mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.Separable

Goal state before:
pp: F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
hp : Nat.Prime p
f : F[X]
hf : Irreducible f
h1 : ¬f.Separable
g : F[X]
hg : Irreducible g
hgf : (expand F p) g = f
N : ℕ
ih : ∀ m < N + 1, ∀ {f : F[X]}, Irreducible f → f.natDegree = m → ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f
hn : f.natDegree = N + 1
hg1 : g.natDegree * p = N.succ
this : g.natDegree = 0
⊢ False

Tactic:
this,

Goal state after:
F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
hp : Nat.Prime p
f : F[X]
hf : Irreducible f
h1 : ¬f.Separable
g : F[X]
hg : Irreducible g
hgf : (expand F p) g = f
N : ℕ
ih : ∀ m < N + 1, ∀ {f : F[X]}, Irreducible f → f.natDegree = m → ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f
hn : f.natDegree = N + 1
hg1 : 0 * p = N.succ
this : g.natDegree = 0
⊢ False
Module: Mathlib.FieldTheory.Perfect

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : PerfectField K
p : K[X]
h : Irreducible p
q : K[X]
sqf : Squarefree (p * q)
dvd : p ∣ derivative (p * q)
⊢ p ∣ q

Tactic:
rw [derivative_mul, dvd_add_left (dvd_mul_right p _)] at dvd

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : PerfectField K
p : K[X]
h : Irreducible p
q : K[X]
sqf : Squarefree (p * q)
dvd : p ∣ derivative p * q
⊢ p ∣ q
Module: Mathlib.FieldTheory.Perfect

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
p n : ℕ
inst✝¹ : ExpChar R p
f : R[X]
inst✝ : PerfectRing R p
⊢ Multiset.map (⇑(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots = p ^ n • f.roots

Tactic:
simp_rw [← coe_iterateFrobeniusEquiv, roots_expand_pow, Multiset.map_nsmul,
    Multiset.map_map, comp_apply, RingEquiv.apply_symm_apply, map_id']

Goal state after:
Module: Mathlib.FieldTheory.Separable

Goal state before:
pp: F : Type u
inst✝ : Field F
f : F[X]
hf : f ≠ 0
h : f.Splits
hnsep : ¬(GCDMonoid.gcd f (derivative f)).degree = 0
x : F
hx : eval x (GCDMonoid.gcd f (derivative f)) = 0
⊢ ¬f.roots.Nodup

Tactic:
Multiset.nodup_iff_count_le_one,

Goal state after:
F : Type u
inst✝ : Field F
f : F[X]
hf : f ≠ 0
h : f.Splits
hnsep : ¬(GCDMonoid.gcd f (derivative f)).degree = 0
x : F
hx : eval x (GCDMonoid.gcd f (derivative f)) = 0
⊢ ¬∀ (a : F), Multiset.count a f.roots ≤ 1
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : Free R M
inst✝ : Module.Finite R M
f : ι → M →ₗ[R] M
b : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M
B : Basis (Free.ChooseBasisIndex R M × ι) R (ι → M) :=
  (Pi.basis fun x ↦ b).reindex
    ((Equiv.sigmaEquivProd ι (Free.ChooseBasisIndex R M)).trans (Equiv.prodComm ι (Free.ChooseBasisIndex R M)))
this : (toMatrix B B) (pi fun i ↦ f i ∘ₗ proj i) = blockDiagonal fun i ↦ (toMatrix b b) (f i)
⊢ ((toMatrix B B) (pi fun i ↦ f i ∘ₗ proj i)).det = ∏ x, ((toMatrix b b) (f x)).det

