awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) x
inst_3 : Invertible (α := R) 2
m : M
⊢ HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have mem : ∀ (x : (CliffordAlgebra Q)ˣ)
  (hx :
    Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
      (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x)
  (m : M),
  HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
Tactic: induction hx using Subgroup.closure_induction'' generalizing m with
  | mem x hx =>
    obtain ⟨a, ha⟩ := hx
    letI := x.invertible
    letI : Invertible (ι Q a) := by rwa [ha]
    letI : Invertible (Q a) := invertibleOfInvertibleι Q a
    simp_rw [← invOf_units x, ← ha, ι_mul_ι_mul_invOf_ι, LinearMap.mem_range_self]
  | inv_mem x hx =>
    obtain ⟨a, ha⟩ := hx
    letI := x.invertible
    letI : Invertible (ι Q a) := by rwa [ha]
    letI : Invertible (Q a) := invertibleOfInvertibleι Q a
    simp_rw [← invOf_units x, inv_inv, ← ha, invOf_ι_mul_ι_mul_ι, LinearMap.mem_range_self]
  | one => simp_rw [inv_one, Units.val_one, one_mul, mul_one, LinearMap.mem_range_self]
  | mul y z _ _ hy hz =>
    simp_rw [mul_inv_rev, Units.val_mul]
    suffices ↑y * (↑z * ι Q m * ↑z⁻¹) * ↑y⁻¹ ∈ _ by
      simpa only [mul_assoc] using this
    obtain ⟨z', hz'⟩ := hz m
    obtain ⟨y', hy'⟩ := hy z'
    simp_rw [← hz', ← hy', LinearMap.mem_range_self]
Cleared hypotheses: hx, m
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) x
inst_3 : Invertible (α := R) 2
m : M
mem :
  ∀ (x : (CliffordAlgebra Q)ˣ)
    (hx :
      Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
        (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x)
    (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑x⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
⊢ HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have inv_mem : ∀ (x : (CliffordAlgebra Q)ˣ)
  (hx :
    Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
      (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x)
  (m : M),
  HMul.hMul (β := CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
Tactic: induction hx using Subgroup.closure_induction'' generalizing m with
  | mem x hx =>
    obtain ⟨a, ha⟩ := hx
    letI := x.invertible
    letI : Invertible (ι Q a) := by rwa [ha]
    letI : Invertible (Q a) := invertibleOfInvertibleι Q a
    simp_rw [← invOf_units x, ← ha, ι_mul_ι_mul_invOf_ι, LinearMap.mem_range_self]
  | inv_mem x hx =>
    obtain ⟨a, ha⟩ := hx
    letI := x.invertible
    letI : Invertible (ι Q a) := by rwa [ha]
    letI : Invertible (Q a) := invertibleOfInvertibleι Q a
    simp_rw [← invOf_units x, inv_inv, ← ha, invOf_ι_mul_ι_mul_ι, LinearMap.mem_range_self]
  | one => simp_rw [inv_one, Units.val_one, one_mul, mul_one, LinearMap.mem_range_self]
  | mul y z _ _ hy hz =>
    simp_rw [mul_inv_rev, Units.val_mul]
    suffices ↑y * (↑z * ι Q m * ↑z⁻¹) * ↑y⁻¹ ∈ _ by
      simpa only [mul_assoc] using this
    obtain ⟨z', hz'⟩ := hz m
    obtain ⟨y', hy'⟩ := hy z'
    simp_rw [← hz', ← hy', LinearMap.mem_range_self]
Cleared hypotheses: hx, m
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) x
inst_3 : Invertible (α := R) 2
m : M
mem :
  ∀ (x : (CliffordAlgebra Q)ˣ)
    (hx :
      Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
        (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x)
    (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑x⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
inv_mem :
  ∀ (x : (CliffordAlgebra Q)ˣ)
    (hx :
      Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
        (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x)
    (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
⊢ HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have one : ∀ (m : M),
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q) ((↑1 : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑1⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
Tactic: induction hx using Subgroup.closure_induction'' generalizing m with
  | mem x hx =>
    obtain ⟨a, ha⟩ := hx
    letI := x.invertible
    letI : Invertible (ι Q a) := by rwa [ha]
    letI : Invertible (Q a) := invertibleOfInvertibleι Q a
    simp_rw [← invOf_units x, ← ha, ι_mul_ι_mul_invOf_ι, LinearMap.mem_range_self]
  | inv_mem x hx =>
    obtain ⟨a, ha⟩ := hx
    letI := x.invertible
    letI : Invertible (ι Q a) := by rwa [ha]
    letI : Invertible (Q a) := invertibleOfInvertibleι Q a
    simp_rw [← invOf_units x, inv_inv, ← ha, invOf_ι_mul_ι_mul_ι, LinearMap.mem_range_self]
  | one => simp_rw [inv_one, Units.val_one, one_mul, mul_one, LinearMap.mem_range_self]
  | mul y z _ _ hy hz =>
    simp_rw [mul_inv_rev, Units.val_mul]
