mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.Charpoly.ToMatrix

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
inst✝³ : Free R M
inst✝² : Module.Finite R M
f : M →ₗ[R] M
ι : Type w
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι R M
a✝ : Nontrivial R
b' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M
P : Matrix ι (ChooseBasisIndex R M) R := b.toMatrix ⇑b'
A : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f
Q : Matrix (ChooseBasisIndex R M) ι R := b'.toMatrix ⇑b
e : ι ≃ ChooseBasisIndex R M := b.indexEquiv b'
ι' : Type u_2 := ChooseBasisIndex R M
φ : Matrix ι ι R ≃ₗ[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e
φ₁ : Matrix ι ι' R ≃ₗ[R] Matrix (ChooseBasisIndex R M) ι' R := reindexLinearEquiv R R e (Equiv.refl ι')
φ₂ : Matrix ι' ι' R ≃ₗ[R] Matrix ι' ι' R := reindexLinearEquiv R R (Equiv.refl ι') (Equiv.refl ι')
φ₃ : Matrix ι' ι R ≃ₗ[R] Matrix ι' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl ι') e
⊢ (φ₁ P * φ₂ A * φ₃ Q).charpoly = A.charpoly

Tactic:
rw [charpoly_mul_comm, ← mul_assoc]

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Free R M
inst✝² : Module.Finite R M
f : M →ₗ[R] M
ι : Type w
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι R M
a✝ : Nontrivial R
b' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M
P : Matrix ι (ChooseBasisIndex R M) R := b.toMatrix ⇑b'
A : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f
Q : Matrix (ChooseBasisIndex R M) ι R := b'.toMatrix ⇑b
e : ι ≃ ChooseBasisIndex R M := b.indexEquiv b'
ι' : Type u_2 := ChooseBasisIndex R M
φ : Matrix ι ι R ≃ₗ[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e
φ₁ : Matrix ι ι' R ≃ₗ[R] Matrix (ChooseBasisIndex R M) ι' R := reindexLinearEquiv R R e (Equiv.refl ι')
φ₂ : Matrix ι' ι' R ≃ₗ[R] Matrix ι' ι' R := reindexLinearEquiv R R (Equiv.refl ι') (Equiv.refl ι')
φ₃ : Matrix ι' ι R ≃ₗ[R] Matrix ι' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl ι') e
⊢ (φ₃ Q * φ₁ P * φ₂ A).charpoly = A.charpoly
Module: Mathlib.LinearAlgebra.Lagrange

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v : ι → F
i : ι
hvs : Set.InjOn v ↑s
hi : i ∈ s
⊢ nodalWeight s v i ≠ 0

Tactic:
by
  rw [nodalWeight, prod_ne_zero_iff]
  intro j hj
  rcases mem_erase.mp hj with ⟨hij, hj⟩
  exact inv_ne_zero (sub_ne_zero_of_ne (mt (hvs.eq_iff hi hj).mp hij.symm))

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.Homogeneous.Submodule

Goal state before:
pp: ιA : Type u_1
ιM : Type u_2
σA : Type u_3
σM : Type u_4
A : Type u_5
M : Type u_6
inst✝¹³ : Semiring A
inst✝¹² : AddCommMonoid M
inst✝¹¹ : Module A M
𝒜 : ιA → σA
ℳ : ιM → σM
inst✝¹⁰ : DecidableEq ιA
inst✝⁹ : AddMonoid ιA
inst✝⁸ : SetLike σA A
inst✝⁷ : AddSubmonoidClass σA A
inst✝⁶ : GradedRing 𝒜
inst✝⁵ : DecidableEq ιM
inst✝⁴ : SetLike σM M
inst✝³ : AddSubmonoidClass σM M
inst✝² : Decomposition ℳ
inst✝¹ : VAdd ιA ιM
inst✝ : GradedSMul 𝒜 ℳ
I J : HomogeneousSubmodule 𝒜 ℳ
h : ∀ (i : ιM), ∀ x ∈ ℳ i, x ∈ I ↔ x ∈ J
x✝ : M
⊢ ∀ (a : ιM), ↑(((decompose ℳ) x✝) a) ∈ I.toSubmodule ↔ ↑(((decompose ℳ) x✝) a) ∈ J.toSubmodule

