l3lab/ntp-mathlib huggingface dataset:
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    

Proof state:
case neg.refine'_2.intro.intro
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : ∑ i : Fin (Nat.succ (Nat.succ 0)), m i • Matrix.vecCons x ![y] i = 0
hmne : ∃ i, m i ≠ 0
⊢ SameRay R x y ∨ SameRay R x (-y)

Next tactic:
rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    

Proof state:
case neg.refine'_2.intro.intro
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
⊢ SameRay R x y ∨ SameRay R x (-y)

Next tactic:
rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0)
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      

Proof state:
case neg.refine'_2.intro.intro.inl
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : m 0 < 0
⊢ SameRay R x y ∨ SameRay R x (-y)

Next tactic:
rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      

Proof state:
case neg.refine'_2.intro.intro.inr.inl
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : m 0 = 0
⊢ SameRay R x y ∨ SameRay R x (-y)

Next tactic:
rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      

Proof state:
case neg.refine'_2.intro.intro.inr.inr
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : 0 < m 0
⊢ SameRay R x y ∨ SameRay R x (-y)

Next tactic:
rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
    · 

Proof state:
case neg.refine'_2.intro.intro.inl.inl
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : m 0 < 0
hm1 : m 1 < 0
⊢ SameRay R x y ∨ SameRay R x (-y)

Next tactic:
refine'
        Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, _⟩))
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
    · refine'
        Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, _⟩))
      

Proof state:
case neg.refine'_2.intro.intro.inl.inl
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : m 0 < 0
hm1 : m 1 < 0
⊢ -m 0 • x = -m 1 • -y

Next tactic:
rw [neg_smul_neg, neg_smul, hm, neg_neg]
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
    · refine'
        Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, _⟩))
      rw [neg_smul_neg, neg_smul, hm, neg_neg]
    · 

Proof state:
case neg.refine'_2.intro.intro.inl.inr.inl
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : m 0 < 0
hm1 : m 1 = 0
⊢ SameRay R x y ∨ SameRay R x (-y)

Next tactic:
exfalso
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
    · refine'
        Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, _⟩))
      rw [neg_smul_neg, neg_smul, hm, neg_neg]
    · exfalso
      

Proof state:
case neg.refine'_2.intro.intro.inl.inr.inl.h
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : m 0 < 0
hm1 : m 1 = 0
⊢ False

Next tactic:
simp [hm1, hx, hm0.ne] at hm
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
    · refine'
        Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, _⟩))
      rw [neg_smul_neg, neg_smul, hm, neg_neg]
    · exfalso
      simp [hm1, hx, hm0.ne] at hm
    · 

Proof state:
case neg.refine'_2.intro.intro.inl.inr.inr
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : m 0 < 0
hm1 : 0 < m 1
⊢ SameRay R x y ∨ SameRay R x (-y)

Next tactic:
refine' Or.inl (Or.inr (Or.inr ⟨-m 0, m 1, Left.neg_pos_iff.2 hm0, hm1, _⟩))
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
    · refine'
        Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, _⟩))
      rw [neg_smul_neg, neg_smul, hm, neg_neg]
    · exfalso
      simp [hm1, hx, hm0.ne] at hm
    · refine' Or.inl (Or.inr (Or.inr ⟨-m 0, m 1, Left.neg_pos_iff.2 hm0, hm1, _⟩))
      

Proof state:
case neg.refine'_2.intro.intro.inl.inr.inr
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : m 0 < 0
hm1 : 0 < m 1
⊢ -m 0 • x = m 1 • y

Next tactic:
rw [neg_smul, hm, neg_neg]
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
    · refine'
        Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, _⟩))
      rw [neg_smul_neg, neg_smul, hm, neg_neg]
    · exfalso
      simp [hm1, hx, hm0.ne] at hm
    · refine' Or.inl (Or.inr (Or.inr ⟨-m 0, m 1, Left.neg_pos_iff.2 hm0, hm1, _⟩))
      rw [neg_smul, hm, neg_neg]
    · 

Proof state:
case neg.refine'_2.intro.intro.inr.inl.inl
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : m 0 = 0
hm1 : m 1 < 0
⊢ SameRay R x y ∨ SameRay R x (-y)

Next tactic:
exfalso
/-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
  by_cases hx : x = 0; · simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl
  by_cases hy : y = 0; · simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl
  simp_rw [Fintype.not_linearIndependent_iff]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
    · exact False.elim (hx hx0)
    · exact False.elim (hy hy0)
    · refine' ⟨![r₁, -r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
    · exact False.elim (hx hx0)
    · exact False.elim (hy (neg_eq_zero.1 hy0))
    · refine' ⟨![r₁, r₂], _⟩
      rw [Fin.sum_univ_two, Fin.exists_fin_two]
      simp [h, hr₁.ne.symm]
  · rcases h with ⟨m, hm, hmne⟩
    rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,
      Matrix.cons_val_one, Matrix.head_cons] at hm
    rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>
      rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
    · refine'
        Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, _⟩))
      rw [neg_smul_neg, neg_smul, hm, neg_neg]
    · exfalso
      simp [hm1, hx, hm0.ne] at hm
    · refine' Or.inl (Or.inr (Or.inr ⟨-m 0, m 1, Left.neg_pos_iff.2 hm0, hm1, _⟩))
      rw [neg_smul, hm, neg_neg]
    · exfalso
      

Proof state:
case neg.refine'_2.intro.intro.inr.inl.inl.h
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ (Nat.succ 0)) → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : m 0 = 0
hm1 : m 1 < 0
⊢ False

Next tactic:
simp [hm0, hy, hm1.ne] at hm