Tactic:
this,

Goal state after:
R : Type u_1
inst✝⁵ : CommRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : Free R M
inst✝ : Module.Finite R M
f : ι → M →ₗ[R] M
b : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M
B : Basis (Free.ChooseBasisIndex R M × ι) R (ι → M) :=
  (Pi.basis fun x ↦ b).reindex
    ((Equiv.sigmaEquivProd ι (Free.ChooseBasisIndex R M)).trans (Equiv.prodComm ι (Free.ChooseBasisIndex R M)))
this : (toMatrix B B) (pi fun i ↦ f i ∘ₗ proj i) = blockDiagonal fun i ↦ (toMatrix b b) (f i)
⊢ (blockDiagonal fun i ↦ (toMatrix b b) (f i)).det = ∏ x, ((toMatrix b b) (f x)).det
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
h : IsUnit (LinearMap.det f)
x : M
⊢ (equivOfIsUnitDet h) x = f x

Tactic:
by
  nontriviality M
  simp [equivOfIsUnitDet, dif_pos (Module.nontrivial R M)]

Goal state after:
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : M ≃ₗ[R] M
f f' : M →ₗ[R] M
h : ∀ (x : M), f x = f' (e x)
this : Associated (LinearMap.det (f' ∘ₗ ↑e)) (LinearMap.det f')
⊢ Associated (LinearMap.det f) (LinearMap.det f')

Tactic:
convert! this using 2
    ext x
    exact h x

Goal state after:
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
M' : Type u_3
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
v : ι → M
i j : ι
h : v i = v j
hij : i ≠ j
⊢ (e.toMatrix (update v i (v j)))ᵀ.det = 0

Tactic:
e.toMatrix_update,

Goal state after:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
M' : Type u_3
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
v : ι → M
i j : ι
h : v i = v j
hij : i ≠ j
⊢ ((e.toMatrix v).updateCol i ⇑(e.repr (v j)))ᵀ.det = 0
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
f : M [⋀^ι]→ₗ[R] R
i : ι → ι
h : Injective i
⊢ (f fun i_1 ↦ e (i i_1)) = (f ⇑e • e.det) fun i_1 ↦ e (i i_1)

Tactic:
let σ : Equiv.Perm ι := Equiv.ofBijective i (Finite.injective_iff_bijective.1 h)

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
f : M [⋀^ι]→ₗ[R] R
i : ι → ι
h : Injective i
σ : Equiv.Perm ι := Equiv.ofBijective i ⋯
⊢ (f fun i_1 ↦ e (i i_1)) = (f ⇑e • e.det) fun i_1 ↦ e (i i_1)
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: M : Type u_2
inst✝⁴ : AddCommGroup M
ι : Type u_4
inst✝³ : DecidableEq ι
inst✝² : Fintype ι
A : Type u_5
inst✝¹ : CommRing A
inst✝ : Module A M
b b' b'' : Basis ι A M
this : ⇑b'' = ⇑↑(b'.equiv b'' (Equiv.refl ι)) ∘ ⇑b'
| b.det ⇑b''

Tactic:
this,

Goal state after:
M : Type u_2
inst✝⁴ : AddCommGroup M
ι : Type u_4
inst✝³ : DecidableEq ι
inst✝² : Fintype ι
A : Type u_5
inst✝¹ : CommRing A
inst✝ : Module A M
b b' b'' : Basis ι A M
this : ⇑b'' = ⇑↑(b'.equiv b'' (Equiv.refl ι)) ∘ ⇑b'
| b.det (⇑↑(b'.equiv b'' (Equiv.refl ι)) ∘ ⇑b')
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: R : Type u_6
V : Type u_7
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : Module.Finite R V
W : Submodule R V
inst✝² : Free R ↥W
inst✝¹ : Module.Finite R ↥W
inst✝ : Free R (V ⧸ W)
e : V →ₗ[R] V
he : W ≤ comap e W
m : Type u_7 := Free.ChooseBasisIndex R ↥W
bW : Basis m R ↥W := Free.chooseBasis R ↥W
n : Type u_7 := Free.ChooseBasisIndex R (V ⧸ W)
bQ : Basis n R (V ⧸ W) := Free.chooseBasis R (V ⧸ W)
b : Basis (m ⊕ n) R V := bW.sumQuot bQ
A : Matrix m m R := (toMatrix bW bW) (e.restrict he)
B : Matrix m n R := of fun i l ↦ ((bW.sumQuot bQ).repr (e ((bW.sumQuot bQ) (Sum.inr l)))) (Sum.inl i)
D : Matrix n n R := (toMatrix bQ bQ) (W.mapQ W e he)
this : (toMatrix b b) e = fromBlocks A B 0 D
⊢ ((toMatrix b b) e).det = LinearMap.det (e.restrict he) * LinearMap.det (W.mapQ W e he)

