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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ ((Algebra.ofId R (CliffordAlgebra 0)).comp ((lift 0) ⟨0, ⋯⟩)).toLinearMap ∘ₗ ι 0 =
    (AlgHom.id R (CliffordAlgebra 0)).toLinearMap ∘ₗ ι 0

Tactic:
;

Goal state after:
R : Type u_1
inst✝ : CommRing R
⊢ ((Algebra.ofId R (CliffordAlgebra 0)).comp ((lift 0) ⟨0, ⋯⟩)).toLinearMap ∘ₗ ι 0 =
    (AlgHom.id R (CliffordAlgebra 0)).toLinearMap ∘ₗ ι 0
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case pos
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
hfv : f v = 0
hv : ¬v = 0
hf : ¬f = 0
ι : Set V
b : Basis (↑ι) K V
i j : ↑ι
hi : i ≠ j
hj : v = b i ∧ f = b.coord j
this : Fintype ↑ι
x y : ↑ι
h : i = x ∧ j = y
⊢ (Finsupp.single x ((Finsupp.single y 1) y)) x = Matrix.single x y 1 x y

Tactic:
;

Goal state after:
case pos
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
hfv : f v = 0
hv : ¬v = 0
hf : ¬f = 0
ι : Set V
b : Basis (↑ι) K V
i j : ↑ι
hi : i ≠ j
hj : v = b i ∧ f = b.coord j
this : Fintype ↑ι
x y : ↑ι
h : i = x ∧ j = y
⊢ (Finsupp.single x ((Finsupp.single y 1) y)) x = Matrix.single x y 1 x y
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
c₁ c₂ r : R
v : R × R
⊢ { re := 0, imI := r * v.1, imJ := r * v.2, imK := 0 } = { re := r * 0, imI := r * v.1, imJ := r * v.2, imK := r * 0 }

Tactic:
;

Goal state after:
R : Type u_1
inst✝ : CommRing R
c₁ c₂ r : R
v : R × R
⊢ { re := 0, imI := r * v.1, imJ := r * v.2, imK := 0 } = { re := r * 0, imI := r * v.1, imJ := r * v.2, imK := r * 0 }
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Goal state before:
pp: case re
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
v : R × R
⊢ 0 * 0 + c₁ * v.1 * v.1 + c₂ * v.2 * v.2 + 0 * c₂ * v.2 * 0 - c₁ * c₂ * 0 * 0 = c₁ * (v.1 * v.1) + c₂ * (v.2 * v.2)

Tactic:
;

Goal state after:
case re
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
v : R × R
⊢ 0 * 0 + c₁ * v.1 * v.1 + c₂ * v.2 * v.2 + 0 * c₂ * v.2 * 0 - c₁ * c₂ * 0 * 0 = c₁ * (v.1 * v.1) + c₂ * (v.2 * v.2)
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Goal state before:
pp: case imI
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
v : R × R
⊢ 0 * v.1 + v.1 * 0 + 0 * v.1 * v.1 - c₂ * v.2 * 0 + c₂ * 0 * v.2 = 0

Tactic:
;

Goal state after:
case imI
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
v : R × R
⊢ 0 * v.1 + v.1 * 0 + 0 * v.1 * v.1 - c₂ * v.2 * 0 + c₂ * 0 * v.2 = 0
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Goal state before:
pp: case imJ
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
v : R × R
⊢ 0 * v.2 + c₁ * v.1 * 0 + v.2 * 0 + 0 * v.2 * v.1 - c₁ * 0 * v.1 = 0

Tactic:
;

Goal state after:
case imJ
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
v : R × R
⊢ 0 * v.2 + c₁ * v.1 * 0 + v.2 * 0 + 0 * v.2 * v.1 - c₁ * 0 * v.1 = 0
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Goal state before:
pp: case imK
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
v : R × R
⊢ 0 * 0 + v.1 * v.2 + 0 * v.1 * 0 - v.2 * v.1 + 0 * 0 = 0

Tactic:
;