    suffices ↑y * (↑z * ι Q m * ↑z⁻¹) * ↑y⁻¹ ∈ _ by
      simpa only [mul_assoc] using this
    obtain ⟨z', hz'⟩ := hz m
    obtain ⟨y', hy'⟩ := hy z'
    simp_rw [← hz', ← hy', LinearMap.mem_range_self]
Cleared hypotheses: hx, m
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
    (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x
b : M
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have mem : ∀ (a : M) (ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)),
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
Tactic: obtain ⟨a, ha⟩ := hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
b a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this_1 : Invertible (↑x : CliffordAlgebra Q) := x.invertible
this_2 : Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  Eq.mpr ⋯ this_1
this : Invertible (α := R) ((Q : (a : M) → R) a : R) := invertibleOfInvertibleι Q a
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have mem : HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
      ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
    @Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _ ∈
  (ι Q).range
Tactic: simp_rw [← invOf_units x, ← ha, involute_ι, neg_mul, ι_mul_ι_mul_invOf_ι Q a b, ← map_neg,
      LinearMap.mem_range_self]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
b a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this_1 : Invertible (↑x : CliffordAlgebra Q) := x.invertible
this_2 : Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  Eq.mpr ⋯ this_1
this : Invertible (α := R) ((Q : (a : M) → R) a : R) := invertibleOfInvertibleι Q a
mem :
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      @Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _ ∈
    (ι Q).range
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have mem₁ : HMul.hMul (β := CliffordAlgebra Q)
    (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a) :
        CliffordAlgebra Q)
      ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q))
    (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q) ∈
  (ι Q).range
Tactic: simp_rw [← invOf_units x, ← ha, involute_ι, neg_mul, ι_mul_ι_mul_invOf_ι Q a b, ← map_neg,
      LinearMap.mem_range_self]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
b a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this_1 : Invertible (↑x : CliffordAlgebra Q) := x.invertible
this_2 : Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  Eq.mpr ⋯ this_1
this : Invertible (α := R) ((Q : (a : M) → R) a : R) := invertibleOfInvertibleι Q a
mem :
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      @Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _ ∈
    (ι Q).range
mem₁ :
  HMul.hMul (β := CliffordAlgebra Q)
      (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a) :
          CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q))
      (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q) ∈
    (ι Q).range
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have mem₂ : HMul.hMul (β := CliffordAlgebra Q)
    (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      (-(ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q)
      ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q))
    (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q) ∈
  (ι Q).range
Tactic: simp_rw [← invOf_units x, ← ha, involute_ι, neg_mul, ι_mul_ι_mul_invOf_ι Q a b, ← map_neg,
      LinearMap.mem_range_self]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
b a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this_1 : Invertible (↑x : CliffordAlgebra Q) := x.invertible
this_2 : Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  Eq.mpr ⋯ this_1
this : Invertible (α := R) ((Q : (a : M) → R) a : R) := invertibleOfInvertibleι Q a
mem :
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      @Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _ ∈
    (ι Q).range
mem₁ :
  HMul.hMul (β := CliffordAlgebra Q)
      (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a) :
          CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q))
      (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q) ∈
    (ι Q).range
mem₂ :
  HMul.hMul (β := CliffordAlgebra Q)
      (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        (-(ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q))
      (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q) ∈
    (ι Q).range
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have mem₃ : -HMul.hMul (β := CliffordAlgebra Q)
      (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q))
      (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q) ∈
  (ι Q).range
