mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Even

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
Q : QuadraticForm R M
A : Type u_3
B : Type u_4
inst✝³ : Ring A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
m₁ m₂ m₃ : M
⊢ (CliffordAlgebra.ι Q) m₁ * ((CliffordAlgebra.ι Q) m₂ * (CliffordAlgebra.ι Q) m₂ * (CliffordAlgebra.ι Q) m₃) =
    Q m₂ • ((CliffordAlgebra.ι Q) m₁ * (CliffordAlgebra.ι Q) m₃)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Even

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra R A
f g : ↥(even Q) →ₐ[R] A
h : (ι Q).compr₂ f = (ι Q).compr₂ g
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ CliffordAlgebra Q →ₐ[R] ↥(even (Q' Q))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ ↑((toEven Q) ((ι Q) m)) = e0 Q * (v Q) m

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
  ((Algebra.lmul R (CliffordAlgebra Q)).toLinearMap ∘ₗ
        (ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ LinearMap.snd R M R)).compl₂
    (ι Q ∘ₗ LinearMap.fst R M R - Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ LinearMap.snd R M R)
x y : M × R
⊢ (f x) y = ((ι Q) x.1 + (algebraMap R (CliffordAlgebra Q)) x.2) * ((ι Q) y.1 - (algebraMap R (CliffordAlgebra Q)) y.2)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
B : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
d : Module.Dual R M
x : CliffordAlgebra Q
⊢ (changeForm h) ((contractLeft d) x) = (contractLeft d) ((changeForm h) x)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
⊢ (changeForm ⋯) x = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: K : Type u_3
V : Type u_4
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : Dual K V
v : V
⊢ LinearMap.det (transvection f v) = 1 + f v

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
  ((Algebra.lmul R (CliffordAlgebra Q)).toLinearMap ∘ₗ
        (ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ LinearMap.snd R M R)).compl₂
    (ι Q ∘ₗ LinearMap.fst R M R - Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ LinearMap.snd R M R)
f_apply :
  ∀ (x y : M × R),
    (f x) y =
      ((ι Q) x.1 + (algebraMap R (CliffordAlgebra Q)) x.2) * ((ι Q) y.1 - (algebraMap R (CliffordAlgebra Q)) y.2)
hc : ∀ (r : R) (x : CliffordAlgebra Q), Commute ((algebraMap R (CliffordAlgebra Q)) r) x
⊢ ∀ (m : M × R),
    (ι Q) m.1 * (ι Q) m.1 - (algebraMap R (CliffordAlgebra Q)) m.2 * (algebraMap R (CliffordAlgebra Q)) m.2 =
      (algebraMap R (CliffordAlgebra Q)) ((Q' Q) m)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_3
V : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup V
inst✝³ : Module R V
inst✝² : Free R V
inst✝¹ : Module.Finite R V
inst✝ : IsDomain R
f : Dual R V
v : V
⊢ LinearMap.det (transvection f v) = 1 + f v

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Even

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra R A
f : EvenHom Q A
m : M
x : A × ↥(S f)
⊢ ((fFold f) m) (((fFold f) m) x) = Q m • x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Even

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
Q : QuadraticForm R M
A : Type u_3
B : Type u_4
inst✝³ : Ring A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
f✝ f : EvenHom Q A
⊢ ↥(even Q) →ₗ[R] A

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Even

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra R A
f : EvenHom Q A
m₁ m₂ : M
⊢ (aux f) (((ι Q).bilin m₁) m₂) = (f.bilin m₁) m₂

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_3
V : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : Free R V
inst✝ : Module.Finite R V
f : Dual R V
v : V
hR : Nontrivial R
n : ℕ := finrank R V
b : Basis (Fin n) R V
S : Type := MvPolynomial (Fin n ⊕ Fin n) ℤ
γ : S →+* R := ↑(MvPolynomial.aeval (Sum.elim (fun i ↦ f (b i)) fun i ↦ (b.coord i) v))
this : IsDomain S
x✝² : Algebra S R := γ.toAlgebra
x✝¹ : Module S V := compHom V γ
x✝ : IsScalarTower S R V
ε : (Fin n → S) →ₗ[S] V := Fintype.linearCombination S fun i ↦ b i
ibc : IsBaseChange R ε
M : Type := Fin n → S
i : Fin n
⊢ ε (Pi.single i 1) = b i

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Even

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra R A
f : EvenHom Q A
x y : ↥(even Q)
⊢ (aux f) (x * y) = (aux f) x * (aux f) y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_3
V : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : Free R V
inst✝ : Module.Finite R V
f : Dual R V
v : V
hR : Nontrivial R
n : ℕ := finrank R V
b : Basis (Fin n) R V
S : Type := MvPolynomial (Fin n ⊕ Fin n) ℤ
γ : S →+* R := ↑(MvPolynomial.aeval (Sum.elim (fun i ↦ f (b i)) fun i ↦ (b.coord i) v))
this : IsDomain S
x✝² : Algebra S R := γ.toAlgebra
x✝¹ : Module S V := compHom V γ
x✝ : IsScalarTower S R V
ε : (Fin n → S) →ₗ[S] V := Fintype.linearCombination S fun i ↦ b i
ibc : IsBaseChange R ε
M : Type := Fin n → S
hε : ∀ (i : Fin n), ε (Pi.single i 1) = b i
fM : Dual S M := Fintype.linearCombination S fun i ↦ MvPolynomial.X (Sum.inl i)
vM : M := fun i ↦ MvPolynomial.X (Sum.inr i)
⊢ ibc.toDual fM = f