Goal state after:
case imK
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
v : R × R
⊢ 0 * 0 + v.1 * v.2 + 0 * v.1 * 0 - v.2 * v.1 + 0 * 0 = 0
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Goal state before:
pp: case hl
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
⊢ (((ofQuaternion.comp toQuaternion).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inl R R R) 1 =
    (((AlgHom.id R (CliffordAlgebra (Q c₁ c₂))).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inl R R R) 1

Tactic:
;

Goal state after:
case hl
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
⊢ (((ofQuaternion.comp toQuaternion).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inl R R R) 1 =
    (((AlgHom.id R (CliffordAlgebra (Q c₁ c₂))).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inl R R R) 1
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Goal state before:
pp: case hr
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
⊢ (((ofQuaternion.comp toQuaternion).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inr R R R) 1 =
    (((AlgHom.id R (CliffordAlgebra (Q c₁ c₂))).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inr R R R) 1

Tactic:
;

Goal state after:
case hr
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R
⊢ (((ofQuaternion.comp toQuaternion).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inr R R R) 1 =
    (((AlgHom.id R (CliffordAlgebra (Q c₁ c₂))).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inr R R R) 1
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ (((CliffordAlgebra.lift 0) ⟨inrHom R R, ⋯⟩).comp (DualNumber.lift ⟨(Algebra.ofId R (CliffordAlgebra 0), (ι 0) 1), ⋯⟩))
      ε =
    (AlgHom.id R R[ε]) ε

Tactic:
;

Goal state after:
R : Type u_1
inst✝ : CommRing R
⊢ (((CliffordAlgebra.lift 0) ⟨inrHom R R, ⋯⟩).comp (DualNumber.lift ⟨(Algebra.ofId R (CliffordAlgebra 0), (ι 0) 1), ⋯⟩))
      ε =
    (AlgHom.id R R[ε]) ε
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ (((DualNumber.lift ⟨(Algebra.ofId R (CliffordAlgebra 0), (ι 0) 1), ⋯⟩).comp
            ((CliffordAlgebra.lift 0) ⟨inrHom R R, ⋯⟩)).toLinearMap ∘ₗ
        ι 0)
      1 =
    ((AlgHom.id R (CliffordAlgebra 0)).toLinearMap ∘ₗ ι 0) 1

Tactic:
;

Goal state after:
R : Type u_1
inst✝ : CommRing R
⊢ (((DualNumber.lift ⟨(Algebra.ofId R (CliffordAlgebra 0), (ι 0) 1), ⋯⟩).comp
            ((CliffordAlgebra.lift 0) ⟨inrHom R R, ⋯⟩)).toLinearMap ∘ₗ
        ι 0)
      1 =
    ((AlgHom.id R (CliffordAlgebra 0)).toLinearMap ∘ₗ ι 0) 1
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
m : M
x : CliffordAlgebra Q
⊢ ((changeFormAux Q' B) m) (((foldr Q (changeFormAux Q' B) ⋯) 1) x) =
    (ι Q') m * (changeForm h) x - (contractLeft (B m)) ((changeForm h) x)

Tactic:
;

Goal state after:
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
m : M
x : CliffordAlgebra Q
⊢ ((changeFormAux Q' B) m) (((foldr Q (changeFormAux Q' B) ⋯) 1) x) =
    (ι Q') m * (changeForm h) x - (contractLeft (B m)) ((changeForm h) x)
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
x✝² x✝¹ x✝ : M
⊢ ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝ + (CliffordAlgebra.ι Q) x✝¹ * (CliffordAlgebra.ι Q) x✝, ⋯⟩ =
    ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝, ⋯⟩ +
      ⟨(CliffordAlgebra.ι Q) x✝¹ * (CliffordAlgebra.ι Q) x✝, ⋯⟩

Tactic:
;

