awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : PartialOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
N : Type u_3
inst_3 : AddCommGroup M
inst_4 : AddCommGroup N
inst_5 : Module R M
inst_6 : Module R N
ι : Type u_4
inst_7 : Fintype ι
inst_8 : DecidableEq ι
e : Basis ι R M
f : M ≃ₗ[R] N
⊢ Eq (α := Orientation R N ι) (e.map f).orientation
    ((Orientation.map ι f : (a : Orientation R M ι) → Orientation R N ι) e.orientation)
have simp_rw : Eq (α := Orientation R N ι) (rayOfNeZero R (e.map f).det sorry)
  ((Orientation.map ι f : (a : Orientation R M ι) → Orientation R N ι) (rayOfNeZero R e.det sorry))
Tactic: simp_rw [Basis.orientation, Orientation.map_apply, Basis.det_map']
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : PartialOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
N : Type u_3
inst_3 : AddCommGroup M
inst_4 : AddCommGroup N
inst_5 : Module R M
inst_6 : Module R N
ι : Type u_4
inst_7 : Fintype ι
inst_8 : DecidableEq ι
e : Basis ι R M
f : M ≃ₗ[R] N
simp_rw :
  Eq (α := Orientation R N ι) (rayOfNeZero R (e.map f).det ⋯)
    ((Orientation.map ι f : (a : Orientation R M ι) → Orientation R N ι) (rayOfNeZero R e.det ⋯))
⊢ Eq (α := Orientation R N ι) (e.map f).orientation
    ((Orientation.map ι f : (a : Orientation R M ι) → Orientation R N ι) e.orientation)
have simp_rw₁ : Eq (α := Orientation R N ι) (rayOfNeZero R (e.map f).det sorry)
  (rayOfNeZero R (e.det.compLinearMap (↑f.symm : N →ₗ[R] M)) sorry)
Tactic: simp_rw [Basis.orientation, Orientation.map_apply, Basis.det_map']
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : PartialOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_4
ι' : Type u_5
inst_5 : Fintype ι
inst_6 : DecidableEq ι
inst_7 : Fintype ι'
inst_8 : DecidableEq ι'
e : Basis ι R M
eι : ι ≃ ι'
⊢ Eq (α := Orientation R M ι') (e.reindex eι).orientation
    ((Orientation.reindex R M eι : (a : Orientation R M ι) → Orientation R M ι') e.orientation)
have simp_rw : Eq (α := Orientation R M ι') (rayOfNeZero R (e.reindex eι).det sorry)
  ((Orientation.reindex R M eι : (a : Orientation R M ι) → Orientation R M ι') (rayOfNeZero R e.det sorry))
Tactic: simp_rw [Basis.orientation, Orientation.reindex_apply, Basis.det_reindex']
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : PartialOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_4
ι' : Type u_5
inst_5 : Fintype ι
inst_6 : DecidableEq ι
inst_7 : Fintype ι'
inst_8 : DecidableEq ι'
e : Basis ι R M
eι : ι ≃ ι'
simp_rw :
  Eq (α := Orientation R M ι') (rayOfNeZero R (e.reindex eι).det ⋯)
    ((Orientation.reindex R M eι : (a : Orientation R M ι) → Orientation R M ι') (rayOfNeZero R e.det ⋯))
⊢ Eq (α := Orientation R M ι') (e.reindex eι).orientation
    ((Orientation.reindex R M eι : (a : Orientation R M ι) → Orientation R M ι') e.orientation)
have simp_rw₁ : Eq (α := Orientation R M ι') (rayOfNeZero R (e.reindex eι).det sorry)
  (rayOfNeZero R (AlternatingMap.domDomCongr eι e.det) sorry)
Tactic: simp_rw [Basis.orientation, Orientation.reindex_apply, Basis.det_reindex']
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : PartialOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_4
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
w : ι → Rˣ
⊢ (e.unitsSMul w).orientation = HSMul.hSMul (α := Rˣ) (β := Orientation R M ι) (∏ i : ι, w i)⁻¹ e.orientation
have rw : SameRay R (e.unitsSMul w).det (HMul.hMul (α := R) (↑(∏ i : ι, w i)⁻¹ : R) (∏ i : ι, (↑(w i) : R)) • (e.unitsSMul w).det)
Tactic: rw [Basis.orientation, Basis.orientation, smul_rayOfNeZero, ray_eq_iff,