Tactic:
exact fun i ↦ h i _ (decompose ℳ _ i).2

Goal state after:
Module: Mathlib.LinearAlgebra.Lagrange

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v r : ι → F
x : F
hx : ∀ i ∈ s, x ≠ v i
⊢ ∑ x_1 ∈ s, eval x (C (r x_1)) * eval x (Lagrange.basis s v x_1) =
    ∑ i ∈ s, eval x (nodal s v) * (nodalWeight s v i * (x - v i)⁻¹ * r i)

Tactic:
eval_C

Goal state after:
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v r : ι → F
x : F
hx : ∀ i ∈ s, x ≠ v i
⊢ ∑ x_1 ∈ s, r x_1 * eval x (Lagrange.basis s v x_1) =
    ∑ i ∈ s, eval x (nodal s v) * (nodalWeight s v i * (x - v i)⁻¹ * r i)
Module: Mathlib.Algebra.Lie.CartanCriterion

Goal state before:
pp: L : Type u_2
M : Type u_3
inst✝⁹ : LieRing L
inst✝⁸ : AddCommGroup M
inst✝⁷ : LieRingModule L M
K : Type u_4
inst✝⁶ : Field K
inst✝⁵ : CharZero K
inst✝⁴ : IsAlgClosed K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
h : traceForm K L M = 0
φ : L →ₗ⁅K⁆ End K M := toEnd K L M
x : L
hx : x ∈ derivedSeries K L 1
n : End K M
hn_adj : n ∈ K[φ x]
s : End K M
hn_nil : _root_.IsNilpotent n
hs_ss : s.IsSemisimple
hX_ns : φ x = n + s
hns : Commute n s
eigenDecomp : DirectSum.IsInternal s.eigenspace :=
  DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top (eigenspaces_iSupIndep s)
    (IsSemisimple.iSup_eigenspace_eq_top hs_ss)
I : Type (max u_4 0) := (ν : K) × Fin (finrank K ↥(s.eigenspace ν))
v : Basis I K M := eigenDecomp.collectedBasis fun μ ↦ finBasis K ↥(s.eigenspace μ)
this : Fintype I
μ : I → K := Sigma.fst
hsv : ∀ (i : I), s (v i) = μ i • v i
E : Submodule ℚ K := Submodule.span ℚ (Set.range μ)
hμ : ∀ (i : I), μ i ∈ E
f : Dual ℚ ↥E
fμ : I → K := fun i ↦ ↑(f ⟨μ i, ⋯⟩)
y : End K M := (Matrix.toLin v v) (Matrix.diagonal fμ)
hyv : ∀ (i : I), y (v i) = fμ i • v i
hy_range : ∀ z ∈ φ.range, ⁅y, z⁆ ∈ φ.range
⊢ y ∈ (aeval s).range

Tactic:
obtain ⟨q, hq⟩ : ∃ q : K[X], ∀ i, q.eval (μ i) = fμ i :=
      (Polynomial.exists_eval_eq_iff μ fμ).mpr <| by aesop

Goal state after:
L : Type u_2
M : Type u_3
inst✝⁹ : LieRing L
inst✝⁸ : AddCommGroup M
inst✝⁷ : LieRingModule L M
K : Type u_4
inst✝⁶ : Field K
inst✝⁵ : CharZero K
inst✝⁴ : IsAlgClosed K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
h : traceForm K L M = 0
φ : L →ₗ⁅K⁆ End K M := toEnd K L M
x : L
hx : x ∈ derivedSeries K L 1
n : End K M
hn_adj : n ∈ K[φ x]
s : End K M
hn_nil : _root_.IsNilpotent n
hs_ss : s.IsSemisimple
hX_ns : φ x = n + s
hns : Commute n s
eigenDecomp : DirectSum.IsInternal s.eigenspace :=
  DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top (eigenspaces_iSupIndep s)
    (IsSemisimple.iSup_eigenspace_eq_top hs_ss)
I : Type (max u_4 0) := (ν : K) × Fin (finrank K ↥(s.eigenspace ν))
v : Basis I K M := eigenDecomp.collectedBasis fun μ ↦ finBasis K ↥(s.eigenspace μ)
this : Fintype I
μ : I → K := Sigma.fst
hsv : ∀ (i : I), s (v i) = μ i • v i
E : Submodule ℚ K := Submodule.span ℚ (Set.range μ)
hμ : ∀ (i : I), μ i ∈ E
f : Dual ℚ ↥E
fμ : I → K := fun i ↦ ↑(f ⟨μ i, ⋯⟩)
y : End K M := (Matrix.toLin v v) (Matrix.diagonal fμ)
hyv : ∀ (i : I), y (v i) = fμ i • v i
hy_range : ∀ z ∈ φ.range, ⁅y, z⁆ ∈ φ.range
q : K[X]
hq : ∀ (i : I), Polynomial.eval (μ i) q = fμ i
⊢ y ∈ (aeval s).range
Module: Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : SemilatticeSup M
w : σ → M
p : MvPolynomial σ R
⊢ weightedTotalDegree' w p = ⊥ ↔ p = 0

