l3lab/ntp-mathlib huggingface dataset:
/-- **Converse of pons asinorum**, vector angle form. -/
theorem norm_eq_of_angle_sub_eq_angle_sub_rev_of_angle_ne_pi {x y : V}
    (h : angle x (x - y) = angle y (y - x)) (hpi : angle x y ≠ π) : ‖x‖ = ‖y‖ := by
  replace h := Real.arccos_injOn (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x (x - y)))
    (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one y (y - x))) h
  by_cases hxy : x = y
  · rw [hxy]
  · rw [← norm_neg (y - x), neg_sub, mul_comm, mul_comm ‖y‖, div_eq_mul_inv, div_eq_mul_inv,
      mul_inv_rev, mul_inv_rev, ← mul_assoc, ← mul_assoc] at h
    replace h :=
      mul_right_cancel₀ (inv_ne_zero fun hz => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 hz))) h
    rw [inner_sub_right, inner_sub_right, real_inner_comm x y, real_inner_self_eq_norm_mul_norm,
      real_inner_self_eq_norm_mul_norm, mul_sub_right_distrib, mul_sub_right_distrib,
      mul_self_mul_inv, mul_self_mul_inv, sub_eq_sub_iff_sub_eq_sub, ← mul_sub_left_distrib] at h
    by_cases hx0 : x = 0
    · rw [hx0, norm_zero, inner_zero_left, zero_mul, zero_sub, neg_eq_zero] at h
      rw [hx0, norm_zero, h]
    · by_cases hy0 : y = 0
      · rw [hy0, norm_zero, inner_zero_right, zero_mul, sub_zero] at h
        rw [hy0, norm_zero, h]
      · rw [inv_sub_inv (fun hz => hx0 (norm_eq_zero.1 hz)) fun hz => hy0 (norm_eq_zero.1 hz), ←
          neg_sub, ← mul_div_assoc, mul_comm, mul_div_assoc, ← mul_neg_one] at h
        symm
        by_contra hyx
        replace h := (mul_left_cancel₀ (sub_ne_zero_of_ne hyx) h).symm
        

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hpi : angle x y ≠ π
hxy : ¬x = y
hx0 : ¬x = 0
hy0 : ¬y = 0
hyx : ¬‖y‖ = ‖x‖
h : inner x y / (‖x‖ * ‖y‖) = -1
⊢ False

Next tactic:
rw [real_inner_div_norm_mul_norm_eq_neg_one_iff, ← angle_eq_pi_iff] at h
/-- **Converse of pons asinorum**, vector angle form. -/
theorem norm_eq_of_angle_sub_eq_angle_sub_rev_of_angle_ne_pi {x y : V}
    (h : angle x (x - y) = angle y (y - x)) (hpi : angle x y ≠ π) : ‖x‖ = ‖y‖ := by
  replace h := Real.arccos_injOn (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x (x - y)))
    (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one y (y - x))) h
  by_cases hxy : x = y
  · rw [hxy]
  · rw [← norm_neg (y - x), neg_sub, mul_comm, mul_comm ‖y‖, div_eq_mul_inv, div_eq_mul_inv,
      mul_inv_rev, mul_inv_rev, ← mul_assoc, ← mul_assoc] at h
    replace h :=
      mul_right_cancel₀ (inv_ne_zero fun hz => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 hz))) h
    rw [inner_sub_right, inner_sub_right, real_inner_comm x y, real_inner_self_eq_norm_mul_norm,
      real_inner_self_eq_norm_mul_norm, mul_sub_right_distrib, mul_sub_right_distrib,
      mul_self_mul_inv, mul_self_mul_inv, sub_eq_sub_iff_sub_eq_sub, ← mul_sub_left_distrib] at h
    by_cases hx0 : x = 0
    · rw [hx0, norm_zero, inner_zero_left, zero_mul, zero_sub, neg_eq_zero] at h
      rw [hx0, norm_zero, h]
    · by_cases hy0 : y = 0
      · rw [hy0, norm_zero, inner_zero_right, zero_mul, sub_zero] at h
        rw [hy0, norm_zero, h]
      · rw [inv_sub_inv (fun hz => hx0 (norm_eq_zero.1 hz)) fun hz => hy0 (norm_eq_zero.1 hz), ←
          neg_sub, ← mul_div_assoc, mul_comm, mul_div_assoc, ← mul_neg_one] at h
        symm
        by_contra hyx
        replace h := (mul_left_cancel₀ (sub_ne_zero_of_ne hyx) h).symm
        rw [real_inner_div_norm_mul_norm_eq_neg_one_iff, ← angle_eq_pi_iff] at h
        

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hpi : angle x y ≠ π
hxy : ¬x = y
hx0 : ¬x = 0
hy0 : ¬y = 0
hyx : ¬‖y‖ = ‖x‖
h : angle x y = π
⊢ False

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

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

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

Next tactic:
rw [hxy, angle_self hy]
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    

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

Next tactic:
simp
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · 

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
⊢ Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y)

Next tactic:
rw [Real.cos_add, cos_angle, cos_angle, cos_angle]
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, cos_angle, cos_angle]
    

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
⊢ inner x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) =
    -(inner x y / (‖x‖ * ‖y‖))

Next tactic:
have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    

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
⊢ inner x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) =
    -(inner x y / (‖x‖ * ‖y‖))

Next tactic:
have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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)
    

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
⊢ inner x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) =
    -(inner x y / (‖x‖ * ‖y‖))

Next tactic:
have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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))
    

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
⊢ inner x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) =
    -(inner x y / (‖x‖ * ‖y‖))

Next tactic:
apply mul_right_cancel₀ hxn
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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
    

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
⊢ (inner x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
        Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *
      ‖x‖ =
    -(inner x y / (‖x‖ * ‖y‖)) * ‖x‖

Next tactic:
apply mul_right_cancel₀ hyn
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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
    

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
⊢ (inner x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
          Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *
        ‖x‖ *
      ‖y‖ =
    -(inner x y / (‖x‖ * ‖y‖)) * ‖x‖ * ‖y‖

Next tactic:
apply mul_right_cancel₀ hxyn
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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
    

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
⊢ (inner x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
            Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *
          ‖x‖ *
        ‖y‖ *
      ‖x - y‖ =
    -(inner x y / (‖x‖ * ‖y‖)) * ‖x‖ * ‖y‖ * ‖x - y‖

Next tactic:
apply mul_right_cancel₀ hxyn
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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
    

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
⊢ (inner x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
              Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    -(inner x y / (‖x‖ * ‖y‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Next tactic:
have H1 :
      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) *
          (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖)) := by
      ring
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) *
          (Real.sin (angle 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
⊢ Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖))

Next tactic:
ring
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) *
          (Real.sin (angle 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)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖))
⊢ (inner x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
              Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    -(inner x y / (‖x‖ * ‖y‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Next tactic:
have H2 :
      ⟪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
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) *
          (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖)) := by
      ring
    have H2 :
      ⟪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)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖))
⊢ 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 cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) *
          (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖)) := by
      ring
    have H2 :
      ⟪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)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖))
H2 :
  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 x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
              Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    -(inner x y / (‖x‖ * ‖y‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Next tactic:
have H3 :
      ⟪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 cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, 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)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) *
          (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖)) := by
      ring
    have H2 :
      ⟪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 H3 :
      ⟪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)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖))
H2 :
  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