    e.det.eq_smul_basis_det (e.unitsSMul w), det_unitsSMul_self, Units.smul_def, smul_smul]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : PartialOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_4
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
w : ι → Rˣ
⊢ SameRay R (e.unitsSMul w).det
    (HMul.hMul (α := R) (↑(∏ i : ι, w i)⁻¹ : R) (∏ i : ι, (↑(w i) : R)) • (e.unitsSMul w).det)
have norm_cast : SameRay R (e.unitsSMul w).det ((↑(HMul.hMul (α := Rˣ) (∏ i : ι, w i)⁻¹ (∏ i : ι, w i)) : R) • (e.unitsSMul w).det)
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : PartialOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_4
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
w : ι → Rˣ
⊢ SameRay R (e.unitsSMul w).det ((↑(HMul.hMul (α := Rˣ) (∏ i : ι, w i)⁻¹ (∏ i : ι, w i)) : R) • (e.unitsSMul w).det)
have simp : SameRay R (e.unitsSMul w).det (e.unitsSMul w).det
Tactic: simp only [inv_mul_cancel, Units.val_one, one_smul]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : PartialOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_4
inst_5 : Fintype ι
inst_6 : DecidableEq ι
inst_7 : IsEmpty ι
b : Basis ι R M
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R b.det ⋯) positiveOrientation
have e_v : b.det = (AlternatingMap.constLinearEquivOfIsEmpty : (a : R) → M [⋀^ι]→ₗ[R] R) 1
Tactic: congr
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R x hx) positiveOrientation ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx) (-positiveOrientation)
have h : Eq (α := Orientation R M ι) (rayOfNeZero R x hx) (rayOfNeZero R (AlternatingMap.constOfIsEmpty R M ι 1) sorry) ∨
  Eq (α := Orientation R M ι) (rayOfNeZero R x hx) (rayOfNeZero R (-AlternatingMap.constOfIsEmpty R M ι 1) sorry)
Tactic: dsimp [positiveOrientation]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R x hx) (rayOfNeZero R (AlternatingMap.constOfIsEmpty R M ι 1) ⋯) ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx) (rayOfNeZero R (-AlternatingMap.constOfIsEmpty R M ι 1) ⋯)
have h : SameRay R x (AlternatingMap.constOfIsEmpty R M ι 1) ∨ SameRay R x (-AlternatingMap.constOfIsEmpty R M ι 1)
Tactic: simp only [ray_eq_iff]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
⊢ SameRay R x (AlternatingMap.constOfIsEmpty R M ι 1) ∨ SameRay R x (-AlternatingMap.constOfIsEmpty R M ι 1)
have h : ¬LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
Tactic: rw [sameRay_or_sameRay_neg_iff_not_linearIndependent]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : M [⋀^ι]→ₗ[R] R ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
⊢ LinearIndependent R (M := R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] : Fin (Nat.succ 0).succ → R)
have h₁ : Eq.{u_1 + 1} (α := Fin (Nat.succ 0).succ → R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] : Fin (Nat.succ 0).succ → R)
  ((⇑(↑f : M [⋀^ι]→ₗ[R] R →ₗ[R] R) : (a : M [⋀^ι]→ₗ[R] R) → R) ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1] :
    Fin (Nat.succ 0).succ → R)
Tactic: convert! h.map' f.toLinearMap f.ker
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : M [⋀^ι]→ₗ[R] R ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
⊢ Eq.{u_1 + 1} (α := Fin (Nat.succ 0).succ → R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] : Fin (Nat.succ 0).succ → R)
    ((⇑(↑f : M [⋀^ι]→ₗ[R] R →ₗ[R] R) : (a : M [⋀^ι]→ₗ[R] R) → R) ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1] :
      Fin (Nat.succ 0).succ → R)
have h₁ : ∀ (i : Fin (Nat.succ 0).succ),
  Eq.{u_1 + 1} (α := R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] i : R)
    (((⇑(↑f : M [⋀^ι]→ₗ[R] R →ₗ[R] R) : (a : M [⋀^ι]→ₗ[R] R) → R) ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1]) i : R)