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
m : M
inst✝ : Invertible (Q m)
⊢ (ι Q) (⅟(Q m) • m) * (ι Q) m = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
m : M
inst✝ : Invertible (Q m)
⊢ (ι Q) m * (ι Q) (⅟(Q m) • m) = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
m : M
inst✝¹ : Invertible (Q m)
inst✝ : Invertible ((ι Q) m)
⊢ ⅟((ι Q) m) = (ι Q) (⅟(Q m) • m)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
h : IsUnit (Q m)
⊢ IsUnit ((ι Q) m)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
x : CliffordAlgebra Q
⊢ (changeForm h') ((changeForm h) x) = (changeForm ⋯) x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
x : CliffordAlgebra Q
⊢ (changeForm ⋯) x = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
x : CliffordAlgebra Q
⊢ (changeForm ⋯) ((changeForm h) x) = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
x : CliffordAlgebra Q'
⊢ (changeForm ⋯) x = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
a b : M
inst✝¹ : Invertible ((ι Q) a)
inst✝ : Invertible (Q a)
⊢ (ι Q) a * (ι Q) b * ⅟((ι Q) a) = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
x : CliffordAlgebra Q'
⊢ (changeForm h) ((changeForm ⋯) x) = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_3
V : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : Free R V
inst✝ : Module.Finite R V
f : Dual R V
v : V
hR : Nontrivial R
n : ℕ := finrank R V
b : Basis (Fin n) R V
S : Type := MvPolynomial (Fin n ⊕ Fin n) ℤ
γ : S →+* R := ↑(MvPolynomial.aeval (Sum.elim (fun i ↦ f (b i)) fun i ↦ (b.coord i) v))
this : IsDomain S
x✝² : Algebra S R := γ.toAlgebra
x✝¹ : Module S V := compHom V γ
x✝ : IsScalarTower S R V
ε : (Fin n → S) →ₗ[S] V := Fintype.linearCombination S fun i ↦ b i
ibc : IsBaseChange R ε
M : Type := Fin n → S
hε : ∀ (i : Fin n), ε (Pi.single i 1) = b i
fM : Dual S M := Fintype.linearCombination S fun i ↦ MvPolynomial.X (Sum.inl i)
vM : M := fun i ↦ MvPolynomial.X (Sum.inr i)
hf : ibc.toDual fM = f
⊢ ε vM = v

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
a b : M
inst✝¹ : Invertible ((ι Q) a)
inst✝ : Invertible (Q a)
⊢ ⅟((ι Q) a) * (ι Q) b * (ι Q) a = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
inst✝¹ : Invertible 2
m : M
inst✝ : Invertible ((ι Q) m)
⊢ ↑(equivExterior Q) 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
inst✝ : Invertible 2
m : M
h : IsUnit ((ι Q) m)
⊢ IsUnit (Q m)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_3
V : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : Free R V
inst✝ : Module.Finite R V
f : Dual R V
v : V
⊢ LinearMap.det (transvection f v) = 1 + f v

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_3
V : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
f : Dual R V
v : V
hfv : f v = 0
⊢ LinearEquiv.det (LinearEquiv.transvection hfv) = 1

Tactic:
rw [← Units.val_inj, LinearEquiv.coe_det,
    LinearEquiv.transvection.coe_toLinearMap hfv, Units.val_one]

Goal state after:
R : Type u_3
V : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
f : Dual R V
v : V
hfv : f v = 0
⊢ LinearMap.det (transvection f v) = 1
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_3
V : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
f : Dual R V
v : V
hfv : f v = 0
⊢ LinearEquiv.det (LinearEquiv.transvection hfv) = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ ↥(even (Q' Q)) →ₐ[R] CliffordAlgebra Q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
⊢ ↑((toEven Q) ((ofEven Q) (((even.ι (Q' Q)).bilin (m₁, r₁)) (m₂, r₂)))) =
    (e0 Q * (v Q) m₁ + (algebraMap R (CliffordAlgebra (Q' Q))) r₁) *
      (e0 Q * (v Q) m₂ - (algebraMap R (CliffordAlgebra (Q' Q))) r₂)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i j : ι
a : 𝒜 i
b : ℬ j
⊢ (gradedCommAux R 𝒜 ℬ) ((lof R (ι × ι) (fun i ↦ 𝒜 i.1 ⊗[R] ℬ i.2) (i, j)) (a ⊗ₜ[R] b)) =
    (-1) ^ (j * i) • (lof R (ι × ι) (fun i ↦ ℬ i.1 ⊗[R] 𝒜 i.2) (j, i)) (b ⊗ₜ[R] a)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
⊢ gradedCommAux R 𝒜 ℬ ∘ₗ gradedCommAux R ℬ 𝒜 = LinearMap.id

