awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
⊢ ¬Eq.{u_1 + 1} (α := Orientation ℝ E (Fin 0))
      (rayOfNeZero ℝ (-(AlternatingMap.constLinearEquivOfIsEmpty : (a : ℝ) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 1) ⋯ :
        Ray ℝ (E [⋀^Fin 0]→ₗ[ℝ] ℝ))
      (rayOfNeZero ℝ ((AlternatingMap.constLinearEquivOfIsEmpty : (a : ℝ) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 1) ⋯ :
        Ray ℝ (E [⋀^Fin 0]→ₗ[ℝ] ℝ))
have hnc : ¬SameRay.{0, u_1} ℝ (M := E [⋀^Fin 0]→ₗ[ℝ] ℝ)
    ((AlternatingMap.constLinearEquivOfIsEmpty : (a : ℝ) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 1 : E [⋀^Fin 0]→ₗ[ℝ] ℝ)
    (-(AlternatingMap.constLinearEquivOfIsEmpty : (a : ℝ) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 1 : E [⋀^Fin 0]→ₗ[ℝ] ℝ)
Tactic: rw [ray_eq_iff, SameRay.sameRay_comm]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n)
o : Orientation ℝ E (Fin n)
b : OrthonormalBasis (Fin n) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det
have zero : ∀ [_i : Fact (finrank ℝ E = 0)] (o : Orientation ℝ E (Fin 0)) (b : OrthonormalBasis (Fin 0) ℝ E)
  (hb : b.toBasis.orientation = o), o.volumeForm = b.toBasis.det
Tactic: cases n
Cleared hypotheses: n, _i, o, b, hb
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n)
o : Orientation ℝ E (Fin n)
b : OrthonormalBasis (Fin n) ℝ E
hb : b.toBasis.orientation = o
zero :
  ∀ [_i : Fact (finrank ℝ E = 0)] (o : Orientation ℝ E (Fin 0)) (b : OrthonormalBasis (Fin 0) ℝ E)
    (hb : b.toBasis.orientation = o), o.volumeForm = b.toBasis.det
⊢ o.volumeForm = b.toBasis.det
have succ : ∀ (n : ℕ) [_i : Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) (b : OrthonormalBasis (Fin (n + 1)) ℝ E)
  (hb : b.toBasis.orientation = o), o.volumeForm = b.toBasis.det
Tactic: cases n
Cleared hypotheses: n, _i, o, b, hb
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det
have this : o = positiveOrientation
Tactic: have : o = positiveOrientation := hb.symm.trans b.toBasis.orientation_isEmpty
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation = o
this : o = positiveOrientation
⊢ o.volumeForm = b.toBasis.det
have zero : Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
    (fun (h : Eq (α := ℕ) 0 0) ↦
      sorry.by_cases (fun (x : o = positiveOrientation) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
        fun (x : o = -positiveOrientation) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
    (fun (n : ℕ) (h : 0 = n + 1) ↦
      Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
        [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
        (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
          (Orientation.finOrthonormalBasis sorry sorry o).toBasis.det)
        sorry o)
    sorry =
  b.toBasis.det
Tactic: simp_rw [volumeForm, Or.by_cases, dif_pos this, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation = o
this : o = positiveOrientation
zero :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦
        ⋯.by_cases (fun (x : o = positiveOrientation) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
          fun (x : o = -positiveOrientation) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    b.toBasis.det
⊢ o.volumeForm = b.toBasis.det
have zero₁ : Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
    (fun (h : Eq (α := ℕ) 0 0) ↦
      if h : o = positiveOrientation then AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1
      else -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
    (fun (n : ℕ) (h : 0 = n + 1) ↦
      Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
        [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
        (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
          (Orientation.finOrthonormalBasis sorry sorry o).toBasis.det)
        sorry o)
    sorry =
  b.toBasis.det
Tactic: simp_rw [volumeForm, Or.by_cases, dif_pos this, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation = o
this : o = positiveOrientation
zero :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦
        ⋯.by_cases (fun (x : o = positiveOrientation) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
          fun (x : o = -positiveOrientation) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    b.toBasis.det
zero₁ :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦
        if h : o = positiveOrientation then AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1
        else -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    b.toBasis.det
⊢ o.volumeForm = b.toBasis.det
have zero₂ : Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
    (fun (h : Eq (α := ℕ) 0 0) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
    (fun (n : ℕ) (h : 0 = n + 1) ↦
      Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
        [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
        (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
          (Orientation.finOrthonormalBasis sorry sorry o).toBasis.det)
        sorry o)
    sorry =
  b.toBasis.det
Tactic: simp_rw [volumeForm, Or.by_cases, dif_pos this, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation = o
this : o = positiveOrientation
zero :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦
        ⋯.by_cases (fun (x : o = positiveOrientation) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
          fun (x : o = -positiveOrientation) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    b.toBasis.det
zero₁ :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦
        if h : o = positiveOrientation then AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1
        else -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    b.toBasis.det
zero₂ :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    b.toBasis.det
⊢ o.volumeForm = b.toBasis.det
have zero₃ : AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1 = b.toBasis.det
Tactic: simp_rw [volumeForm, Or.by_cases, dif_pos this, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det
have zero : ∀ (this : o = positiveOrientation), o.volumeForm = b.toBasis.det
Tactic: classical
      have : o = positiveOrientation := hb.symm.trans b.toBasis.orientation_isEmpty
      simp_rw [volumeForm, Or.by_cases, dif_pos this, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
b : OrthonormalBasis (Fin (n + 1)) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det
have succ : Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : n + 1 = a) → E [⋀^Fin (n + 1)]→ₗ[ℝ] ℝ) (n + 1)
    (fun (h : n + 1 = 0) ↦
      Eq.rec (motive := fun (x : ℕ) (x_1 : 0 = x) ↦
        [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
        (fun [Fact (finrank ℝ E = 0)] (o : Orientation ℝ E (Fin 0)) ↦
          sorry.by_cases (fun (x : o = positiveOrientation) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
            fun (x : o = -positiveOrientation) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
        sorry o)
    (fun (n_1 : ℕ) (h : n + 1 = n_1 + 1) ↦
      Eq.rec (motive := fun (x : ℕ) (x_1 : n_1 + 1 = x) ↦
        [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
        (fun [Fact (finrank ℝ E = n_1 + 1)] (o : Orientation ℝ E (Fin (n_1 + 1))) ↦
          (Orientation.finOrthonormalBasis sorry sorry o).toBasis.det)
        sorry o)
    sorry =
  b.toBasis.det
Tactic: simp_rw [volumeForm]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n)
o : Orientation ℝ E (Fin n)
b : OrthonormalBasis (Fin n) ℝ E
hb : b.toBasis.orientation ≠ o
⊢ o.volumeForm = -b.toBasis.det
have zero : ∀ [_i : Fact (finrank ℝ E = 0)] (o : Orientation ℝ E (Fin 0)) (b : OrthonormalBasis (Fin 0) ℝ E)
  (hb : b.toBasis.orientation ≠ o), o.volumeForm = -b.toBasis.det
Tactic: rcases n with - | n
Cleared hypotheses: n, _i, o, b, hb
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n)
o : Orientation ℝ E (Fin n)
b : OrthonormalBasis (Fin n) ℝ E
hb : b.toBasis.orientation ≠ o
zero :
  ∀ [_i : Fact (finrank ℝ E = 0)] (o : Orientation ℝ E (Fin 0)) (b : OrthonormalBasis (Fin 0) ℝ E)
    (hb : b.toBasis.orientation ≠ o), o.volumeForm = -b.toBasis.det
⊢ o.volumeForm = -b.toBasis.det
have succ : ∀ (n : ℕ) [_i : Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) (b : OrthonormalBasis (Fin (n + 1)) ℝ E)
  (hb : b.toBasis.orientation ≠ o), o.volumeForm = -b.toBasis.det
Tactic: rcases n with - | n
Cleared hypotheses: n, _i, o, b, hb
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
⊢ positiveOrientation ≠ o
have rwa : ∀ (hb : positiveOrientation ≠ o), positiveOrientation ≠ o
Tactic: rwa [b.toBasis.orientation_isEmpty] at hb
Cleared hypotheses: hb
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
⊢ o.volumeForm = -b.toBasis.det
have this : positiveOrientation ≠ o
Tactic: have : positiveOrientation ≠ o := by rwa [b.toBasis.orientation_isEmpty] at hb
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
this : positiveOrientation ≠ o
⊢ o.volumeForm = -b.toBasis.det
have zero : Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
    (fun (h : Eq (α := ℕ) 0 0) ↦
      sorry.by_cases (fun (x : o = positiveOrientation) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
        fun (x : o = -positiveOrientation) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
    (fun (n : ℕ) (h : 0 = n + 1) ↦
      Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
        [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
        (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
          (Orientation.finOrthonormalBasis sorry sorry o).toBasis.det)
        sorry o)
    sorry =
  -b.toBasis.det
Tactic: simp_rw [volumeForm, Or.by_cases, dif_neg this.symm, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
this : positiveOrientation ≠ o
zero :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦
        ⋯.by_cases (fun (x : o = positiveOrientation) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
          fun (x : o = -positiveOrientation) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    -b.toBasis.det
⊢ o.volumeForm = -b.toBasis.det
have zero₁ : Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
    (fun (h : Eq (α := ℕ) 0 0) ↦
      if h : o = positiveOrientation then AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1
      else -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
    (fun (n : ℕ) (h : 0 = n + 1) ↦
      Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
        [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
        (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
          (Orientation.finOrthonormalBasis sorry sorry o).toBasis.det)
        sorry o)
    sorry =
  -b.toBasis.det
Tactic: simp_rw [volumeForm, Or.by_cases, dif_neg this.symm, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
this : positiveOrientation ≠ o
zero :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦
        ⋯.by_cases (fun (x : o = positiveOrientation) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
          fun (x : o = -positiveOrientation) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    -b.toBasis.det
zero₁ :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦
        if h : o = positiveOrientation then AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1
        else -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    -b.toBasis.det
⊢ o.volumeForm = -b.toBasis.det
have zero₂ : Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
    (fun (h : Eq (α := ℕ) 0 0) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
    (fun (n : ℕ) (h : 0 = n + 1) ↦
      Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
        [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
        (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
          (Orientation.finOrthonormalBasis sorry sorry o).toBasis.det)
        sorry o)
    sorry =
  -b.toBasis.det
Tactic: simp_rw [volumeForm, Or.by_cases, dif_neg this.symm, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
this : positiveOrientation ≠ o
zero :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦
        ⋯.by_cases (fun (x : o = positiveOrientation) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
          fun (x : o = -positiveOrientation) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    -b.toBasis.det
zero₁ :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦
        if h : o = positiveOrientation then AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1
        else -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    -b.toBasis.det
zero₂ :
  Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : 0 = a) → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0
      (fun (h : Eq (α := ℕ) 0 0) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
      (fun (n : ℕ) (h : 0 = n + 1) ↦
        Eq.rec (motive := fun (x : ℕ) (x_1 : n + 1 = x) ↦
          [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] (o : Orientation ℝ E (Fin (n + 1))) ↦
            (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
          ⋯ o)
      ⋯ =
    -b.toBasis.det
⊢ o.volumeForm = -b.toBasis.det
have zero₃ : -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1 = -b.toBasis.det
Tactic: simp_rw [volumeForm, Or.by_cases, dif_neg this.symm, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
⊢ o.volumeForm = -b.toBasis.det
have have : positiveOrientation ≠ o
Tactic: classical
      have : positiveOrientation ≠ o := by rwa [b.toBasis.orientation_isEmpty] at hb
      simp_rw [volumeForm, Or.by_cases, dif_neg this.symm, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
have : positiveOrientation ≠ o
⊢ o.volumeForm = -b.toBasis.det
have zero : ∀ (this : positiveOrientation ≠ o), o.volumeForm = -b.toBasis.det
Tactic: classical
      have : positiveOrientation ≠ o := by rwa [b.toBasis.orientation_isEmpty] at hb
      simp_rw [volumeForm, Or.by_cases, dif_neg this.symm, Nat.rec_zero, Basis.det_isEmpty]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.Orientation (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
b : OrthonormalBasis (Fin (n + 1)) ℝ E
hb : b.toBasis.orientation ≠ o
e : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o
⊢ o.volumeForm = -b.toBasis.det
have succ : Nat.casesAuxOn (motive := fun (a : ℕ) ↦ (h : n + 1 = a) → E [⋀^Fin (n + 1)]→ₗ[ℝ] ℝ) (n + 1)
    (fun (h : n + 1 = 0) ↦
      Eq.rec (motive := fun (x : ℕ) (x_1 : 0 = x) ↦
        [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
        (fun [Fact (finrank ℝ E = 0)] (o : Orientation ℝ E (Fin 0)) ↦
          sorry.by_cases (fun (x : o = positiveOrientation) ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
            fun (x : o = -positiveOrientation) ↦ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
        sorry o)
    (fun (n_1 : ℕ) (h : n + 1 = n_1 + 1) ↦
      Eq.rec (motive := fun (x : ℕ) (x_1 : n_1 + 1 = x) ↦
        [_i : Fact (finrank ℝ E = x)] → (o : Orientation ℝ E (Fin x)) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
        (fun [Fact (finrank ℝ E = n_1 + 1)] (o : Orientation ℝ E (Fin (n_1 + 1))) ↦
          (Orientation.finOrthonormalBasis sorry sorry o).toBasis.det)
        sorry o)
    sorry =
  -b.toBasis.det
Tactic: simp_rw [volumeForm]