Tactic:
simp only [weightedTotalDegree', Finset.sup_eq_bot_iff, mem_support_iff, WithBot.coe_ne_bot,
    MvPolynomial.eq_zero_iff]

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : SemilatticeSup M
w : σ → M
p : MvPolynomial σ R
⊢ (∀ (s : σ →₀ ℕ), coeff s p ≠ 0 → False) ↔ ∀ (d : σ →₀ ℕ), coeff d p = 0
Module: Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : GradedRing 𝒜
I : HomogeneousIdeal 𝒜
⊢ Ideal.homogeneousCore 𝒜 I.toIdeal = I

Tactic:
ext1
  convert! Ideal.IsHomogeneous.toIdeal_homogeneousCore_eq_self I.isHomogeneous

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I J : Ideal A
HI : ∃ S, I = span (Subtype.val '' S)
HJ : ∃ S, J = span (Subtype.val '' S)
⊢ ∃ S, I ⊔ J = span (Subtype.val '' S)

Tactic:
obtain ⟨⟨s₁, rfl⟩, ⟨s₂, rfl⟩⟩ := HI, HJ

Goal state after:
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
s₁ s₂ : Set ↥(homogeneousSubmonoid 𝒜)
⊢ ∃ S, span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' S)
Module: Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : CommSemiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I J : Ideal A
HI : ∃ S, I = span (Subtype.val '' S)
HJ : ∃ S, J = span (Subtype.val '' S)
⊢ ∃ S, I * J = span (Subtype.val '' S)

Tactic:
obtain ⟨⟨s₁, rfl⟩, ⟨s₂, rfl⟩⟩ := HI, HJ

Goal state after:
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : CommSemiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
s₁ s₂ : Set ↥(homogeneousSubmonoid 𝒜)
⊢ ∃ S, span (Subtype.val '' s₁) * span (Subtype.val '' s₂) = span (Subtype.val '' S)
Module: Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
⊢ ∑ᶠ (m : M), (weightedHomogeneousComponent w m) φ = φ

Tactic:
rw [finsum_eq_sum _ (weightedHomogeneousComponent_finsupp φ)]

Goal state after:
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
⊢ ∑ i ∈ Finite.toFinset ⋯, (weightedHomogeneousComponent w i) φ = φ
Module: Mathlib.RingTheory.MvPolynomial.Homogeneous

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
φ : MvPolynomial σ R
m : ℕ
hφ : φ.IsHomogeneous m
n : ℕ
⊢ (∏ _i ∈ Finset.range n, φ).IsHomogeneous (m * n)

Tactic:
rw [show m * n = ∑ _i ∈ Finset.range n, m by simp [mul_comm]]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
φ : MvPolynomial σ R
m : ℕ
hφ : φ.IsHomogeneous m
n : ℕ
⊢ (∏ _i ∈ Finset.range n, φ).IsHomogeneous (∑ _i ∈ Finset.range n, m)
Module: Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁷ : Semiring A
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : PartialOrder ι
inst✝³ : CanonicallyOrderedAdd ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
P : AddSubmonoid A
⊢ ⨆ i, ⨆ (_ : i > 0), AddSubmonoid.ofClass (𝒜 i) ≤ P ↔ ∀ i > 0, AddSubmonoid.ofClass (𝒜 i) ≤ P

Tactic:
iSup₂_le_iff

Goal state after:
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁷ : Semiring A
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : PartialOrder ι
inst✝³ : CanonicallyOrderedAdd ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
P : AddSubmonoid A
⊢ (∀ i > 0, AddSubmonoid.ofClass (𝒜 i) ≤ P) ↔ ∀ i > 0, AddSubmonoid.ofClass (𝒜 i) ≤ P
Module: Mathlib.RingTheory.MvPolynomial.Homogeneous