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
⊢ ((⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) ≃ₗ[R] (⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), 𝒜 i

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
⊢ (gradedComm R 𝒜 ℬ).symm = gradedComm R ℬ 𝒜

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i j : ι
a : 𝒜 i
b : ℬ j
⊢ (gradedComm R 𝒜 ℬ) ((lof R ι 𝒜 i) a ⊗ₜ[R] (lof R ι ℬ j) b) = (-1) ^ (j * i) • (lof R ι ℬ j) b ⊗ₜ[R] (lof R ι 𝒜 i) a

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
⊢ (e0 Q * (v Q) m₁ + (algebraMap R (CliffordAlgebra (Q' Q))) r₁) *
      (e0 Q * (v Q) m₂ - (algebraMap R (CliffordAlgebra (Q' Q))) r₂) =
    e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) + r₁ • e0 Q * (v Q) m₂ - r₂ • e0 Q * (v Q) m₁ -
      (algebraMap R (CliffordAlgebra (Q' Q))) (r₁ * r₂)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
a : ⨁ (i : ι), 𝒜 i
b : ℬ 0
⊢ (gradedComm R 𝒜 ℬ) (a ⊗ₜ[R] (lof R ι ℬ 0) b) = (lof R ι ℬ 0) b ⊗ₜ[R] a

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
⊢ e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) = (v Q) m₁ * (v Q) m₂

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
h1 : e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) = (v Q) m₁ * (v Q) m₂
⊢ -(r₂ • e0 Q * (v Q) m₁) = (v Q) m₁ * r₂ • e0 Q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
a : 𝒜 0
b : ⨁ (i : ι), ℬ i
⊢ (gradedComm R 𝒜 ℬ) ((lof R ι 𝒜 0) a ⊗ₜ[R] b) = b ⊗ₜ[R] (lof R ι 𝒜 0) a

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
h1 : e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) = (v Q) m₁ * (v Q) m₂
h2 : -(r₂ • e0 Q * (v Q) m₁) = (v Q) m₁ * r₂ • e0 Q
⊢ -(algebraMap R (CliffordAlgebra (Q' Q))) (r₁ * r₂) = r₁ • e0 Q * r₂ • e0 Q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
⊢ DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
⊢ e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) + r₁ • e0 Q * (v Q) m₂ - r₂ • e0 Q * (v Q) m₁ -
      (algebraMap R (CliffordAlgebra (Q' Q))) (r₁ * r₂) =
    (v Q) m₁ * (v Q) m₂ + r₁ • e0 Q * (v Q) m₂ + (v Q) m₁ * r₂ • e0 Q + r₁ • e0 Q * r₂ • e0 Q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
⊢ (v Q) m₁ * (v Q) m₂ + r₁ • e0 Q * (v Q) m₂ + (v Q) m₁ * r₂ • e0 Q + r₁ • e0 Q * r₂ • e0 Q =
    (ι (Q' Q)) (m₁, r₁) * (ι (Q' Q)) (m₂, r₂)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (ofEven Q) ((toEven Q) ((ι Q) m)) = (ofEven Q) ⟨↑((toEven Q) ((ι Q) m)), ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (ofEven Q) ⟨↑((toEven Q) ((ι Q) m)), ⋯⟩ =
    ((ι Q) 0 + (algebraMap R (CliffordAlgebra Q)) 1) * ((ι Q) m - (algebraMap R (CliffordAlgebra Q)) 0)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ ((ι Q) 0 + (algebraMap R (CliffordAlgebra Q)) 1) * ((ι Q) m - (algebraMap R (CliffordAlgebra Q)) 0) = (ι Q) m

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
⊢ ↑((toEven Q) (reverse (involute x))) = reverse ↑((toEven Q) x)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
h : Q' = -Q
m : M
⊢ ((-(even.ι Q').bilin) m) m = (algebraMap R ↥(even Q')) (Q m)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
h : Q' = -Q
m₁ m₂ m₃ : M
⊢ ((-(even.ι Q').bilin) m₁) m₂ * ((-(even.ι Q').bilin) m₂) m₃ = Q m₂ • ((-(even.ι Q').bilin) m₁) m₃

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
j₁ i₂ : ι
a₁ : ⨁ (i : ι), 𝒜 i
b₁ : ℬ j₁
a₂ : 𝒜 i₂
b₂ : ⨁ (i : ι), ℬ i
⊢ ((gradedMul R 𝒜 ℬ) (a₁ ⊗ₜ[R] (lof R ι ℬ j₁) b₁)) ((lof R ι 𝒜 i₂) a₂ ⊗ₜ[R] b₂) =
    (-1) ^ (j₁ * i₂) • (a₁ * (lof R ι 𝒜 i₂) a₂) ⊗ₜ[R] ((lof R ι ℬ j₁) b₁ * b₂)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
h : Q' = -Q
h' : Q = -Q'
⊢ (evenToNeg Q' Q h').comp (evenToNeg Q Q' h) = AlgHom.id R ↥(even Q)

Tactic:
by

Goal state after: