mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.BilinearForm.Isometry

Goal state before:
pp: R : Type u_1
M : Type u_2
M₁ : Type u_3
M₂ : Type u_4
M₃ : Type u_5
M₄ : Type u_6
N : Type u_7
inst✝¹³ : CommSemiring R
inst✝¹² : AddCommMonoid M
inst✝¹¹ : AddCommMonoid M₁
inst✝¹⁰ : AddCommMonoid M₂
inst✝⁹ : AddCommMonoid M₃
inst✝⁸ : AddCommMonoid M₄
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R M
inst✝⁵ : Module R M₁
inst✝⁴ : Module R M₂
inst✝³ : Module R M₃
inst✝² : Module R M₄
inst✝¹ : Module R N
B₁ : LinearMap.BilinForm R M₁
B₂ : LinearMap.BilinForm R M₂
B₃ : LinearMap.BilinForm R M₃
B₄ : LinearMap.BilinForm R M₄
inst✝ : Subsingleton M₁
x y : M₁
⊢ (B₂ ((toAddHom 0).toFun 0)) ((toAddHom 0).toFun y) = (B₁ 0) y

Tactic:
Subsingleton.elim y 0

Goal state after:
R : Type u_1
M : Type u_2
M₁ : Type u_3
M₂ : Type u_4
M₃ : Type u_5
M₄ : Type u_6
N : Type u_7
inst✝¹³ : CommSemiring R
inst✝¹² : AddCommMonoid M
inst✝¹¹ : AddCommMonoid M₁
inst✝¹⁰ : AddCommMonoid M₂
inst✝⁹ : AddCommMonoid M₃
inst✝⁸ : AddCommMonoid M₄
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R M
inst✝⁵ : Module R M₁
inst✝⁴ : Module R M₂
inst✝³ : Module R M₃
inst✝² : Module R M₄
inst✝¹ : Module R N
B₁ : LinearMap.BilinForm R M₁
B₂ : LinearMap.BilinForm R M₂
B₃ : LinearMap.BilinForm R M₃
B₄ : LinearMap.BilinForm R M₄
inst✝ : Subsingleton M₁
x y : M₁
⊢ (B₂ ((toAddHom 0).toFun 0)) ((toAddHom 0).toFun 0) = (B₁ 0) 0
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
σ : Perm.ModSumCongr ιa ιb
σ₁ σ₂ : Perm (ιa ⊕ ιb)
sl : Perm ιa
sr : Perm ιb
v : ιa ⊕ ιb → Mᵢ
h : σ₂ = σ₁ * (Perm.sumCongrHom ιa ιb) (sl, sr)
this : Perm.sign (σ₁ * (Perm.sumCongrHom ιa ιb) (sl, sr)) = Perm.sign σ₁ * (Perm.sign sl * Perm.sign sr)
⊢ (Perm.sign σ₁ • (a fun i₁ ↦ v (σ₁ (Sum.inl i₁))) ⊗ₜ[R'] b fun i₂ ↦ v (σ₁ (Sum.inr i₂))) =
    Perm.sign σ₂ • (a fun i₁ ↦ v (σ₂ (Sum.inl i₁))) ⊗ₜ[R'] b fun i₂ ↦ v (σ₂ (Sum.inr i₂))

Tactic:
rw [h, this, mul_smul, mul_smul, smul_left_cancel_iff, ← TensorProduct.tmul_smul,
      TensorProduct.smul_tmul', a.map_congr_perm _ sl, b.map_congr_perm _ sr]

Goal state after:
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
σ : Perm.ModSumCongr ιa ιb
σ₁ σ₂ : Perm (ιa ⊕ ιb)
sl : Perm ιa
sr : Perm ιb
v : ιa ⊕ ιb → Mᵢ
h : σ₂ = σ₁ * (Perm.sumCongrHom ιa ιb) (sl, sr)
this : Perm.sign (σ₁ * (Perm.sumCongrHom ιa ιb) (sl, sr)) = Perm.sign σ₁ * (Perm.sign sl * Perm.sign sr)
⊢ (Perm.sign sl • a ((fun i₁ ↦ v (σ₁ (Sum.inl i₁))) ∘ ⇑sl)) ⊗ₜ[R']
      (Perm.sign sr • b ((fun i₂ ↦ v (σ₁ (Sum.inr i₂))) ∘ ⇑sr)) =
    (Perm.sign sl • a fun i₁ ↦ v ((σ₁ * (Perm.sumCongrHom ιa ιb) (sl, sr)) (Sum.inl i₁))) ⊗ₜ[R']
      (Perm.sign sr • b fun i₂ ↦ v ((σ₁ * (Perm.sumCongrHom ιa ιb) (sl, sr)) (Sum.inr i₂)))
Module: Mathlib.LinearAlgebra.Dual.BaseChange