Goal state after:
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
x✝² x✝¹ x✝ : M
⊢ ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝ + (CliffordAlgebra.ι Q) x✝¹ * (CliffordAlgebra.ι Q) x✝, ⋯⟩ =
    ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝, ⋯⟩ +
      ⟨(CliffordAlgebra.ι Q) x✝¹ * (CliffordAlgebra.ι Q) x✝, ⋯⟩
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
x✝² : R
x✝¹ x✝ : M
⊢ ⟨x✝² • ((CliffordAlgebra.ι Q) x✝¹ * (CliffordAlgebra.ι Q) x✝), ⋯⟩ =
    x✝² • ⟨(CliffordAlgebra.ι Q) x✝¹ * (CliffordAlgebra.ι Q) x✝, ⋯⟩

Tactic:
;

Goal state after:
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
x✝² : R
x✝¹ x✝ : M
⊢ ⟨x✝² • ((CliffordAlgebra.ι Q) x✝¹ * (CliffordAlgebra.ι Q) x✝), ⋯⟩ =
    x✝² • ⟨(CliffordAlgebra.ι Q) x✝¹ * (CliffordAlgebra.ι Q) x✝, ⋯⟩
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
x✝² x✝¹ x✝ : M
⊢ ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝¹ + (CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝, ⋯⟩ =
    ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝¹, ⋯⟩ +
      ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝, ⋯⟩

Tactic:
;

Goal state after:
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
x✝² x✝¹ x✝ : M
⊢ ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝¹ + (CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝, ⋯⟩ =
    ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝¹, ⋯⟩ +
      ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) x✝, ⋯⟩
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case neg
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
hfv : ¬f v = 0
ι : Set V
b : Basis (↑ι) K V
i : ↑ι
hv : v = b i
hf : f = f v • b.coord i
this : Fintype ↑ι
x y : ↑ι
hxy : x = y
hxi : ¬x = i
⊢ (Finsupp.single y 1) x + f v * (Finsupp.single y 1) i * 0 = 1

Tactic:
;

Goal state after:
case neg
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
hfv : ¬f v = 0
ι : Set V
b : Basis (↑ι) K V
i : ↑ι
hv : v = b i
hf : f = f v • b.coord i
this : Fintype ↑ι
x y : ↑ι
hxy : x = y
hxi : ¬x = i
⊢ (Finsupp.single y 1) x + f v * (Finsupp.single y 1) i * 0 = 1
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:
(neg_add_cancel B),

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

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ ↑((-1) ^ (j₁ * i₂)) •
      (of R 𝒜 ℬ)
        ((TensorProduct.congr ↑(decomposeAlgEquiv 𝒜) ↑(decomposeAlgEquiv ℬ)).symm
          (((decompose 𝒜) a₁ * (DirectSum.of (fun x ↦ ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
            ((DirectSum.of (fun x ↦ ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))

Tactic:
← @Units.smul_def _ _ (_) (_)

Goal state after:
R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (-1) ^ (j₁ * i₂) •
      (of R 𝒜 ℬ)
        ((TensorProduct.congr ↑(decomposeAlgEquiv 𝒜) ↑(decomposeAlgEquiv ℬ)).symm
          (((decompose 𝒜) a₁ * (DirectSum.of (fun x ↦ ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
            ((DirectSum.of (fun x ↦ ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : CommRing R
inst✝⁸ : Ring A
inst✝⁷ : Ring B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝⁴ : GradedAlgebra 𝒜
inst✝³ : GradedAlgebra ℬ
inst✝² : Module ι (Additive ℤˣ)
C : Type ?u.46
inst✝¹ : Ring C
inst✝ : Algebra R C
f : A →ₐ[R] C
g : B →ₐ[R] C
h_anti_commutes : ∀ ⦃i j : ι⦄ (a : ↥(𝒜 i)) (b : ↥(ℬ j)), f ↑a * g ↑b = (-1) ^ (j * i) • (g ↑b * f ↑a)
a₁ : A
j₁ : ι
b₁ : ↥(ℬ j₁)
i₂ : ι
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ ↑((-1) ^ (j₁ * i₂)) •
      (LinearMap.mul' R C)
        ((TensorProduct.map f.toLinearMap g.toLinearMap) ((of R 𝒜 ℬ).symm ((a₁ * ↑a₂) ᵍ⊗ₜ[R] (↑b₁ * b₂)))) =
    f a₁ * g ↑b₁ * (f ↑a₂ * g b₂)