Goal state before:
pp: case zero
R : Type u_5
inst✝¹ : CommRing R
inst✝ : IsDomain R
F : MvPolynomial (Fin 0) R
n : ℕ
hF : F.IsHomogeneous n
hF₀ : F ≠ 0
hnR : ↑n ≤ #R
⊢ ∃ r, (eval r) F ≠ 0

Tactic:
use 0

Goal state after:
case h
R : Type u_5
inst✝¹ : CommRing R
inst✝ : IsDomain R
F : MvPolynomial (Fin 0) R
n : ℕ
hF : F.IsHomogeneous n
hF₀ : F ≠ 0
hnR : ↑n ≤ #R
⊢ (eval 0) F ≠ 0
Module: Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
⊢ (DirectSum.coeAddMonoidHom (weightedHomogeneousSubmodule R w)) (decompose' R w φ) = φ

Tactic:
conv_rhs => rw [← sum_weightedHomogeneousComponent w φ]
    rw [← DirectSum.sum_support_of (decompose' R w φ)]
    simp only [DirectSum.coeAddMonoidHom_of, map_sum,
      finsum_eq_sum _ (weightedHomogeneousComponent_finsupp φ)]
    apply Finset.sum_congr _ (fun m _ ↦ by rw [decompose'_apply])
    ext m
    simp only [DFinsupp.mem_support_toFun, ne_eq, Set.Finite.mem_toFinset, Function.mem_support,
      not_iff_not]
    conv_lhs => rw [← Subtype.coe_inj]
    rw [decompose'_apply, Submodule.coe_zero]

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Zero

Goal state before:
pp: K : Type u_2
M : Type u_3
inst✝³ : Field K
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : End K M
V : Submodule K M := φ.maxGenEigenspace 0
hV : V = ⨆ n, ker (φ ^ n)
W : Submodule K M := ⨅ n, range (φ ^ n)
hVW : IsCompl V W
hφV : ∀ x ∈ V, φ x ∈ V
hφW : ∀ x ∈ W, φ x ∈ W
F : ↥V →ₗ[K] ↥V := restrict φ hφV
G : ↥W →ₗ[K] ↥W := restrict φ hφW
ψ : ↥V × ↥W →ₗ[K] ↥V × ↥W := F.prodMap G
e : (↥V × ↥W) ≃ₗ[K] M := V.prodEquivOfIsCompl W hVW
bV : Basis (ChooseBasisIndex K ↥V) K ↥V := chooseBasis K ↥V
bW : Basis (ChooseBasisIndex K ↥W) K ↥W := chooseBasis K ↥W
b : Basis (ChooseBasisIndex K ↥V ⊕ ChooseBasisIndex K ↥W) K (↥V × ↥W) := bV.prod bW
⊢ ψ = e.symm.conj φ

Tactic:
apply b.ext

Goal state after:
K : Type u_2
M : Type u_3
inst✝³ : Field K
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : End K M
V : Submodule K M := ⋯
hV : V = ⨆ n, ker (φ ^ n)
W : Submodule K M := ⋯
hVW : IsCompl V W
hφV : ∀ x ∈ V, φ x ∈ V
hφW : ∀ x ∈ W, φ x ∈ W
F : ↥V →ₗ[K] ↥V := ⋯
G : ↥W →ₗ[K] ↥W := ⋯
ψ : ↥V × ↥W →ₗ[K] ↥V × ↥W := ⋯
e : (↥V × ↥W) ≃ₗ[K] M := ⋯
bV : Basis (ChooseBasisIndex K ↥V) K ↥V := ⋯
bW : Basis (ChooseBasisIndex K ↥W) K ↥W := ⋯
b : Basis (ChooseBasisIndex K ↥V ⊕ ChooseBasisIndex K ↥W) K (↥V × ↥W) := ⋯
⊢ ∀ (i : ChooseBasisIndex K ↥V ⊕ ChooseBasisIndex K ↥W), ψ (b i) = (e.symm.conj φ) (b i)
Module: Mathlib.Algebra.Module.LinearMap.Polynomial

Goal state before:
pp: R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ιM : Type u_7
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ιM
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ιM
b : Basis ι R L
bₘ : Basis ιM R M
ιM' : Type u_8
inst✝¹ : Fintype ιM'
inst✝ : DecidableEq ιM'
bₘ' : Basis ιM' R M
f : Polynomial (MvPolynomial ι R) → Polynomial (MvPolynomial ι R) := ⋯
hf : Function.Injective f
⊢ f (φ.polyCharpolyAux b bₘ) = f (φ.polyCharpolyAux b bₘ')