Tactic: ext i
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : M [⋀^ι]→ₗ[R] R ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
i : Fin (Nat.succ 0).succ
⊢ Eq.{u_1 + 1} (α := R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] i : R)
    (((⇑(↑f : M [⋀^ι]→ₗ[R] R →ₗ[R] R) : (a : M [⋀^ι]→ₗ[R] R) → R) ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1]) i : R)
have h₁ : Eq.{u_1 + 1} (α := R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] ((fun (i : Fin (Nat.succ 0).succ) ↦ i) ⟨0, sorry⟩) : R)
  (((⇑(↑f : M [⋀^ι]→ₗ[R] R →ₗ[R] R) : (a : M [⋀^ι]→ₗ[R] R) → R) ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1])
      ((fun (i : Fin (Nat.succ 0).succ) ↦ i) ⟨0, sorry⟩) :
    R)
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : M [⋀^ι]→ₗ[R] R ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
i : Fin (Nat.succ 0).succ
h₁ :
  Eq.{u_1 + 1} (α := R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] ((fun (i : Fin (Nat.succ 0).succ) ↦ i) ⟨0, ⋯⟩) : R)
    (((⇑(↑f : M [⋀^ι]→ₗ[R] R →ₗ[R] R) : (a : M [⋀^ι]→ₗ[R] R) → R) ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1])
        ((fun (i : Fin (Nat.succ 0).succ) ↦ i) ⟨0, ⋯⟩) :
      R)
⊢ Eq.{u_1 + 1} (α := R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] i : R)
    (((⇑(↑f : M [⋀^ι]→ₗ[R] R →ₗ[R] R) : (a : M [⋀^ι]→ₗ[R] R) → R) ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1]) i : R)
have h₂ : Eq.{u_1 + 1} (α := R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] ((fun (i : Fin (Nat.succ 0).succ) ↦ i) ⟨1, sorry⟩) : R)
  (((⇑(↑f : M [⋀^ι]→ₗ[R] R →ₗ[R] R) : (a : M [⋀^ι]→ₗ[R] R) → R) ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1])
      ((fun (i : Fin (Nat.succ 0).succ) ↦ i) ⟨1, sorry⟩) :
    R)
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : M [⋀^ι]→ₗ[R] R ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
⊢ False
have H : LinearIndependent R (M := R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] : Fin (Nat.succ 0).succ → R)
Tactic: have H : LinearIndependent R ![f x, 1] := by
    convert! h.map' f.toLinearMap f.ker
    ext i
    fin_cases i <;> simp [f]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : M [⋀^ι]→ₗ[R] R ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
H : LinearIndependent R (M := R) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] : Fin (Nat.succ 0).succ → R)
⊢ False
have H₁ : ∀ (s : Finset (Fin (Nat.succ 0).succ)) (g : Fin (Nat.succ 0).succ → R),
  Eq.{u_1 + 1} (α := R) (∑ i ∈ s, HSMul.hSMul (β := R) (g i) (![(f : (a : M [⋀^ι]→ₗ[R] R) → R) x, 1] i : R)) 0 →
    ∀ i ∈ s, g i = 0
Tactic: rw [linearIndependent_iff'] at H
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : IsEmpty ι
o : Orientation R M ι
⊢ o = positiveOrientation ∨ o = -positiveOrientation
have h : ∀ (x : M [⋀^ι]→ₗ[R] R) (hx : x ≠ 0),
  Eq (α := Orientation R M ι) (rayOfNeZero R x hx) positiveOrientation ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx) (-positiveOrientation)
Tactic: induction o using Module.Ray.ind with | h x hx =>
  dsimp [positiveOrientation]
  simp only [ray_eq_iff]
  rw [sameRay_or_sameRay_neg_iff_not_linearIndependent]
  intro h
  set f : (M [⋀^ι]→ₗ[R] R) ≃ₗ[R] R := AlternatingMap.constLinearEquivOfIsEmpty.symm
  have H : LinearIndependent R ![f x, 1] := by
    convert! h.map' f.toLinearMap f.ker
    ext i
    fin_cases i <;> simp [f]
  rw [linearIndependent_iff'] at H
  simpa using H Finset.univ ![1, -f x] (by simp [Fin.sum_univ_succ]) 0 (by simp)
Cleared hypotheses: o
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e₁ e₂ : Basis ι R M