Tactic:
← @Units.smul_def _ _ (_) (_)

Goal state after:
R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : CommRing R
inst✝⁸ : Ring A
inst✝⁷ : Ring B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝⁴ : GradedAlgebra 𝒜
inst✝³ : GradedAlgebra ℬ
inst✝² : Module ι (Additive ℤˣ)
C : Type ?u.46
inst✝¹ : Ring C
inst✝ : Algebra R C
f : A →ₐ[R] C
g : B →ₐ[R] C
h_anti_commutes : ∀ ⦃i j : ι⦄ (a : ↥(𝒜 i)) (b : ↥(ℬ j)), f ↑a * g ↑b = (-1) ^ (j * i) • (g ↑b * f ↑a)
a₁ : A
j₁ : ι
b₁ : ↥(ℬ j₁)
i₂ : ι
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (-1) ^ (j₁ * i₂) •
      (LinearMap.mul' R C)
        ((TensorProduct.map f.toLinearMap g.toLinearMap) ((of R 𝒜 ℬ).symm ((a₁ * ↑a₂) ᵍ⊗ₜ[R] (↑b₁ * b₂)))) =
    f a₁ * g ↑b₁ * (f ↑a₂ * g b₂)
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: case fst
R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : CommRing R
inst✝⁸ : Ring A
inst✝⁷ : Ring B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝⁴ : GradedAlgebra 𝒜
inst✝³ : GradedAlgebra ℬ
inst✝² : Module ι (Additive ℤˣ)
C : Type ?u.46
inst✝¹ : Ring C
inst✝ : Algebra R C
fg : { fg // ∀ ⦃i j : ι⦄ (a : ↥(𝒜 i)) (b : ↥(ℬ j)), fg.1 ↑a * fg.2 ↑b = (-1) ^ (j * i) • (fg.2 ↑b * fg.1 ↑a) }
x✝ : A
⊢ (↑fg).1 x✝ * (↑fg).2 1 = (↑fg).1 x✝

Tactic:
;

Goal state after:
case fst
R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : CommRing R
inst✝⁸ : Ring A
inst✝⁷ : Ring B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝⁴ : GradedAlgebra 𝒜
inst✝³ : GradedAlgebra ℬ
inst✝² : Module ι (Additive ℤˣ)
C : Type ?u.46
inst✝¹ : Ring C
inst✝ : Algebra R C
fg : { fg // ∀ ⦃i j : ι⦄ (a : ↥(𝒜 i)) (b : ↥(ℬ j)), fg.1 ↑a * fg.2 ↑b = (-1) ^ (j * i) • (fg.2 ↑b * fg.1 ↑a) }
x✝ : A
⊢ (↑fg).1 x✝ * (↑fg).2 1 = (↑fg).1 x✝
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: case snd
R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : CommRing R
inst✝⁸ : Ring A
inst✝⁷ : Ring B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝⁴ : GradedAlgebra 𝒜
inst✝³ : GradedAlgebra ℬ
inst✝² : Module ι (Additive ℤˣ)
C : Type ?u.46
inst✝¹ : Ring C
inst✝ : Algebra R C
fg : { fg // ∀ ⦃i j : ι⦄ (a : ↥(𝒜 i)) (b : ↥(ℬ j)), fg.1 ↑a * fg.2 ↑b = (-1) ^ (j * i) • (fg.2 ↑b * fg.1 ↑a) }
x✝ : B
⊢ (↑fg).1 1 * (↑fg).2 x✝ = (↑fg).2 x✝

Tactic:
;

Goal state after:
case snd
R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : CommRing R
inst✝⁸ : Ring A
inst✝⁷ : Ring B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝⁴ : GradedAlgebra 𝒜
inst✝³ : GradedAlgebra ℬ
inst✝² : Module ι (Additive ℤˣ)
C : Type ?u.46
inst✝¹ : Ring C
inst✝ : Algebra R C
fg : { fg // ∀ ⦃i j : ι⦄ (a : ↥(𝒜 i)) (b : ↥(ℬ j)), fg.1 ↑a * fg.2 ↑b = (-1) ^ (j * i) • (fg.2 ↑b * fg.1 ↑a) }
x✝ : B
⊢ (↑fg).1 1 * (↑fg).2 x✝ = (↑fg).2 x✝
Module: Mathlib.LinearAlgebra.ExteriorAlgebra.Grading

