l3lab/ntp-mathlib huggingface dataset:
/-- 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
    rcases hxy with ⟨hx, ⟨r, ⟨hr, hxr⟩⟩⟩
    

Proof state:
case intro.intro.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)
hnlt : ¬π + angle x (x - y) + angle y (y - x) < π + π + π
hx : x ≠ 0
r : ℝ
hr : r < 0
hxr : y = r • x
⊢ False

Next tactic:
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
/-- 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
    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
    

Proof state:
case intro.intro.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)
hx : x ≠ 0
r : ℝ
hnlt : ¬π + angle (r • x) (r • x - x) < π + π + π
hr : r < 0
hxr : y = r • x
⊢ False

Next tactic:
apply 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
    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
    

Proof state:
case intro.intro.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)
hx : x ≠ 0
r : ℝ
hnlt : ¬π + angle (r • x) (r • x - x) < π + π + π
hr : r < 0
hxr : y = r • x
⊢ π + angle (r • x) (r • x - x) < π + π + π

Next tactic:
rw [add_assoc]
/-- 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
    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]
    

Proof state:
case intro.intro.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)
hx : x ≠ 0
r : ℝ
hnlt : ¬π + angle (r • x) (r • x - x) < π + π + π
hr : r < 0
hxr : y = r • x
⊢ π + angle (r • x) (r • x - x) < π + (π + π)

Next tactic:
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
    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)) _
  

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)
h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π
⊢ angle x y + angle x (x - y) + angle y (y - x) = π

Next tactic:
have hn0 : 0 ≤ n := by
    rw [hn, mul_nonneg_iff_left_nonneg_of_pos Real.pi_pos] at h0
    norm_cast at h0
/-- 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
    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)) _
  have hn0 : 0 ≤ n := 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)
h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π
⊢ 0 ≤ n

Next tactic:
rw [hn, mul_nonneg_iff_left_nonneg_of_pos Real.pi_pos] at h0
/-- 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
    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)) _
  have hn0 : 0 ≤ n := by
    rw [hn, mul_nonneg_iff_left_nonneg_of_pos Real.pi_pos] at h0
    

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 ≤ ↑n
h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π
⊢ 0 ≤ n

Next tactic:
norm_cast at h0
/-- 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
    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)) _
  have hn0 : 0 ≤ n := by
    rw [hn, mul_nonneg_iff_left_nonneg_of_pos Real.pi_pos] at h0
    norm_cast at h0
  

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)
h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π
hn0 : 0 ≤ n
⊢ angle x y + angle x (x - y) + angle y (y - x) = π

Next tactic:
have hn3 : n < 3 := by
    rw [hn, show π + π + π = 3 * π by ring] at h3lt
    replace h3lt := lt_of_mul_lt_mul_right h3lt (le_of_lt Real.pi_pos)
    norm_cast at h3lt
/-- 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
    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)) _
  have hn0 : 0 ≤ n := by
    rw [hn, mul_nonneg_iff_left_nonneg_of_pos Real.pi_pos] at h0
    norm_cast at h0
  have hn3 : n < 3 := 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)
h3lt : angle x y + angle x (x - y) + angle y (y - x) < π + π + π
hn0 : 0 ≤ n
⊢ n < 3

Next tactic:
rw [hn, show π + π + π = 3 * π by ring] at h3lt
/-- 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
    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)) _
  have hn0 : 0 ≤ n := by
    rw [hn, mul_nonneg_iff_left_nonneg_of_pos Real.pi_pos] at h0
    norm_cast at h0
  have hn3 : n < 3 := by
    rw [hn, show π + π + π = 3 * 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)
h3lt : ↑n * π < π + π + π
hn0 : 0 ≤ n
⊢ π + π + π = 3 * π

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
  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
    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)) _
  have hn0 : 0 ≤ n := by
    rw [hn, mul_nonneg_iff_left_nonneg_of_pos Real.pi_pos] at h0
    norm_cast at h0
  have hn3 : n < 3 := by
    rw [hn, show π + π + π = 3 * π by ring] at h3lt
    

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)
h3lt : ↑n * π < 3 * π
hn0 : 0 ≤ n
⊢ n < 3

Next tactic:
replace h3lt := lt_of_mul_lt_mul_right h3lt (le_of_lt 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
    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)) _
  have hn0 : 0 ≤ n := by
    rw [hn, mul_nonneg_iff_left_nonneg_of_pos Real.pi_pos] at h0
    norm_cast at h0
  have hn3 : n < 3 := by
    rw [hn, show π + π + π = 3 * π by ring] at h3lt
    replace h3lt := lt_of_mul_lt_mul_right h3lt (le_of_lt Real.pi_pos)
    

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)
hn0 : 0 ≤ n
h3lt : ↑n < 3
⊢ n < 3

Next tactic:
norm_cast at h3lt