⊢ SameRay R e₁.det e₂.det ↔
    SameRay R (HSMul.hSMul (α := R) ((e₁.det : (a : ι → M) → R) (⇑e₂ : (a : ι) → M) : R) e₂.det) e₂.det
have rw : SameRay R e₁.det e₂.det ↔ SameRay R e₁.det e₂.det
Tactic: rw [← e₁.det.eq_smul_basis_det e₂]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R x hx) e.orientation ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx) (-e.orientation)
have hx₁ : Ne.{u_1 + 1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
Tactic: rw [← x.map_basis_ne_zero_iff e] at hx
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
x : M [⋀^ι]→ₗ[R] R
hx_1 : x ≠ 0
hx : Ne.{u_1 + 1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) e.orientation ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-e.orientation)
have h : Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R e.det sorry) ∨
  Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det sorry)
Tactic: rwa [Basis.orientation, ray_eq_iff, neg_rayOfNeZero, ray_eq_iff, x.eq_smul_basis_det e,
    sameRay_neg_smul_left_iff_of_ne e.det_ne_zero hx, sameRay_smul_left_iff_of_ne e.det_ne_zero hx,
    lt_or_lt_iff_ne, ne_comm]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
x : M [⋀^ι]→ₗ[R] R
hx_1 : x ≠ 0
hx : Ne.{u_1 + 1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
h :
  Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R e.det ⋯) ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) e.orientation ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-e.orientation)
have h₁ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det sorry)
Tactic: rwa [Basis.orientation, ray_eq_iff, neg_rayOfNeZero, ray_eq_iff, x.eq_smul_basis_det e,
    sameRay_neg_smul_left_iff_of_ne e.det_ne_zero hx, sameRay_smul_left_iff_of_ne e.det_ne_zero hx,
    lt_or_lt_iff_ne, ne_comm]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
x : M [⋀^ι]→ₗ[R] R
hx_1 : x ≠ 0
hx : Ne.{u_1 + 1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
h :
  Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R e.det ⋯) ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
h₁ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) e.orientation ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-e.orientation)
have h₂ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R (-e.det) sorry)
Tactic: rwa [Basis.orientation, ray_eq_iff, neg_rayOfNeZero, ray_eq_iff, x.eq_smul_basis_det e,
    sameRay_neg_smul_left_iff_of_ne e.det_ne_zero hx, sameRay_smul_left_iff_of_ne e.det_ne_zero hx,
    lt_or_lt_iff_ne, ne_comm]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
x : M [⋀^ι]→ₗ[R] R
hx_1 : x ≠ 0
hx : Ne.{u_1 + 1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
h :
  Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R e.det ⋯) ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
h₁ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
h₂ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R (-e.det) ⋯)
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) e.orientation ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-e.orientation)
have h₃ : SameRay R x e.det ∨ SameRay R x (-e.det)
Tactic: rwa [Basis.orientation, ray_eq_iff, neg_rayOfNeZero, ray_eq_iff, x.eq_smul_basis_det e,
    sameRay_neg_smul_left_iff_of_ne e.det_ne_zero hx, sameRay_smul_left_iff_of_ne e.det_ne_zero hx,
    lt_or_lt_iff_ne, ne_comm]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
x : M [⋀^ι]→ₗ[R] R
hx_1 : x ≠ 0
hx : Ne.{u_1 + 1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
h :
  Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R e.det ⋯) ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
h₁ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
h₂ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R (-e.det) ⋯)
h₃ : SameRay R x e.det ∨ SameRay R x (-e.det)
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) e.orientation ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-e.orientation)
have h₄ : SameRay R (HSMul.hSMul (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) e.det) e.det ∨
  SameRay R (HSMul.hSMul (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) e.det) (-e.det)
Tactic: rwa [Basis.orientation, ray_eq_iff, neg_rayOfNeZero, ray_eq_iff, x.eq_smul_basis_det e,
    sameRay_neg_smul_left_iff_of_ne e.det_ne_zero hx, sameRay_smul_left_iff_of_ne e.det_ne_zero hx,
    lt_or_lt_iff_ne, ne_comm]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
x : M [⋀^ι]→ₗ[R] R
hx_1 : x ≠ 0
hx : Ne.{u_1 + 1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
h :
  Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R e.det ⋯) ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
h₁ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
h₂ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R (-e.det) ⋯)
h₃ : SameRay R x e.det ∨ SameRay R x (-e.det)
h₄ :
  SameRay R (HSMul.hSMul (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) e.det) e.det ∨
    SameRay R (HSMul.hSMul (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) e.det) (-e.det)
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) e.orientation ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-e.orientation)
have h₅ : SameRay R (HSMul.hSMul (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) e.det) e.det ∨
  LT.lt.{u_1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
Tactic: rwa [Basis.orientation, ray_eq_iff, neg_rayOfNeZero, ray_eq_iff, x.eq_smul_basis_det e,
    sameRay_neg_smul_left_iff_of_ne e.det_ne_zero hx, sameRay_smul_left_iff_of_ne e.det_ne_zero hx,
    lt_or_lt_iff_ne, ne_comm]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Orientation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
M : Type u_2
inst_3 : AddCommGroup M
inst_4 : Module R M
ι : Type u_3
inst_5 : Fintype ι
inst_6 : DecidableEq ι
e : Basis ι R M
x : M [⋀^ι]→ₗ[R] R
hx_1 : x ≠ 0
hx : Ne.{u_1 + 1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
h :
  Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R e.det ⋯) ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
h₁ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-rayOfNeZero R e.det ⋯)
h₂ : SameRay R x e.det ∨ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (rayOfNeZero R (-e.det) ⋯)
h₃ : SameRay R x e.det ∨ SameRay R x (-e.det)
h₄ :
  SameRay R (HSMul.hSMul (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) e.det) e.det ∨
    SameRay R (HSMul.hSMul (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) e.det) (-e.det)
h₅ :
  SameRay R (HSMul.hSMul (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) e.det) e.det ∨
    LT.lt.{u_1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
⊢ Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) e.orientation ∨
    Eq (α := Orientation R M ι) (rayOfNeZero R x hx_1) (-e.orientation)
have h₆ : LT.lt.{u_1} (α := R) (0 : R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) ∨
  LT.lt.{u_1} (α := R) ((x : (a : ι → M) → R) (⇑e : (a : ι) → M) : R) (0 : R)
Tactic: rwa [Basis.orientation, ray_eq_iff, neg_rayOfNeZero, ray_eq_iff, x.eq_smul_basis_det e,
    sameRay_neg_smul_left_iff_of_ne e.det_ne_zero hx, sameRay_smul_left_iff_of_ne e.det_ne_zero hx,
    lt_or_lt_iff_ne, ne_comm]