Tactic: simp_rw [← invOf_units x, ← ha, involute_ι, neg_mul, ι_mul_ι_mul_invOf_ι Q a b, ← map_neg,
      LinearMap.mem_range_self]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
b a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this_1 : Invertible (↑x : CliffordAlgebra Q) := x.invertible
this_2 : Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  Eq.mpr ⋯ this_1
this : Invertible (α := R) ((Q : (a : M) → R) a : R) := invertibleOfInvertibleι Q a
mem :
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      @Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _ ∈
    (ι Q).range
mem₁ :
  HMul.hMul (β := CliffordAlgebra Q)
      (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a) :
          CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q))
      (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q) ∈
    (ι Q).range
mem₂ :
  HMul.hMul (β := CliffordAlgebra Q)
      (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        (-(ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q))
      (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q) ∈
    (ι Q).range
mem₃ :
  -HMul.hMul (β := CliffordAlgebra Q)
        (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q))
        (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q) ∈
    (ι Q).range
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have mem₄ : Membership.mem (α := CliffordAlgebra Q) (ι Q).range
  (-(ι Q : (a : M) → CliffordAlgebra Q)
        (HMul.hMul (α := R) (β := R)
              (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne ((Q : (a : M) → R) a) _ : R)
              (QuadraticMap.polar (⇑Q : (a : M) → R) a b : R) •
            a -
          b) :
    CliffordAlgebra Q)
Tactic: simp_rw [← invOf_units x, ← ha, involute_ι, neg_mul, ι_mul_ι_mul_invOf_ι Q a b, ← map_neg,
      LinearMap.mem_range_self]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
    (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x
b : M
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have inv_mem : ∀ (a : M) (ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)),
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
Tactic: obtain ⟨a, ha⟩ := hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
b a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this_1 : Invertible (↑x : CliffordAlgebra Q) := x.invertible
this_2 : Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  Eq.mpr ⋯ this_1
this_3 : Invertible (α := R) ((Q : (a : M) → R) a : R) := invertibleOfInvertibleι Q a
this_4 : Invertible (α := CliffordAlgebra Q) (-(ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  invertibleNeg ((ι Q : (a : M) → CliffordAlgebra Q) a)
this : Invertible (α := CliffordAlgebra Q)
  ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a) :
    CliffordAlgebra Q) :=
  Invertible.map involute ((ι Q : (a : M) → CliffordAlgebra Q) a)
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have inv_mem : HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q)
          (@Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _) :
        CliffordAlgebra Q)
      ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
    (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
  (ι Q).range
Tactic: simp_rw [← invOf_units x, inv_inv, ← ha, map_invOf, involute_ι, invOf_neg, neg_mul,
      invOf_ι_mul_ι_mul_ι, ← map_neg, LinearMap.mem_range_self]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
b a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this_1 : Invertible (↑x : CliffordAlgebra Q) := x.invertible
this_2 : Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  Eq.mpr ⋯ this_1
this_3 : Invertible (α := R) ((Q : (a : M) → R) a : R) := invertibleOfInvertibleι Q a
this_4 : Invertible (α := CliffordAlgebra Q) (-(ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  invertibleNeg ((ι Q : (a : M) → CliffordAlgebra Q) a)
this : Invertible (α := CliffordAlgebra Q)
  ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a) :
    CliffordAlgebra Q) :=
  Invertible.map involute ((ι Q : (a : M) → CliffordAlgebra Q) a)
inv_mem :
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q)
            (@Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _) :
          CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have inv_mem₁ : HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q)
          (@Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _) :
        CliffordAlgebra Q)
      ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
    (↑x : CliffordAlgebra Q) ∈
  (ι Q).range
Tactic: simp_rw [← invOf_units x, inv_inv, ← ha, map_invOf, involute_ι, invOf_neg, neg_mul,
      invOf_ι_mul_ι_mul_ι, ← map_neg, LinearMap.mem_range_self]