Goal state before:
pp: case algebraMap
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ℕ
x : ExteriorAlgebra R M
r✝ : R
⊢ (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) 0) (DirectSum.GAlgebra.toFun r✝) =
    (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) 0) ⟨(algebraMap R (ExteriorAlgebra R M)) r✝, ⋯⟩

Tactic:
;

Goal state after:
case algebraMap
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ℕ
x : ExteriorAlgebra R M
r✝ : R
⊢ (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) 0) (DirectSum.GAlgebra.toFun r✝) =
    (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) 0) ⟨(algebraMap R (ExteriorAlgebra R M)) r✝, ⋯⟩
Module: Mathlib.LinearAlgebra.ExteriorAlgebra.Grading

Goal state before:
pp: case add
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ℕ
x x✝ y✝ : ExteriorAlgebra R M
i✝ : ℕ
hx✝ : x✝ ∈ (ι R).range ^ i✝
hy✝ : y✝ ∈ (ι R).range ^ i✝
ihx : (liftι R M) x✝ = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨x✝, hx✝⟩
ihy : (liftι R M) y✝ = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨y✝, hy✝⟩
⊢ (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) (⟨x✝, hx✝⟩ + ⟨y✝, hy✝⟩) = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨x✝ + y✝, ⋯⟩

Tactic:
;

Goal state after:
case add
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ℕ
x x✝ y✝ : ExteriorAlgebra R M
i✝ : ℕ
hx✝ : x✝ ∈ (ι R).range ^ i✝
hy✝ : y✝ ∈ (ι R).range ^ i✝
ihx : (liftι R M) x✝ = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨x✝, hx✝⟩
ihy : (liftι R M) y✝ = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨y✝, hy✝⟩
⊢ (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) (⟨x✝, hx✝⟩ + ⟨y✝, hy✝⟩) = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨x✝ + y✝, ⋯⟩
Module: Mathlib.LinearAlgebra.ExteriorAlgebra.Grading

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ℕ
x : ↥(⋀[R]^i M)
⊢ (liftι R M) ↑x = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i) x

Tactic:
obtain ⟨x, hx⟩ := x
  dsimp only [Subtype.coe_mk, DirectSum.lof_eq_of]
  induction hx using Submodule.pow_induction_on_left' with
  | algebraMap => simp_rw [AlgHom.commutes, DirectSum.algebraMap_apply]; rfl
  | add _ _ _ _ _ ihx ihy => simp_rw [map_add, ihx, ihy, ← map_add]; rfl
  | mem_mul _ hm _ _ _ ih =>
      obtain ⟨_, rfl⟩ := hm
      simp_rw [map_mul, ih, GradedAlgebra.liftι, lift_ι_apply, GradedAlgebra.ι_apply R M,
        DirectSum.of_mul_of]
      exact DirectSum.of_eq_of_gradedMonoid_eq (Sigma.subtype_ext (add_comm _ _) rfl)

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.ModN

Goal state before:
pp: G : Type u_1
H : Type u_2
M : Type u_3
inst✝¹ : AddCommGroup G
n : ℕ
inst✝ : AddMonoid M
φ : { φ // ∀ (g : G), n • φ g = 0 }
g : G
⊢ ((LinearMap.lsmul ℤ G) ↑n) g ∈ (↑φ).ker

Tactic:
;

Goal state after:
G : Type u_1
H : Type u_2
M : Type u_3
inst✝¹ : AddCommGroup G
n : ℕ
inst✝ : AddMonoid M
φ : { φ // ∀ (g : G), n • φ g = 0 }
g : G
⊢ ((LinearMap.lsmul ℤ G) ↑n) g ∈ (↑φ).ker
Module: Mathlib.LinearAlgebra.FreeProduct.Basic

Goal state before:
pp: I : Type u
inst✝³ : DecidableEq I
R : Type v
inst✝² : CommSemiring R
A : I → Type w
inst✝¹ : (i : I) → Semiring (A i)
inst✝ : (i : I) → Algebra R (A i)
i : I
⊢ (ι' R A ∘ₗ DirectSum.lof R I A i) 1 = 1

Tactic:
;

Goal state after:
I : Type u
inst✝³ : DecidableEq I
R : Type v
inst✝² : CommSemiring R
A : I → Type w
inst✝¹ : (i : I) → Semiring (A i)
inst✝ : (i : I) → Algebra R (A i)
i : I
⊢ (ι' R A ∘ₗ DirectSum.lof R I A i) 1 = 1
Module: Mathlib.LinearAlgebra.FreeProduct.Basic

Goal state before:
pp: I : Type u
inst✝³ : DecidableEq I
R : Type v
inst✝² : CommSemiring R
A : I → Type w
inst✝¹ : (i : I) → Semiring (A i)
inst✝ : (i : I) → Algebra R (A i)
i : I
⊢ (ι' R A) ((DirectSum.lof R I A i) 1) = 1

Tactic:
;

Goal state after:
I : Type u
inst✝³ : DecidableEq I
R : Type v
inst✝² : CommSemiring R
A : I → Type w
inst✝¹ : (i : I) → Semiring (A i)
inst✝ : (i : I) → Algebra R (A i)
i : I
⊢ (ι' R A) ((DirectSum.lof R I A i) 1) = 1
Module: Mathlib.LinearAlgebra.FreeModule.ModN

Goal state before:
pp: G : Type u_1
inst✝¹ : AddCommGroup G
n : ℕ
inst✝ : NeZero n
ι : Type u_4
b : Basis ι ℤ G
i : ι
⊢ (basis b).repr ((mkQ n) (b i)) = single i 1

Tactic:
;

Goal state after:
G : Type u_1
inst✝¹ : AddCommGroup G
n : ℕ
inst✝ : NeZero n
ι : Type u_4
b : Basis ι ℤ G
i : ι
⊢ (basis b).repr ((mkQ n) (b i)) = single i 1
Module: Mathlib.LinearAlgebra.FreeProduct.Basic

Goal state before:
pp: I : Type u
inst✝⁵ : DecidableEq I
R : Type v
inst✝⁴ : CommSemiring R
A : I → Type w
inst✝³ : (i : I) → Semiring (A i)
inst✝² : (i : I) → Algebra R (A i)
B : Type w'
inst✝¹ : Semiring B
inst✝ : Algebra R B
maps : {i : I} → A i →ₐ[R] B
f : FreeProduct R A →ₐ[R] B
h : ∀ (i : I), f ∘ₐ ι R A i = maps
i : I
a : A i
⊢ (((f ∘ₐ RingCon.mkₐ R (ringCon R A)).toLinearMap ∘ₗ TensorAlgebra.ι R) ∘ₗ DirectSum.lof R I A i) a =
    (((((lift R A) fun {i} ↦ maps) ∘ₐ RingCon.mkₐ R (ringCon R A)).toLinearMap ∘ₗ TensorAlgebra.ι R) ∘ₗ
        DirectSum.lof R I A i)
      a

Tactic:
;

Goal state after:
I : Type u
inst✝⁵ : DecidableEq I
R : Type v
inst✝⁴ : CommSemiring R
A : I → Type w
inst✝³ : (i : I) → Semiring (A i)
inst✝² : (i : I) → Algebra R (A i)
B : Type w'
inst✝¹ : Semiring B
inst✝ : Algebra R B
maps : {i : I} → A i →ₐ[R] B
f : FreeProduct R A →ₐ[R] B
h : ∀ (i : I), f ∘ₐ ι R A i = maps
i : I
a : A i
⊢ (((f ∘ₐ RingCon.mkₐ R (ringCon R A)).toLinearMap ∘ₗ TensorAlgebra.ι R) ∘ₗ DirectSum.lof R I A i) a =
    (((((lift R A) fun {i} ↦ maps) ∘ₐ RingCon.mkₐ R (ringCon R A)).toLinearMap ∘ₗ TensorAlgebra.ι R) ∘ₗ
        DirectSum.lof R I A i)
      a
Module: Mathlib.LinearAlgebra.FreeProduct.Basic