Tactic:
this,

Goal state after:
R : Type u_6
V : Type u_7
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : Module.Finite R V
W : Submodule R V
inst✝² : Free R ↥W
inst✝¹ : Module.Finite R ↥W
inst✝ : Free R (V ⧸ W)
e : V →ₗ[R] V
he : W ≤ comap e W
m : Type u_7 := Free.ChooseBasisIndex R ↥W
bW : Basis m R ↥W := Free.chooseBasis R ↥W
n : Type u_7 := Free.ChooseBasisIndex R (V ⧸ W)
bQ : Basis n R (V ⧸ W) := Free.chooseBasis R (V ⧸ W)
b : Basis (m ⊕ n) R V := bW.sumQuot bQ
A : Matrix m m R := (toMatrix bW bW) (e.restrict he)
B : Matrix m n R := of fun i l ↦ ((bW.sumQuot bQ).repr (e ((bW.sumQuot bQ) (Sum.inr l)))) (Sum.inl i)
D : Matrix n n R := (toMatrix bQ bQ) (W.mapQ W e he)
this : (toMatrix b b) e = fromBlocks A B 0 D
⊢ (fromBlocks A B 0 D).det = LinearMap.det (e.restrict he) * LinearMap.det (W.mapQ W e he)
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: K : Type u_5
inst✝ : Field K
f : K[X]
hf : f.Monic
hf' : f ≠ 0
⊢ minpoly K (powerBasis hf').gen = f

Tactic:
by
  rw [minpoly_powerBasis_gen hf', hf.leadingCoeff, inv_one, C.map_one, mul_one]

Goal state after:
Module: Mathlib.LinearAlgebra.Semisimple

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
hn : IsNilpotent f
hs : f.IsSemisimple
⊢ f = 0

Tactic:
by
  have ⟨n, h0⟩ := hn
  rw [← aeval_X (R := R) f]; rw [← aeval_X_pow (R := R) f] at h0
  rw [← RingHom.mem_ker, ← AEval.annihilator_eq_ker_aeval (M := M)] at h0 ⊢
  exact hs.annihilator_isRadical _ _ ⟨n, h0⟩

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a : A
k : ℕ
h : a ^ k = 0
h₁ :
  ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i =
    ∑ i ∈ range (nilpotencyClass a), (↑i !)⁻¹ • a ^ i + ∑ i ∈ Ico (nilpotencyClass a) k, (↑i !)⁻¹ • a ^ i
this : ∑ i ∈ Ico (nilpotencyClass a) k, (↑i !)⁻¹ • a ^ i = 0
⊢ ∑ i ∈ range (nilpotencyClass a), (↑i !)⁻¹ • a ^ i =
    ∑ i ∈ range (nilpotencyClass a), (↑i !)⁻¹ • a ^ i + ∑ i ∈ Ico (nilpotencyClass a) k, (↑i !)⁻¹ • a ^ i

Tactic:
this,

Goal state after:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a : A
k : ℕ
h : a ^ k = 0
h₁ :
  ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i =
    ∑ i ∈ range (nilpotencyClass a), (↑i !)⁻¹ • a ^ i + ∑ i ∈ Ico (nilpotencyClass a) k, (↑i !)⁻¹ • a ^ i
this : ∑ i ∈ Ico (nilpotencyClass a) k, (↑i !)⁻¹ • a ^ i = 0
⊢ ∑ i ∈ range (nilpotencyClass a), (↑i !)⁻¹ • a ^ i = ∑ i ∈ range (nilpotencyClass a), (↑i !)⁻¹ • a ^ i + 0
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: A : Type u_1
inst✝⁴ : Ring A
inst✝³ : Module ℚ A
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module A M
inst✝ : Module ℚ M
a : A
m : M
k : ℕ
h : a ^ k • m = 0
hn : IsNilpotent a
h₀ : k < nilpotencyClass a
this : ∑ i ∈ Ico k (nilpotencyClass a), ((↑i !)⁻¹ • a ^ i) • m = 0
⊢ ∑ k ∈ range k, ((↑k !)⁻¹ • a ^ k) • m + ∑ k ∈ Ico k (nilpotencyClass a), ((↑k !)⁻¹ • a ^ k) • m =
    ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i • m