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommSemiring R
V : Type u_2
inst✝⁹ : AddCommMonoid V
inst✝⁸ : Module R V
W : Type u_3
inst✝⁷ : AddCommMonoid W
inst✝⁶ : Module R W
A : Type u_4
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : Module A W
inst✝² : IsScalarTower R A W
j : V →ₗ[R] W
ibc : IsBaseChange A j
inst✝¹ : Free R V
inst✝ : Module.Finite R V
ι : Type u_2 := Free.ChooseBasisIndex R V
b : Basis ι R V
ibc_pow : IsBaseChange A ((Algebra.linearMap R A).compLeft ι)
f : Dual R V
w : W
⊢ (((AlgebraTensorModule.curry ibc.toDualBaseChangeAux) 1) f) w =
    (((AlgebraTensorModule.curry
            ↑(LinearEquiv.baseChange R A (V →ₗ[R] R) (ι → R) (b.constr R).symm ≪≫ₗ ibc_pow.equiv ≪≫ₗ
                (basis b ibc).constr A))
          1)
        f)
      w

Tactic:
simp only [AlgebraTensorModule.curry_apply, curry_apply, LinearMap.coe_restrictScalars,
    LinearEquiv.coe_coe, LinearEquiv.trans_apply]

Goal state after:
R : Type u_1
inst✝¹⁰ : CommSemiring R
V : Type u_2
inst✝⁹ : AddCommMonoid V
inst✝⁸ : Module R V
W : Type u_3
inst✝⁷ : AddCommMonoid W
inst✝⁶ : Module R W
A : Type u_4
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : Module A W
inst✝² : IsScalarTower R A W
j : V →ₗ[R] W
ibc : IsBaseChange A j
inst✝¹ : Free R V
inst✝ : Module.Finite R V
ι : Type u_2 := Free.ChooseBasisIndex R V
b : Basis ι R V
ibc_pow : IsBaseChange A ((Algebra.linearMap R A).compLeft ι)
f : Dual R V
w : W
⊢ (ibc.toDualBaseChangeAux (1 ⊗ₜ[R] f)) w =
    (((basis b ibc).constr A)
        (ibc_pow.equiv ((LinearEquiv.baseChange R A (V →ₗ[R] R) (ι → R) (b.constr R).symm) (1 ⊗ₜ[R] f))))
      w
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
⊢ AlternatingMap.domCoprod' ((Fintype.card ιa).factorial • a ⊗ₜ[R'] ((Fintype.card ιb).factorial • b)) =
    (Fintype.card ιa).factorial • (Fintype.card ιb).factorial • AlternatingMap.domCoprod' (a ⊗ₜ[R'] b)

Tactic:
TensorProduct.tmul_smul,

Goal state after:
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
⊢ AlternatingMap.domCoprod' ((Fintype.card ιa).factorial • (Fintype.card ιb).factorial • a ⊗ₜ[R'] b) =
    (Fintype.card ιa).factorial • (Fintype.card ιb).factorial • AlternatingMap.domCoprod' (a ⊗ₜ[R'] b)
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
e✝ e : V ≃ₗ[R] V
f : Dual R V
v : V
he : ↑e = LinearMap.transvection f v
x : V
this : x = e x - f x • v
⊢ ↑e.symm x = (LinearMap.transvection f (-e.symm v)) x

Tactic:
simpa [LinearMap.transvection.apply, ← sub_eq_add_neg, symm_apply_eq]

Goal state after:
Module: Mathlib.LinearAlgebra.Center

Goal state before:
pp: case h
R : Type u_1
V : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : Free R V
f : V →ₗ[R] V
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
hV1 : Nonempty (Basis Unit R V)
b : Basis Unit R V := Classical.ofNonempty
i : Unit
⊢ f (b i) = ((b.coord ()) (f (b ())) • 1) (b i)

Tactic:
nth_rewrite 1 [← b.linearCombination_repr (f (b i))]

Goal state after:
case h
R : Type u_1
V : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : Free R V
f : V →ₗ[R] V
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
hV1 : Nonempty (Basis Unit R V)
b : Basis Unit R V := Classical.ofNonempty
i : Unit
⊢ (linearCombination R ⇑b) (b.repr (f (b i))) = ((b.coord ()) (f (b ())) • 1) (b i)
Module: Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat

Goal state before:
pp: R : Type u
inst✝ : CommRing R
V W : QuadraticModuleCat R
x✝² x✝¹ : V.Hom W
f g : V.form →qᵢ W.form
x✝ : { toIsometry' := f }.toIsometry = { toIsometry' := g }.toIsometry
⊢ { toIsometry' := f } = { toIsometry' := g }

Tactic:
by congr

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

Goal state before:
pp: case algebraMap
R : Type u_1
inst✝ : CommRing R
r : R
⊢ reverse ((algebraMap R (CliffordAlgebra 0)) r) = (algebraMap R (CliffordAlgebra 0)) r

Tactic:
exact reverse.commutes _

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

Goal state before:
pp: case algebraMap
r : ℝ
⊢ reverse ((algebraMap ℝ (CliffordAlgebra Q)) r) = (algebraMap ℝ (CliffordAlgebra Q)) r

Tactic:
exact reverse.commutes _

Goal state after:
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:
rw [mul_zero]

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)
⊢ ⅟(Q m) • (ι Q) m * (ι Q) m = 1

Tactic:
smul_mul_assoc,

Goal state after:
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)
⊢ ⅟(Q m) • ((ι Q) m * (ι Q) m) = 1
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(Q a) • (ι Q) a * (ι Q) b * (ι Q) a = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)

Tactic:
smul_mul_assoc,

Goal state after:
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(Q a) • ((ι Q) a * (ι Q) b) * (ι Q) a = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)
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(Q a) • ((ι Q) a * (ι Q) b) * (ι Q) a = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)

Tactic:
smul_mul_assoc,

Goal state after:
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(Q a) • ((ι Q) a * (ι Q) b * (ι Q) a) = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)
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) =
    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:
smul_mul_assoc,

Goal state after:
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) =
    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)
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) = r₂ • ((v Q) m₁ * e0 Q)

Tactic:
smul_mul_assoc,

Goal state after:
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)) = r₂ • ((v Q) m₁ * e0 Q)
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 ⊗[R] (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) →ₗ[R]
    ((⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), 𝒜 i) ⊗[R] ((⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i)

Tactic:
refine (assoc R _ _ _).symm.toLinearMap ∘ₗ .lTensor _ ?_ ∘ₗ (assoc R _ _ _).toLinearMap

Goal state after:
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 ℬ ⊗[R] (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) →ₗ[R]
    (⨁ (i : ι), 𝒜 i) ⊗[R] ((⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i)
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

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
Q : QuadraticForm R M
r : R
⊢ ↑((toEven Q) (reverse (involute ((algebraMap R (CliffordAlgebra Q)) r)))) =
    reverse ↑((toEven Q) ((algebraMap R (CliffordAlgebra Q)) r))

Tactic:
simp only [AlgHom.commutes, Subalgebra.coe_algebraMap, reverse.commutes]

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

Goal state before:
pp: case one
R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
inst✝ : Invertible 2
m : M
⊢ ↑1 * (ι Q) m * ↑1⁻¹ ∈ (ι Q).range

Tactic:
simp_rw [inv_one, Units.val_one, one_mul, mul_one, LinearMap.mem_range_self]

Goal state after:
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)
x y z : 𝒜 ᵍ⊗[R] ℬ
⊢ x * (y + z) = x * y + x * z

Tactic:
simp_rw [mul_def, map_add]

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

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x✝ : (CliffordAlgebra Q)ˣ
hx✝ : x✝ ∈ lipschitzGroup Q
inst✝ : Invertible 2
x : (CliffordAlgebra Q)ˣ
hx : x ∈ lipschitzGroup Q
⊢ ConjAct.toConjAct x • (ι Q).range ≤ (ι Q).range

Tactic:
rw [Submodule.pointwise_smul_def, Submodule.map_le_iff_le_comap]

Goal state after:
R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x✝ : (CliffordAlgebra Q)ˣ
hx✝ : x✝ ∈ lipschitzGroup Q
inst✝ : Invertible 2
x : (CliffordAlgebra Q)ˣ
hx : x ∈ lipschitzGroup Q
⊢ (ι Q).range ≤ Submodule.comap (DistribSMul.toLinearMap R (CliffordAlgebra Q) (ConjAct.toConjAct x)) (ι Q).range
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) • ((-1) ^ (j₁ * i₂) • (f a₁ * (g ↑b₁ * f ↑a₂)) * g b₂) = f a₁ * (g ↑b₁ * f ↑a₂) * g b₂

Tactic:
smul_mul_assoc,

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) • (-1) ^ (j₁ * i₂) • (f a₁ * (g ↑b₁ * f ↑a₂) * g b₂) = f a₁ * (g ↑b₁ * f ↑a₂) * g b₂
Module: Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ofDirectSum.comp toDirectSum = AlgHom.id R (TensorAlgebra R M)

Tactic:
by
  ext
  simp [tprod_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x : Fin n → M
⊢ (List.map (⇑toDirectSum) (List.ofFn fun i ↦ (ι R) (x i))).prod =
    (DirectSum.of (fun i ↦ ⨂[R]^i M) n) ((PiTensorProduct.tprod R) x)

Tactic:
List.map_ofFn

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x : Fin n → M
⊢ (List.ofFn (⇑toDirectSum ∘ fun i ↦ (ι R) (x i))).prod =
    (DirectSum.of (fun i ↦ ⨂[R]^i M) n) ((PiTensorProduct.tprod R) x)
Module: Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x : Fin n → M
⊢ (List.map (fun a ↦ GradedMonoid.mk 1 ((PiTensorProduct.tprod R) fun x_1 ↦ x a)) (List.finRange n)).prod =
    GradedMonoid.mk n ((PiTensorProduct.tprod R) x)

Tactic:
rw [TensorPower.list_prod_gradedMonoid_mk_single]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.CharP

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝³ : AddMonoidWithOne R
inst✝² : DecidableEq n
inst✝¹ : Nonempty n
p : ℕ
inst✝ : CharP R p
k : ℕ
⊢ (∀ (i : n), p ∣ k) ↔ p ∣ k

Tactic:
forall_const

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.FiniteField

Goal state before:
pp: case inl
n : Type u_1
inst✝³ : DecidableEq n
inst✝² : Fintype n
K : Type u_2
inst✝¹ : Field K
inst✝ : Fintype K
M : Matrix n n K
h✝ : Nonempty n
p : ℕ
hp✝ : CharP K p
k : ℕ
kpos : 0 < k
hp : Nat.Prime p
hk : Fintype.card K = p ^ k
this : Fact (Nat.Prime p)
⊢ ((expand K (Fintype.card K)).mapMatrix (M ^ Fintype.card K).charmatrix).det = M.charmatrix.det ^ Fintype.card K

Tactic:
← coe_detMonoidHom,

Goal state after:
case inl
n : Type u_1
inst✝³ : DecidableEq n
inst✝² : Fintype n
K : Type u_2
inst✝¹ : Field K
inst✝ : Fintype K
M : Matrix n n K
h✝ : Nonempty n
p : ℕ
hp✝ : CharP K p
k : ℕ
kpos : 0 < k
hp : Nat.Prime p
hk : Fintype.card K = p ^ k
this : Fact (Nat.Prime p)
⊢ detMonoidHom ((expand K (Fintype.card K)).mapMatrix (M ^ Fintype.card K).charmatrix) =
    detMonoidHom M.charmatrix ^ Fintype.card K
Module: Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular

Goal state before:
pp: case h
m : Type u_1
n : Type u_3
R : Type u_5
inst✝ : CommRing R
A : Matrix m n R
hA : A.IsTotallyUnimodular
k : ℕ
f : Fin k → m
g : Fin k → n
hfg : ¬(Function.Injective f ∧ Function.Injective g)
⊢ ↑0 = (A.submatrix f g).det

Tactic:
SignType.coe_zero,

Goal state after:
case h
m : Type u_1
n : Type u_3
R : Type u_5
inst✝ : CommRing R
A : Matrix m n R
hA : A.IsTotallyUnimodular
k : ℕ
f : Fin k → m
g : Fin k → n
hfg : ¬(Function.Injective f ∧ Function.Injective g)
⊢ 0 = (A.submatrix f g).det
Module: Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular

Goal state before:
pp: m : Type u_1
n : Type u_3
R : Type u_5
inst✝¹ : CommRing R
inst✝ : DecidableEq m
A : Matrix m n R
⊢ (fromCols 1 A).IsTotallyUnimodular ↔ A.IsTotallyUnimodular

Tactic:
by
  rw [← transpose_isTotallyUnimodular_iff, transpose_fromCols, transpose_one,
    one_fromRows_isTotallyUnimodular_iff, transpose_isTotallyUnimodular_iff]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.FixedDetMatrices

Goal state before:
pp: m : ℤ
A : FixedDetMatrix (Fin 2) ℤ m
h : ↑A 1 0 ≠ 0
this : ↑(reduceStep A) 1 0 = ↑A 0 0 % ↑A 1 0
⊢ |↑(reduceStep A) 1 0| < |↑A 1 0|

Tactic:
by
    rw [this, abs_eq_self.mpr (Int.emod_nonneg (A.1 0 0) h)]
    exact Int.emod_lt_abs (A.1 0 0) h

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Irreducible.Defs

Goal state before:
pp: case zero.refine_1.inl
n : Type u_1
R : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : LinearOrder R
A : Matrix n n R
inst✝⁴ : Fintype n
inst✝³ : IsOrderedRing R
inst✝² : PosMulStrictMono R
inst✝¹ : Nontrivial R
inst✝ : DecidableEq n
hA : ∀ (i j : n), 0 ≤ A i j
this : Quiver n := A.toQuiver
i : n
h_pos : 0 < (A ^ 0) i i
⊢ Nonempty { p // p.length = 0 }

Tactic:
exact ⟨⟨Quiver.Path.nil, rfl⟩⟩

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Gershgorin

Goal state before:
pp: K : Type u_1
n : Type u_2
inst✝² : NormedField K
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n K
h : A.det = 0
this : ∃ k, 0 ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖)
⊢ ∃ k, ‖A k k‖ ≤ ∑ j ∈ Finset.univ.erase k, ‖A k j‖

Tactic:
exact this.imp (fun a h ↦ by rwa [mem_closedBall_iff_norm', sub_zero] at h)

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.HadamardMatrix

Goal state before:
pp: n : Type u_2
R : Type u_3
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : StarRing R
A : Matrix n n R
hentry : ∀ (i j : n), A i j ∈ unitary R
hmul : A * Aᴴ = ↑(Fintype.card n) • 1
hcard : IsRegular ↑(Fintype.card n)
hdet : IsRegular (A.det * star A.det)
hreg : IsLeftRegular A
⊢ ↑(Fintype.card n) • 1 * A = A * ↑(Fintype.card n) • 1

Tactic:
smul_mul_assoc,

Goal state after:
n : Type u_2
R : Type u_3
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : StarRing R
A : Matrix n n R
hentry : ∀ (i j : n), A i j ∈ unitary R
hmul : A * Aᴴ = ↑(Fintype.card n) • 1
hcard : IsRegular ↑(Fintype.card n)
hdet : IsRegular (A.det * star A.det)
hreg : IsLeftRegular A
⊢ ↑(Fintype.card n) • (1 * A) = A * ↑(Fintype.card n) • 1
Module: Mathlib.LinearAlgebra.Multilinear.DirectSum

Goal state before:
pp: R : Type u_1
ι : Type u_2
M' : Type u_3
κ : ι → Type u_4
M : (i : ι) → κ i → Type u_5
inst✝⁷ : CommSemiring R
inst✝⁶ : (i : ι) → (j : κ i) → AddCommMonoid (M i j)
inst✝⁵ : (i : ι) → (j : κ i) → Module R (M i j)
inst✝⁴ : AddCommMonoid M'
inst✝³ : Module R M'
inst✝² : DecidableEq ι
inst✝¹ : Finite ι
inst✝ : (i : ι) → DecidableEq (κ i)
f : MultilinearMap R (fun i ↦ ⨁ (j : κ i), M i j) M'
p : (i : ι) → κ i
this : Fintype ι
⊢ (fromDFinsuppEquiv κ R).symm f p = f.compLinearMap fun i ↦ DFinsupp.lsingle (p i)

Tactic:
← fromDFinsuppEquiv_symm_apply

Goal state after:
R : Type u_1
ι : Type u_2
M' : Type u_3
κ : ι → Type u_4
M : (i : ι) → κ i → Type u_5
inst✝⁷ : CommSemiring R
inst✝⁶ : (i : ι) → (j : κ i) → AddCommMonoid (M i j)
inst✝⁵ : (i : ι) → (j : κ i) → Module R (M i j)
inst✝⁴ : AddCommMonoid M'
inst✝³ : Module R M'
inst✝² : DecidableEq ι
inst✝¹ : Finite ι
inst✝ : (i : ι) → DecidableEq (κ i)
f : MultilinearMap R (fun i ↦ ⨁ (j : κ i), M i j) M'
p : (i : ι) → κ i
this : Fintype ι
⊢ (fromDFinsuppEquiv κ R).symm f p = (fromDFinsuppEquiv κ R).symm f p
Module: Mathlib.LinearAlgebra.Projectivization.Collinear

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
a b : ℙ K V
h : ¬a = b
⊢ IsCollinear {a, b}

Tactic:
else

Goal state after:
K : Type u_1
V : Type u_2
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
a b : ℙ K V
h : ¬a = b
⊢ IsCollinear {a, b}