, ← one_smul ℝ x, ← mul_smul, mul_one, ← sub_smul, one_smul, sub_eq_add_neg,
      angle_smul_right_of_pos _ _ (add_pos zero_lt_one (neg_pos_of_neg hr)), angle_self hx,
      add_zero] at hnlt
    apply hnlt
    rw [add_assoc]
    exact add_lt_add_left (lt_of_le_of_lt (angle_le_pi _ _) (lt_add_of_pos_right π Real.pi_pos)) _
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  cases' hsin with n hn
  symm at hn
  have h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x) :=
    add_nonneg (add_nonneg (angle_nonneg _ _) (angle_nonneg _ _)) (angle_nonneg _ _)
  have h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π := by
    

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
n : ℤ
hn : angle x y + angle x (x - y) + angle y (y - x) = ↑n * π
h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x)
⊢ angle x y + angle x (x - y) + angle y (y - x) < π + π + π

Next tactic:
by_contra hnlt
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  cases' hsin with n hn
  symm at hn
  have h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x) :=
    add_nonneg (add_nonneg (angle_nonneg _ _) (angle_nonneg _ _)) (angle_nonneg _ _)
  have h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π := by
    by_contra hnlt
    

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
n : ℤ
hn : angle x y + angle x (x - y) + angle y (y - x) = ↑n * π
h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x)
hnlt : ¬angle x y + angle x (x - y) + angle y (y - x) < π + π + π
⊢ False

Next tactic:
have hxy : angle x y = π := by
      by_contra hxy
      exact hnlt (add_lt_add_of_lt_of_le (add_lt_add_of_lt_of_le (lt_of_le_of_ne
        (angle_le_pi _ _) hxy) (angle_le_pi _ _)) (angle_le_pi _ _))
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  cases' hsin with n hn
  symm at hn
  have h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x) :=
    add_nonneg (add_nonneg (angle_nonneg _ _) (angle_nonneg _ _)) (angle_nonneg _ _)
  have h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π := by
    by_contra hnlt
    have hxy : angle x y = π := by
      

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
n : ℤ
hn : angle x y + angle x (x - y) + angle y (y - x) = ↑n * π
h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x)
hnlt : ¬angle x y + angle x (x - y) + angle y (y - x) < π + π + π
⊢ angle x y = π

Next tactic:
by_contra hxy
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  cases' hsin with n hn
  symm at hn
  have h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x) :=
    add_nonneg (add_nonneg (angle_nonneg _ _) (angle_nonneg _ _)) (angle_nonneg _ _)
  have h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π := by
    by_contra hnlt
    have hxy : angle x y = π := by
      by_contra hxy
      

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
n : ℤ
hn : angle x y + angle x (x - y) + angle y (y - x) = ↑n * π
h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x)
hnlt : ¬angle x y + angle x (x - y) + angle y (y - x) < π + π + π
hxy : ¬angle x y = π
⊢ False

Next tactic:
exact hnlt (add_lt_add_of_lt_of_le (add_lt_add_of_lt_of_le (lt_of_le_of_ne
        (angle_le_pi _ _) hxy) (angle_le_pi _ _)) (angle_le_pi _ _))
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  cases' hsin with n hn
  symm at hn
  have h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x) :=
    add_nonneg (add_nonneg (angle_nonneg _ _) (angle_nonneg _ _)) (angle_nonneg _ _)
  have h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π := by
    by_contra hnlt
    have hxy : angle x y = π := by
      by_contra hxy
      exact hnlt (add_lt_add_of_lt_of_le (add_lt_add_of_lt_of_le (lt_of_le_of_ne
        (angle_le_pi _ _) hxy) (angle_le_pi _ _)) (angle_le_pi _ _))
    

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
n : ℤ
hn : angle x y + angle x (x - y) + angle y (y - x) = ↑n * π
h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x)
hnlt : ¬angle x y + angle x (x - y) + angle y (y - x) < π + π + π
hxy : angle x y = π
⊢ False

Next tactic:
rw [hxy] at hnlt
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  cases' hsin with n hn
  symm at hn
  have h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x) :=
    add_nonneg (add_nonneg (angle_nonneg _ _) (angle_nonneg _ _)) (angle_nonneg _ _)
  have h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π := by
    by_contra hnlt
    have hxy : angle x y = π := by
      by_contra hxy
      exact hnlt (add_lt_add_of_lt_of_le (add_lt_add_of_lt_of_le (lt_of_le_of_ne
        (angle_le_pi _ _) hxy) (angle_le_pi _ _)) (angle_le_pi _ _))
    rw [hxy] at hnlt
    

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
n : ℤ
hn : angle x y + angle x (x - y) + angle y (y - x) = ↑n * π
h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x)
hnlt : ¬π + angle x (x - y) + angle y (y - x) < π + π + π
hxy : angle x y = π
⊢ False

Next tactic:
rw [angle_eq_pi_iff] at hxy
/-- The sum of the angles of a possibly degenerate triangle (where the
two given sides are nonzero), vector angle form. -/
theorem angle_add_angle_sub_add_angle_sub_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    angle x y + angle x (x - y) + angle y (y - x) = π := by
  have hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy
  have hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy
  rw [Real.sin_eq_zero_iff] at hsin
  cases' hsin with n hn
  symm at hn
  have h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x) :=
    add_nonneg (add_nonneg (angle_nonneg _ _) (angle_nonneg _ _)) (angle_nonneg _ _)
  have h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π := by
    by_contra hnlt
    have hxy : angle x y = π := by
      by_contra hxy
      exact hnlt (add_lt_add_of_lt_of_le (add_lt_add_of_lt_of_le (lt_of_le_of_ne
        (angle_le_pi _ _) hxy) (angle_le_pi _ _)) (angle_le_pi _ _))
    rw [hxy] at hnlt
    rw [angle_eq_pi_iff] at hxy
    

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1
n : ℤ
hn : angle x y + angle x (x - y) + angle y (y - x) = ↑n * π
h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x)
hnlt : ¬π + angle x (x - y) + angle y (y - x) < π + π + π
hxy : x ≠ 0 ∧ ∃ r < 0, y = r • x
⊢ False

Next tactic:
rcases hxy with ⟨hx, ⟨r, ⟨hr, hxr⟩⟩⟩