Tactic:
this,

Goal state after:
A : Type u_1
inst✝⁴ : Ring A
inst✝³ : Module ℚ A
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module A M
inst✝ : Module ℚ M
a : A
m : M
k : ℕ
h : a ^ k • m = 0
hn : IsNilpotent a
h₀ : k < nilpotencyClass a
this : ∑ i ∈ Ico k (nilpotencyClass a), ((↑i !)⁻¹ • a ^ i) • m = 0
⊢ ∑ k ∈ range k, ((↑k !)⁻¹ • a ^ k) • m + 0 = ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i • m
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
K : Type u_5
inst✝ : CommRing R
I✝ : Ideal R
f✝ f : R[X]
I : Ideal R
x : R
this : (algebraMap R (AdjoinRoot f ⧸ Ideal.map (of f) I)) x = (Ideal.Quotient.mk (Ideal.map (of f) I)) ((mk f) (C x))
⊢ (quotAdjoinRootEquivQuotPolynomialQuot I f) ((algebraMap R (AdjoinRoot f ⧸ Ideal.map (of f) I)) x) =
    (algebraMap R ((R ⧸ I)[X] ⧸ span {Polynomial.map (Ideal.Quotient.mk I) f})) x

Tactic:
this,

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
K : Type u_5
inst✝ : CommRing R
I✝ : Ideal R
f✝ f : R[X]
I : Ideal R
x : R
this : (algebraMap R (AdjoinRoot f ⧸ Ideal.map (of f) I)) x = (Ideal.Quotient.mk (Ideal.map (of f) I)) ((mk f) (C x))
⊢ (quotAdjoinRootEquivQuotPolynomialQuot I f) ((Ideal.Quotient.mk (Ideal.map (of f) I)) ((mk f) (C x))) =
    (algebraMap R ((R ⧸ I)[X] ⧸ span {Polynomial.map (Ideal.Quotient.mk I) f})) x
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: case refine_3
R✝ : Type u_1
S✝ : Type u_2
T✝ : Type u_3
U✝ : Type u_4
K : Type u_5
R : Type u_6
S : Type u_7
T : Type u_8
U : Type u_9
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing T
inst✝³ : Algebra R S
inst✝² : Algebra R T
inst✝¹ : CommRing U
inst✝ : Algebra R U
p✝ p : S[X]
q : (T ⊗[R] S)[X]
h : Polynomial.map includeRight.toRingHom p = q
⊢ (liftAlgHom q (Algebra.TensorProduct.map (AlgHom.id T T) (AlgHom.restrictScalars R (ofId S (AdjoinRoot p))))
          (1 ⊗ₜ[R] root p) ⋯).comp
      (Algebra.TensorProduct.lift (algHom T T (AdjoinRoot q)) (mapAlgHom includeRight p q ⋯) ⋯) =
    AlgHom.id T (T ⊗[R] AdjoinRoot p)

Tactic:
ext : 3 <;> simp

Goal state after:
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: K : Type u_6
L : Type u_7
inst✝⁴ : CommRing K
inst✝³ : IsDomain K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsIntegral K L
f : L[X]
hf : Irreducible f
⊢ ∃ g, g.Monic ∧ Irreducible g ∧ f ∣ Polynomial.map (algebraMap K L) g