Tactic:
let _h1 : Module.Finite (MvPolynomial ι R) (TensorProduct R (MvPolynomial ι R) M) :=
    Module.Finite.of_basis (basis (MvPolynomial ι R) bₘ)

Goal state after:
R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ιM : Type u_7
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ιM
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ιM
b : Basis ι R L
bₘ : Basis ιM R M
ιM' : Type u_8
inst✝¹ : Fintype ιM'
inst✝ : DecidableEq ιM'
bₘ' : Basis ιM' R M
f : Polynomial (MvPolynomial ι R) → Polynomial (MvPolynomial ι R) := Polynomial.map (aeval X).toRingHom
hf : Function.Injective f
_h1 : Module.Finite (MvPolynomial ι R) (TensorProduct R (MvPolynomial ι R) M) :=
  Finite.of_basis (basis (MvPolynomial ι R) bₘ)
⊢ f (φ.polyCharpolyAux b bₘ) = f (φ.polyCharpolyAux b bₘ')
Module: Mathlib.Algebra.Lie.Matrix

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
P A : Matrix n n R
h : Invertible P
⊢ (P.lieConj h) A = P * A * P⁻¹

Tactic:
simp [LinearEquiv.conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp,
    LinearMap.toMatrix'_toLin']

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : Fintype n
J : Matrix n n R
inst✝ : DecidableEq n
P : Matrix n n R
h : Invertible P
⊢ LieSubalgebra.map (P.lieConj h).symm.toLieHom (skewAdjointMatricesLieSubalgebra J) =
    skewAdjointMatricesLieSubalgebra (Pᵀ * J * P)

Tactic:
ext A
    suffices P.lieConj h A ∈ skewAdjointMatricesSubmodule J ↔
        A ∈ skewAdjointMatricesSubmodule (Pᵀ * J * P) by
      simp only [Submodule.mem_map_equiv, LieSubalgebra.mem_map_submodule]
      exact this
    simp [Matrix.IsSkewAdjoint, J.isAdjointPair_equiv _ _ P (isUnit_of_invertible P)]

Goal state after:
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: R : Type u_2
L : Type u_3
M : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : Module.Finite R L
inst✝³ : Free R L
inst✝² : Module.Finite R M
inst✝¹ : Free R M
x y : L
inst✝ : Nontrivial R
i j : ℕ
hij : i + j = finrank R M
⊢ ((lieCharpoly R M x y).coeff i).natDegree ≤ j

Tactic:
rw [← mul_one j, lieCharpoly, coeff_map]
  apply MvPolynomial.aeval_natDegree_le
  · apply (polyCharpoly_coeff_isHomogeneous φ (chooseBasis R L) _ _ hij).totalDegree_le
  intro k
  exact natDegree_linear_le

Goal state after:
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := lieCharpoly K (↥E) x' u
ψ : K[X][X] := lieCharpoly K Q x' u
this : ∀ (m : ↥E), ∃ n, ((toEnd K (↥U) L) y' ^ n) ↑m = 0
z : L
hz : z ∈ ↑Ex
⊢ z ∈ ↑Ey

Tactic:
rw [mem_engel_iff]

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := lieCharpoly K (↥E) x' u
ψ : K[X][X] := lieCharpoly K Q x' u
this : ∀ (m : ↥E), ∃ n, ((toEnd K (↥U) L) y' ^ n) ↑m = 0
z : L
hz : z ∈ ↑Ex
⊢ ∃ n, ((ad K L) y ^ n) z = 0
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := lieCharpoly K (↥E) x' u
ψ : K[X][X] := lieCharpoly K Q x' u
hi : 0 < r
⊢ ↑(χ.coeff 0).natDegree < #K

Tactic:
apply lt_of_lt_of_le _ hLK

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⋯
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⋯
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L := ⋯
hx₀ : x ≠ 0
Q : Type u_2 := ⋯
r : ℕ := ⋯
hr : r < finrank K L
x' : ↥U := ⋯
y' : ↥U := ⋯
u : ↥U := ⋯
χ : K[X][X] := ⋯
ψ : K[X][X] := ⋯
hi : 0 < r
⊢ ↑(χ.coeff 0).natDegree < ↑(finrank K L)
Module: Mathlib.LinearAlgebra.Matrix.Rank