Goal state before:
pp: I : Type u
inst✝⁵ : DecidableEq I
R : Type v
inst✝⁴ : CommSemiring R
A : I → Type w
inst✝³ : (i : I) → Semiring (A i)
inst✝² : (i : I) → Algebra R (A i)
B : Type w'
inst✝¹ : Semiring B
inst✝ : Algebra R B
maps : {i : I} → A i →ₐ[R] B
f : FreeProduct R A →ₐ[R] B
i : I
a : A i
h : ∀ (i : I) (x : A i), (f ∘ₐ ι R A i) x = maps x
⊢ (((f ∘ₐ RingCon.mkₐ R (ringCon R A)).toLinearMap ∘ₗ TensorAlgebra.ι R) ∘ₗ DirectSum.lof R I A i) a =
    (((((lift R A) fun {i} ↦ maps) ∘ₐ RingCon.mkₐ R (ringCon R A)).toLinearMap ∘ₗ TensorAlgebra.ι R) ∘ₗ
        DirectSum.lof R I A i)
      a

Tactic:
;

Goal state after:
I : Type u
inst✝⁵ : DecidableEq I
R : Type v
inst✝⁴ : CommSemiring R
A : I → Type w
inst✝³ : (i : I) → Semiring (A i)
inst✝² : (i : I) → Algebra R (A i)
B : Type w'
inst✝¹ : Semiring B
inst✝ : Algebra R B
maps : {i : I} → A i →ₐ[R] B
f : FreeProduct R A →ₐ[R] B
i : I
a : A i
h : ∀ (i : I) (x : A i), (f ∘ₐ ι R A i) x = maps x
⊢ (((f ∘ₐ RingCon.mkₐ R (ringCon R A)).toLinearMap ∘ₗ TensorAlgebra.ι R) ∘ₗ DirectSum.lof R I A i) a =
    (((((lift R A) fun {i} ↦ maps) ∘ₐ RingCon.mkₐ R (ringCon R A)).toLinearMap ∘ₗ TensorAlgebra.ι R) ∘ₗ
        DirectSum.lof R I A i)
      a
Module: Mathlib.LinearAlgebra.LinearIndependent.BaseChange

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
inst✝³ : Finite ι'
K : Type u_3
L : Type u_4
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
v : ι → ι' → K
hv : LinearIndependent K v
this : Fintype ι' := Fintype.ofFinite ι'
I : Set (ι' → K) := ⋯.extend ⋯
b : Module.Basis (↑I) K (ι' → K) := Module.Basis.extend ⋯
b' : Module.Basis (↑I) L (ι' → L) := (Module.Basis.baseChange L b).map (piScalarRight K L L ι')
v' : ι → ↑I := fun i ↦ ⟨v i, ⋯⟩
x✝¹ : ι
x✝ : ι'
⊢ (⇑b' ∘ v') x✝¹ x✝ = (⇑(algebraMap K L) ∘ v x✝¹) x✝

Tactic:
;

Goal state after:
ι : Type u_1
ι' : Type u_2
inst✝³ : Finite ι'
K : Type u_3
L : Type u_4
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
v : ι → ι' → K
hv : LinearIndependent K v
this : Fintype ι' := Fintype.ofFinite ι'
I : Set (ι' → K) := ⋯.extend ⋯
b : Module.Basis (↑I) K (ι' → K) := Module.Basis.extend ⋯
b' : Module.Basis (↑I) L (ι' → L) := (Module.Basis.baseChange L b).map (piScalarRight K L L ι')
v' : ι → ↑I := fun i ↦ ⟨v i, ⋯⟩
x✝¹ : ι
x✝ : ι'
⊢ (⇑b' ∘ v') x✝¹ x✝ = (⇑(algebraMap K L) ∘ v x✝¹) x✝