Tactic:
by
  have := Fact.mk hf
  have h := hf.ne_zero
  have h2 := isIntegral_trans (R := K) _ (AdjoinRoot.isIntegral_root h)
  have h3 := (AdjoinRoot.minpoly_root h) ▸ minpoly.dvd_map_of_isScalarTower K L (AdjoinRoot.root f)
  exact ⟨_, minpoly.monic h2, minpoly.irreducible h2, dvd_of_mul_right_dvd h3⟩

Goal state after:
Module: Mathlib.LinearAlgebra.JordanChevalley

Goal state before:
pp: case refine_3
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k✝ : ℕ
sep : P.Separable
nil : minpoly K f ∣ P ^ k✝
ff : ↥K[f] := ⟨f, ⋯⟩
P' : K[X] := derivative P
nil' : IsNilpotent ((aeval ff) P)
sep' : IsUnit ((aeval ff) P')
s : End K V
mem : s ∈ K[f]
hss : (aeval ⟨s, mem⟩) P = 0
k : ℕ
hk : (ff - ⟨s, mem⟩) ^ k = 0
⊢ s.IsSemisimple

Tactic:
· replace hss : aeval s P = 0 := by rwa [Subtype.ext_iff, coe_aeval_mk_apply] at hss
    exact isSemisimple_of_squarefree_aeval_eq_zero sep.squarefree hss

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

Goal state before:
pp: case zero
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I J : LieIdeal R L
h₁ : I ≤ J
l : ℕ
h₂ : l ≤ 0
⊢ D 0 I ≤ D l J

Tactic:
rw [le_zero_iff] at h₂

Goal state after:
case zero
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I J : LieIdeal R L
h₁ : I ≤ J
l : ℕ
h₂ : l = 0
⊢ D 0 I ≤ D l J
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a : A
ha : IsNilpotent a
a✝ : Nontrivial A
⊢ IsNilpotent (∑ i ∈ range (nilpotencyClass a - Nat.succ 0 + Nat.succ 0), (↑i !)⁻¹ • a ^ i - 1)

Tactic:
Finset.sum_range_succ'

Goal state after:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a : A
ha : IsNilpotent a
a✝ : Nontrivial A
⊢ IsNilpotent (∑ k ∈ range (nilpotencyClass a - Nat.succ 0), (↑(k + 1)!)⁻¹ • a ^ (k + 1) + (↑0!)⁻¹ • a ^ 0 - 1)
Module: Mathlib.Algebra.Lie.Solvable

Goal state before:
pp: case h
R : Type u
L : Type v
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : LieAlgebra R L
A : Type u_1
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Module.FaithfullyFlat R A
k : ℕ
h : LieSubmodule.baseChange A (derivedSeries R L k) ≤ ⊥
x : L
hx : x ∈ derivedSeries R L k
⊢ x ∈ ⊥

Tactic:
specialize h <| Submodule.tmul_mem_baseChange_of_mem 1 hx

Goal state after:
case h
R : Type u
L : Type v
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : LieAlgebra R L
A : Type u_1
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Module.FaithfullyFlat R A
k : ℕ
x : L
h : 1 ⊗ₜ[R] x ∈ ⊥
hx : x ∈ derivedSeries R L k
⊢ x ∈ ⊥
Module: Mathlib.Algebra.Lie.Solvable

Goal state before:
pp: case h
R : Type u
L : Type v
M : Type w
L' : Type w₁
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : LieAlgebra R L
inst✝² : LieRing L'
inst✝¹ : LieAlgebra R L'
I J : LieIdeal R L
f : L' →ₗ⁅R⁆ L
inst✝ : IsLieAbelian L
⊢ derivedSeries ℤ L 1 = ⊥

Tactic:
← abelian_iff_derived_one_eq_bot,

Goal state after:
case h
R : Type u
L : Type v
M : Type w
L' : Type w₁
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : LieAlgebra R L
inst✝² : LieRing L'
inst✝¹ : LieAlgebra R L'
I J : LieIdeal R L
f : L' →ₗ⁅R⁆ L
inst✝ : IsLieAbelian L
⊢ IsLieAbelian ↥⊤
Module: Mathlib.Algebra.Lie.Engel

Goal state before:
pp: case succ.refine_1
R : Type u₁
L : Type u₂
M : Type u₄
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
I : LieIdeal R L
x : L
hxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤
n i j : ℕ
hxn : (toEnd R L M) x ^ n = 0
hIM : lowerCentralSeries R L M i ≤ I.lcs M j
l : ℕ
ih : ↑(⊤.lcs M (i + l)) ≤ Submodule.map ((toEnd R L M) x ^ l) ↑(I.lcs M j) ⊔ ↑(I.lcs M (j + 1))
⊢ Submodule.map ((toEnd R L M) x) (Submodule.map ((toEnd R L M) x ^ l) ↑(I.lcs M j)) ⊔
      Submodule.map ((toEnd R L M) x) ↑(I.lcs M (j + 1)) ≤
    Submodule.map ((toEnd R L M) x ^ (l + 1)) ↑(I.lcs M j) ⊔ ↑⁅I, I.lcs M j⁆

Tactic:
← Submodule.map_comp,

Goal state after:
case succ.refine_1
R : Type u₁
L : Type u₂
M : Type u₄
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
I : LieIdeal R L
x : L
hxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤
n i j : ℕ
hxn : (toEnd R L M) x ^ n = 0
hIM : lowerCentralSeries R L M i ≤ I.lcs M j
l : ℕ
ih : ↑(⊤.lcs M (i + l)) ≤ Submodule.map ((toEnd R L M) x ^ l) ↑(I.lcs M j) ⊔ ↑(I.lcs M (j + 1))
⊢ Submodule.map ((toEnd R L M) x ∘ₗ (toEnd R L M) x ^ l) ↑(I.lcs M j) ⊔
      Submodule.map ((toEnd R L M) x) ↑(I.lcs M (j + 1)) ≤
    Submodule.map ((toEnd R L M) x ^ (l + 1)) ↑(I.lcs M j) ⊔ ↑⁅I, I.lcs M j⁆
Module: Mathlib.Algebra.Lie.Nilpotent

Goal state before:
pp: case succ
R : Type u
L : Type v
M : Type w
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
k : ℕ
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : LieModule R L M
f : M →ₗ⁅R,L⁆ M₂
hf : Function.Surjective ⇑f
n✝ : ℕ
a✝ : lowerCentralSeries R L M₂ n✝ ≤ LieSubmodule.map f (lowerCentralSeries R L M n✝)
⊢ ⁅⊤, lowerCentralSeries R L M₂ n✝⁆ ≤ ⁅⊤, LieSubmodule.map f (lowerCentralSeries R L M n✝)⁆

Tactic:
apply LieSubmodule.mono_lie_right

Goal state after:
case succ
R : Type u
L : Type v
M : Type w
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
k : ℕ
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : LieModule R L M
f : M →ₗ⁅R,L⁆ M₂
hf : Function.Surjective ⇑f
n✝ : ℕ
a✝ : lowerCentralSeries R L M₂ n✝ ≤ LieSubmodule.map f (lowerCentralSeries R L M n✝)
⊢ lowerCentralSeries R L M₂ n✝ ≤ LieSubmodule.map f (lowerCentralSeries R L M n✝)
Module: Mathlib.Algebra.Lie.BaseChange

Goal state before:
pp: R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : LieIdeal R L
N : LieSubmodule R L M
s : Set (A ⊗[R] M) := {m | ∃ x ∈ I, ∃ n ∈ N, 1 ⊗ₜ[R] ⁅x, n⁆ = m}
this : ⇑((TensorProduct.mk R A M) 1) '' {m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m} = s
⊢ Submodule.span A (⇑((TensorProduct.mk R A M) 1) '' {x | ∃ x_1 ∈ I, ∃ n ∈ N, ⁅x_1, n⁆ = x}) =
    ↑⁅baseChange A I, baseChange A N⁆

Tactic:
this,

Goal state after:
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : LieIdeal R L
N : LieSubmodule R L M
s : Set (A ⊗[R] M) := {m | ∃ x ∈ I, ∃ n ∈ N, 1 ⊗ₜ[R] ⁅x, n⁆ = m}
this : ⇑((TensorProduct.mk R A M) 1) '' {m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m} = s
⊢ Submodule.span A s = ↑⁅baseChange A I, baseChange A N⁆
Module: Mathlib.Algebra.Lie.Engel

Goal state before:
pp: R : Type u₁
L : Type u₂
M : Type u₄
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
I : LieIdeal R L
x : L
hxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤
hIM : LieModule.IsNilpotent (↥I) M
n : ℕ
hn : (toEnd R L M) x ^ n = 0
k : ℕ
hk : lowerCentralSeries R (↥I) M k = ⊥
hk' : I.lcs M k = ⊥
this : ∀ (l : ℕ), lowerCentralSeries R L M (l * n) ≤ I.lcs M l
⊢ LieModule.IsNilpotent L M

Tactic:
isNilpotent_iff R

Goal state after:
R : Type u₁
L : Type u₂
M : Type u₄
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
I : LieIdeal R L
x : L
hxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤
hIM : LieModule.IsNilpotent (↥I) M
n : ℕ
hn : (toEnd R L M) x ^ n = 0
k : ℕ
hk : lowerCentralSeries R (↥I) M k = ⊥
hk' : I.lcs M k = ⊥
this : ∀ (l : ℕ), lowerCentralSeries R L M (l * n) ≤ I.lcs M l
⊢ ∃ k, lowerCentralSeries R L M k = ⊥
Module: Mathlib.Algebra.Lie.Nilpotent

Goal state before:
pp: case succ
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
k : ℕ
h : nilpotencyLength L M = k + 1
⊢ (match k + 1 with
    | 0 => ⊥
    | k.succ => lowerCentralSeries R L M k) ≤
    maxTrivSubmodule R L M

Tactic:
rw [le_max_triv_iff_bracket_eq_bot]
    rw [nilpotencyLength_eq_succ_iff R, lowerCentralSeries_succ] at h
    exact h.1

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

Goal state before:
pp: R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
N : LieSubmodule R L M
inst✝ : LieModule R L M
⊢ LieModule.IsNilpotent L ↥N ↔ ∃ k, N ≤ ucs k ⊥

Tactic:
simp_rw [LieModule.isNilpotent_iff_exists_ucs_eq_top R, ← ucs_comap_incl, comap_incl_eq_top]

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

Goal state before:
pp: R : Type u
L : Type v
M : Type w
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
L₂ : Type u_1
M₂ : Type u_2
inst✝⁶ : LieRing L₂
inst✝⁵ : LieAlgebra R L₂
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L₂ M₂
f : L →ₗ⁅R⁆ L₂
g : M →ₗ[R] M₂
hfg : ∀ (x : L) (m : M), ⁅f x, g m⁆ = g ⁅x, m⁆
inst✝¹ : LieModule R L₂ M₂
hg_inj : Injective ⇑g
inst✝ : LieModule.IsNilpotent L₂ M₂
k : ℕ
hk : lowerCentralSeries R L₂ M₂ k = ⊥
⊢ LieModule.IsNilpotent L M

Tactic:
isNilpotent_iff R

Goal state after:
R : Type u
L : Type v
M : Type w
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
L₂ : Type u_1
M₂ : Type u_2
inst✝⁶ : LieRing L₂
inst✝⁵ : LieAlgebra R L₂
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L₂ M₂
f : L →ₗ⁅R⁆ L₂
g : M →ₗ[R] M₂
hfg : ∀ (x : L) (m : M), ⁅f x, g m⁆ = g ⁅x, m⁆
inst✝¹ : LieModule R L₂ M₂
hg_inj : Injective ⇑g
inst✝ : LieModule.IsNilpotent L₂ M₂
k : ℕ
hk : lowerCentralSeries R L₂ M₂ k = ⊥
⊢ ∃ k, lowerCentralSeries R L M k = ⊥