Goal state before:
pp: m : Type um
R : Type uR
inst✝³ : Semiring R
inst✝² : Nontrivial R
inst✝¹ : DecidableEq m
inst✝ : StrongRankCondition R
⊢ cRank 1 = lift.{uR, um} #m

Tactic:
have h : LinearIndependent R (1 : Matrix m m R).col := by
    convert! Pi.linearIndependent_single_one m R
    simp [funext_iff, one_apply, Pi.single_apply]

Goal state after:
m : Type um
R : Type uR
inst✝³ : Semiring R
inst✝² : Nontrivial R
inst✝¹ : DecidableEq m
inst✝ : StrongRankCondition R
h : LinearIndependent R (col 1)
⊢ cRank 1 = lift.{uR, um} #m
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := lieCharpoly K (↥E) x' u
ψ : K[X][X] := lieCharpoly K Q x' u
i : ℕ
hi : i < r
hi0 : i ≠ 0
hψ : constantCoeff ψ ≠ 0
⊢ ∃ s, r ≤ s.card ∧ ∀ α ∈ s, eval α (constantCoeff ψ) ≠ 0

Tactic:
by
    classical
    -- Let `t` denote the set of roots of `constantCoeff ψ`.
    let t := (constantCoeff ψ).roots.toFinset
    -- We show that `t` has cardinality at most `finrank K L - r`.
    have ht : t.card ≤ finrank K L - r := by
      refine (Multiset.toFinset_card_le _).trans ?_
      refine (card_roots' _).trans ?_
      rw [constantCoeff_apply]
      -- Indeed, `constantCoeff ψ` has degree at most `finrank K Q = finrank K L - r`.
      apply lieCharpoly_coeff_natDegree
      suffices finrank K Q + r = finrank K L by rw [← this, zero_add, Nat.add_sub_cancel]
      apply Submodule.finrank_quotient_add_finrank
    -- Hence there exists a subset of size `≥ r` in the complement of `t`,
    -- and `constantCoeff ψ` takes non-zero values on all of this subset.
    obtain ⟨s, hs⟩ := exists_finset_le_card K _ hLK
    use s \ t
    refine ⟨?_, ?_⟩
    · refine le_trans ?_ (Finset.le_card_sdiff _ _)
      omega
    · intro α hα
      simp only [Finset.mem_sdiff, Multiset.mem_toFinset, mem_roots', IsRoot.def, not_and, t] at hα
      exact hα.2 hψ

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

Goal state before:
pp: m : Type um
R : Type uR
inst✝¹ : Field R
inst✝ : DecidableEq m
w : m → R
w' : { i // w i ≠ 0 } → m → R := fun i ↦ diagonal w ↑i
h : LinearIndependent R w'
⊢ (diagonal w).cRank = lift.{uR, um} #{ i // w i ≠ 0 }

Tactic:
have hrw : insert 0 (range (diagonal w).col) = insert 0 (range w') := by
    suffices ∀ a, diagonal w a = 0 ∨ ∃ b, w b ≠ 0 ∧ diagonal w b = diagonal w a
      by aesop (add simp [col_eq_transpose, subset_def])
    simp_rw [or_iff_not_imp_right, not_exists, not_and, not_imp_not]
    simp +contextual [funext_iff, diagonal]

Goal state after:
m : Type um
R : Type uR
inst✝¹ : Field R
inst✝ : DecidableEq m
w : m → R
w' : { i // w i ≠ 0 } → m → R := fun i ↦ diagonal w ↑i
h : LinearIndependent R w'
hrw : insert 0 (range (diagonal w).col) = insert 0 (range w')
⊢ (diagonal w).cRank = lift.{uR, um} #{ i // w i ≠ 0 }
Module: Mathlib.Algebra.Lie.Derivation.BaseChange

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
A : Type u_2
inst✝³ : CommRing A
inst✝² : Algebra R A
L : Type u_3
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x✝¹ x✝ : Derivation R A A
z : A ⊗[R] L
⊢ ∀ (x : A) (y : L),
    { toFun := ⇑(LinearMap.rTensor L ↑⁅x✝¹, x✝⁆), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ } (x ⊗ₜ[R] y) =
      ⁅{ toFun := ⇑(LinearMap.rTensor L ↑x✝¹), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ },
          { toFun := ⇑(LinearMap.rTensor L ↑x✝), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ }⁆
        (x ⊗ₜ[R] y)

Tactic:
by simp [sub_tmul]

Goal state after: