mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
⊢ ?m.193

Tactic:
calc ∑ i ∈ R2N, (i ! : ℚ)⁻¹ • (a + b) ^ i
        = ∑ i ∈ R2N, (i ! : ℚ)⁻¹ • ∑ j ∈ range (i + 1), a ^ j * b ^ (i - j) * i.choose j := ?_
      _ = ∑ i ∈ R2N, (∑ j ∈ range (i + 1),
            ((j ! : ℚ)⁻¹ * ((i - j) ! : ℚ)⁻¹) • (a ^ j * b ^ (i - j))) := ?_
      _ = ∑ ij ∈ R2N ×ˢ R2N with ij.1 + ij.2 ≤ 2 * N,
            ((ij.1 ! : ℚ)⁻¹ * (ij.2 ! : ℚ)⁻¹) • (a ^ ij.1 * b ^ ij.2) := ?_

Goal state after:
case calc_1
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = ∑ i ∈ R2N, (↑i !)⁻¹ • ∑ j ∈ range (i + 1), a ^ j * b ^ (i - j) * ↑(i.choose j)

case calc_2
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • ∑ j ∈ range (i + 1), a ^ j * b ^ (i - j) * ↑(i.choose j) =
    ∑ i ∈ R2N, ∑ j ∈ range (i + 1), ((↑j !)⁻¹ * (↑(i - j)!)⁻¹) • (a ^ j * b ^ (i - j))

case calc_3
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
⊢ ∑ i ∈ R2N, ∑ j ∈ range (i + 1), ((↑j !)⁻¹ * (↑(i - j)!)⁻¹) • (a ^ j * b ^ (i - j)) =
    ∑ ij ∈ R2N ×ˢ R2N with ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: case calc_3
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
⊢ ∑ i ∈ R2N, ∑ j ∈ range (i + 1), ((↑j !)⁻¹ * (↑(i - j)!)⁻¹) • (a ^ j * b ^ (i - j)) =
    ∑ ij ∈ R2N ×ˢ R2N with ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)

Tactic:
rw [hR2N, sum_sigma']

Goal state after:
case calc_3
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
⊢ ∑ x ∈ (range (2 * N + 1)).sigma fun i ↦ range (i + 1),
      ((↑x.snd !)⁻¹ * (↑(x.fst - x.snd)!)⁻¹) • (a ^ x.snd * b ^ (x.fst - x.snd)) =
    ∑ ij ∈ range (2 * N + 1) ×ˢ range (2 * N + 1) with ij.1 + ij.2 ≤ 2 * N,
      ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)
Module: Mathlib.LinearAlgebra.JordanChevalley

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : FiniteDimensional K V
inst✝ : PerfectField K
n₁ s₁ n₂ s₂ : End K V
hn₁ : IsNilpotent n₁
hs₁ : s₁.IsSemisimple
hn₂ : IsNilpotent n₂
hs₂ : s₂.IsSemisimple
hc₁ : Commute n₁ s₁
hc₂ : Commute n₂ s₂
h : n₁ + s₁ = n₂ + s₂
n₀ : End K V
hn₀ : n₀ ∈ K[n₁ + s₁]
s₀ : End K V
hs₀ : s₀ ∈ K[n₁ + s₁]
hn₀_nil : IsNilpotent n₀
hs₀_ss : s₀.IsSemisimple
h₀ : n₁ + s₁ = n₀ + s₀
n s : End K V
hn : IsNilpotent n
⊢ s.IsSemisimple → Commute n s → n₁ + s₁ = n + s → s = s₀

Tactic:
hs

Goal state after:
K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : FiniteDimensional K V
inst✝ : PerfectField K
n₁ s₁ n₂ s₂ : End K V
hn₁ : IsNilpotent n₁
hs₁ : s₁.IsSemisimple
hn₂ : IsNilpotent n₂
hs₂ : s₂.IsSemisimple
hc₁ : Commute n₁ s₁
hc₂ : Commute n₂ s₂
h : n₁ + s₁ = n₂ + s₂
n₀ : End K V
hn₀ : n₀ ∈ K[n₁ + s₁]
s₀ : End K V
hs₀ : s₀ ∈ K[n₁ + s₁]
hn₀_nil : IsNilpotent n₀
hs₀_ss : s₀.IsSemisimple
h₀ : n₁ + s₁ = n₀ + s₀
n s : End K V
hn : IsNilpotent n
hs : s.IsSemisimple
⊢ Commute n s → n₁ + s₁ = n + s → s = s₀
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ : ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₂ : ∑ ij ∈ R2N ×ˢ R2N with ¬(ij.1 ≤ N ∧ ij.2 ≤ N), ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₂ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 ≤ N ∧ x.2 ≤ N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i

Tactic:
← split₂

Goal state after:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ :
  ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i =
    ∑ x ∈ R2N ×ˢ R2N with x.1 ≤ N ∧ x.2 ≤ N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₂ : ∑ ij ∈ R2N ×ˢ R2N with ¬(ij.1 ≤ N ∧ ij.2 ≤ N), ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₂ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 ≤ N ∧ x.2 ≤ N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i
Module: Mathlib.Algebra.Lie.Quotient

Goal state before:
pp: R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
N N' : LieSubmodule R L M
inst✝¹ : LieAlgebra R L
inst✝ : LieModule R L M
I J : LieIdeal R L
x y : L ⧸ I
x₁ : L
⊢ ∀ (a₂ b₁ b₂ : L), (↑I).quotientRel x₁ b₁ → (↑I).quotientRel a₂ b₂ → mk ⁅x₁, a₂⁆ = mk ⁅b₁, b₂⁆

Tactic:
x₂

Goal state after:
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
N N' : LieSubmodule R L M
inst✝¹ : LieAlgebra R L
inst✝ : LieModule R L M
I J : LieIdeal R L
x y : L ⧸ I
x₁ x₂ : L
⊢ ∀ (b₁ b₂ : L), (↑I).quotientRel x₁ b₁ → (↑I).quotientRel x₂ b₂ → mk ⁅x₁, x₂⁆ = mk ⁅b₁, b₂⁆
Module: Mathlib.Algebra.Lie.Normalizer

Goal state before:
pp: R : Type u_1
L : Type u_2
M : Type u_3
M' : Type u_4
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
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
inst✝¹ : LieRingModule L M'
inst✝ : LieModule R L M'
N N₁ N₂ : LieSubmodule R L M
x y : L
hx : x ∈ {x | ∀ (m : M), ⁅x, m⁆ ∈ N}
hy : y ∈ {x | ∀ (m : M), ⁅x, m⁆ ∈ N}
m : M
⊢ ⁅x + y, m⁆ ∈ N

Tactic:
rw [add_lie]; exact N.add_mem (hx m) (hy m)

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

Goal state before:
pp: case refine_3.refine_2.refine_3
R : Type u_1
A : Type u_2
L : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x z₁ z₂ : A ⊗[R] L
h₁ : ((bracket' R A L L) z₁) z₁ = 0
h₂ : ((bracket' R A L L) z₂) z₂ = 0
a₁ : A
l₁ : L
⊢ ∀ (x y : A ⊗[R] L),
    ((bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) x + ((bracket' R A L L) x) (a₁ ⊗ₜ[R] l₁) = 0 →
      ((bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) y + ((bracket' R A L L) y) (a₁ ⊗ₜ[R] l₁) = 0 →
        ((bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) (x + y) + ((bracket' R A L L) (x + y)) (a₁ ⊗ₜ[R] l₁) = 0

Tactic:
· intro y₁ y₂ hy₁ hy₂
        simp only [hy₁, hy₂, add_add_add_comm, add_zero, LinearMap.add_apply, map_add]

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

Goal state before:
pp: case 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 : LieSubmodule R L M
y : L
n : M
hn : n ∈ N
⊢ ⁅y, n⁆ ∈ ↑(Submodule.span R ((fun a ↦ ⁅x, a⁆) '' ↑↑N ∪ {x | ∃ x_1 ∈ I, ∃ n ∈ N, ⁅x_1, n⁆ = x}))

Tactic:
obtain ⟨t, z, hz, rfl⟩ := exists_smul_add_of_span_sup_eq_top hxI y

Goal state after:
case 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 : LieSubmodule R L M
n : M
hn : n ∈ N
t : R
z : L
hz : z ∈ I
⊢ ⁅t • x + z, n⁆ ∈ ↑(Submodule.span R ((fun a ↦ ⁅x, a⁆) '' ↑↑N ∪ {x | ∃ x_1 ∈ I, ∃ n ∈ N, ⁅x_1, n⁆ = x}))
Module: Mathlib.Algebra.Lie.Engel

Goal state before:
pp: R : Type u₁
L : Type u₂
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : Subsingleton L
M : Type u_1
_i1 : AddCommGroup M
_i2 : Module R M
⊢ ∀ [inst : LieRingModule L M] [inst_1 : LieModule R L M],
    (∀ (x : L), IsNilpotent ((toEnd R L M) x)) → LieModule.IsNilpotent L M

Tactic:
_i3

Goal state after:
R : Type u₁
L : Type u₂
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : Subsingleton L
M : Type u_1
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule L M
⊢ ∀ [inst : LieModule R L M], (∀ (x : L), IsNilpotent ((toEnd R L M) x)) → LieModule.IsNilpotent L M
Module: Mathlib.Algebra.Lie.Engel

Goal state before:
pp: R : Type u₁
L : Type u₂
L₂ : Type u₃
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
hf : Surjective ⇑f
h : LieAlgebra.IsEngelian R L
M : Type u₄
_i1 : AddCommGroup M
_i2 : Module R M
⊢ ∀ [inst : LieRingModule L₂ M] [inst_1 : LieModule R L₂ M],
    (∀ (x : L₂), IsNilpotent ((toEnd R L₂ M) x)) → LieModule.IsNilpotent L₂ M

Tactic:
_i3

Goal state after:
R : Type u₁
L : Type u₂
L₂ : Type u₃
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
hf : Surjective ⇑f
h : LieAlgebra.IsEngelian R L
M : Type u₄
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule L₂ M
⊢ ∀ [inst : LieModule R L₂ M], (∀ (x : L₂), IsNilpotent ((toEnd R L₂ M) x)) → LieModule.IsNilpotent L₂ M
Module: Mathlib.Algebra.Lie.Engel

Goal state before:
pp: R : Type u₁
L : Type u₂
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
K : LieSubalgebra R L
hK₁ : IsEngelian R ↥K
hK₂ : K < K.normalizer
x : L
hx₁ : x ∈ K.normalizer
hx₂ : x ∉ K
K' : LieSubalgebra R L :=
  let __src := R ∙ x ⊔ K.toSubmodule;
  { toSubmodule := __src, lie_mem' := ⋯ }
hxK' : x ∈ K'
hKK' : K ≤ K'
hK' : K' ≤ K.normalizer
M : Type u₄
_i1 : AddCommGroup M
_i2 : Module R M
⊢ ∀ [inst : LieRingModule (↥K') M] [inst_1 : LieModule R (↥K') M],
    (∀ (x : ↥K'), IsNilpotent ((toEnd R (↥K') M) x)) → LieModule.IsNilpotent (↥K') M

Tactic:
_i3

Goal state after:
R : Type u₁
L : Type u₂
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
K : LieSubalgebra R L
hK₁ : IsEngelian R ↥K
hK₂ : K < K.normalizer
x : L
hx₁ : x ∈ K.normalizer
hx₂ : x ∉ K
K' : LieSubalgebra R L :=
  let __src := R ∙ x ⊔ K.toSubmodule;
  { toSubmodule := __src, lie_mem' := ⋯ }
hxK' : x ∈ K'
hKK' : K ≤ K'
hK' : K' ≤ K.normalizer
M : Type u₄
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule (↥K') M
⊢ ∀ [inst : LieModule R (↥K') M], (∀ (x : ↥K'), IsNilpotent ((toEnd R (↥K') M) x)) → LieModule.IsNilpotent (↥K') M
Module: Mathlib.Algebra.Lie.Engel

Goal state before:
pp: R : Type u₁
L : Type u₂
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsNoetherian R L
M : Type u_1
_i1 : AddCommGroup M
_i2 : Module R M
⊢ ∀ [inst : LieRingModule L M] [inst_1 : LieModule R L M],
    (∀ (x : L), IsNilpotent ((toEnd R L M) x)) → LieModule.IsNilpotent L M

Tactic:
_i3

Goal state after:
R : Type u₁
L : Type u₂
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsNoetherian R L
M : Type u_1
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule L M
⊢ ∀ [inst : LieModule R L M], (∀ (x : L), IsNilpotent ((toEnd R L M) x)) → LieModule.IsNilpotent L M
Module: Mathlib.Algebra.Lie.Engel

Goal state before:
pp: R : Type u₁
L : Type u₂
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsNoetherian R L
M : Type u_1
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule L M
_i4 : LieModule R L M
L' : LieSubalgebra R (Module.End R M) := (toEnd R L M).range
h : ∀ (y : ↥L'), IsNilpotent ↑y
s : Set (LieSubalgebra R ↥L') := {K | IsEngelian R ↥K}
hs : s.Nonempty
K : LieSubalgebra R ↥L'
hK₁ : IsEngelian R ↥K
hK₂ : K ≠ ⊤
this : Nontrivial (↥L' ⧸ K.toLieSubmodule)
x : ↥K
hx : IsNilpotent ((ad R ↥L') ↑x)
X : ↥L'
HX : X ∈ ↑K.toLieSubmodule
⊢ X ∈ Submodule.comap ((ad R ↥L') ↑x) ↑K.toLieSubmodule

Tactic:
simp only [LieSubalgebra.coe_toLieSubmodule, LieSubalgebra.mem_toSubmodule] at HX

Goal state after:
R : Type u₁
L : Type u₂
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsNoetherian R L
M : Type u_1
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule L M
_i4 : LieModule R L M
L' : LieSubalgebra R (Module.End R M) := (toEnd R L M).range
h : ∀ (y : ↥L'), IsNilpotent ↑y
s : Set (LieSubalgebra R ↥L') := {K | IsEngelian R ↥K}
hs : s.Nonempty
K : LieSubalgebra R ↥L'
hK₁ : IsEngelian R ↥K
hK₂ : K ≠ ⊤
this : Nontrivial (↥L' ⧸ K.toLieSubmodule)
x : ↥K
hx : IsNilpotent ((ad R ↥L') ↑x)
X : ↥L'
HX : X ∈ K
⊢ X ∈ Submodule.comap ((ad R ↥L') ↑x) ↑K.toLieSubmodule
Module: Mathlib.Algebra.Lie.Nilpotent

Goal state before:
pp: case refine_1
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
h : LieRing.IsNilpotent ↥(ad R L).range
⊢ LieRing.IsNilpotent L

Tactic:
have : (ad R L).ker = center R L := by simp
    exact
      LieAlgebra.nilpotent_of_nilpotent_quotient (le_of_eq this)
        ((ad R L).quotKerEquivRange.nilpotent_iff_equiv_nilpotent.mpr h)

Goal state after:
Module: Mathlib.FieldTheory.Normal.Defs

Goal state before:
pp: F : Type u_1
inst✝⁵ : Field F
E : Type u_3
inst✝⁴ : Field E
inst✝³ : Algebra F E
M : Type u_5
N : Type u_6
inst✝² : Field N
inst✝¹ : Field M
inst✝ : Algebra M N
h : ∀ (x : E), IsIntegral F x ∧ (map (algebraMap F E) (minpoly F x)).Splits
f : F ≃+* M
g : E ≃+* N
hcomp : (algebraMap M N).comp ↑f = (↑g).comp (algebraMap F E)
this : Normal F E
x : N
⊢ (map ((algebraMap M N).comp ↑f) (minpoly F (g.symm x))).Splits

Tactic:
hcomp,

Goal state after:
F : Type u_1
inst✝⁵ : Field F
E : Type u_3
inst✝⁴ : Field E
inst✝³ : Algebra F E
M : Type u_5
N : Type u_6
inst✝² : Field N
inst✝¹ : Field M
inst✝ : Algebra M N
h : ∀ (x : E), IsIntegral F x ∧ (map (algebraMap F E) (minpoly F x)).Splits
f : F ≃+* M
g : E ≃+* N
hcomp : (algebraMap M N).comp ↑f = (↑g).comp (algebraMap F E)
this : Normal F E
x : N
⊢ (map ((↑g).comp (algebraMap F E)) (minpoly F (g.symm x))).Splits
Module: Mathlib.FieldTheory.Normal.Defs

Goal state before:
pp: K₁ : Type u_6
K₂ : Type u_7
F : Type u_8
K₃ : Type u_9
inst✝¹⁵ : Field F
inst✝¹⁴ : Field K₁
inst✝¹³ : Field K₂
inst✝¹² : Field K₃
inst✝¹¹ : Algebra F K₁
inst✝¹⁰ : Algebra F K₂
inst✝⁹ : Algebra F K₃
inst✝⁸ : Algebra K₁ K₂
inst✝⁷ : Algebra K₁ K₃
inst✝⁶ : Algebra K₂ K₃
inst✝⁵ : IsScalarTower F K₁ K₃
inst✝⁴ : IsScalarTower F K₁ K₂
inst✝³ : IsScalarTower F K₂ K₃
inst✝² : IsScalarTower K₁ K₂ K₃
inst✝¹ : Normal F K₁
inst✝ : Normal F K₂
f : Gal(K₃/F)
⊢ ((AlgEquiv.restrictNormalHom K₁).comp (AlgEquiv.restrictNormalHom K₂)) f =
    (AlgEquiv.restrictNormalHom K₁) ((AlgEquiv.restrictNormalHom K₂) f)

Tactic:
MonoidHom.comp_apply

Goal state after:
K₁ : Type u_6
K₂ : Type u_7
F : Type u_8
K₃ : Type u_9
inst✝¹⁵ : Field F
inst✝¹⁴ : Field K₁
inst✝¹³ : Field K₂
inst✝¹² : Field K₃
inst✝¹¹ : Algebra F K₁
inst✝¹⁰ : Algebra F K₂
inst✝⁹ : Algebra F K₃
inst✝⁸ : Algebra K₁ K₂
inst✝⁷ : Algebra K₁ K₃
inst✝⁶ : Algebra K₂ K₃
inst✝⁵ : IsScalarTower F K₁ K₃
inst✝⁴ : IsScalarTower F K₁ K₂
inst✝³ : IsScalarTower F K₂ K₃
inst✝² : IsScalarTower K₁ K₂ K₃
inst✝¹ : Normal F K₁
inst✝ : Normal F K₂
f : Gal(K₃/F)
⊢ (AlgEquiv.restrictNormalHom K₁) ((AlgEquiv.restrictNormalHom K₂) f) =
    (AlgEquiv.restrictNormalHom K₁) ((AlgEquiv.restrictNormalHom K₂) f)
Module: Mathlib.FieldTheory.Fixed

Goal state before:
pp: M : Type u
inst✝⁸ : Monoid M
G : Type u
inst✝⁷ : Group G
K : Type u_1
F : Type v
inst✝⁶ : Field F
inst✝⁵ : MulSemiringAction M F
inst✝⁴ : MulSemiringAction G F
m : M
inst✝³ : Field K
inst✝² : Algebra K F
inst✝¹ : SMulCommClass M K F
inst✝ : Finite G
⊢ Algebra.IsSeparable (↥(subfield G F)) F

Tactic:
classical
  exact ⟨fun x => by
    cases nonempty_fintype G
    rw [IsSeparable, ← minpoly_eq_minpoly,
      ← Polynomial.separable_map (FixedPoints.subfield G F).subtype, minpoly,
      ← Subfield.toSubring_subtype_eq_subtype, Polynomial.map_toSubring _ (subfield G F).toSubring]
    exact Polynomial.separable_prod_X_sub_C_iff.2 (injective_ofQuotientStabilizer G x)⟩

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra

Goal state before:
pp: case mp
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
s : Finset E
hs : adjoin F ↑s = ⊤
this✝¹ : Algebra.FiniteType F ↥(Algebra.adjoin F ↑s)
this✝ : Algebra.EssFiniteType ↥(Algebra.adjoin F ↑s) ↥(adjoin F ↑s)
this : Algebra.EssFiniteType F ↥(adjoin F ↑s)
⊢ Algebra.EssFiniteType F E

Tactic:
hs

Goal state after:
case mp
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
s : Finset E
hs : adjoin F ↑s = ⊤
this✝¹ : Algebra.FiniteType F ↥(Algebra.adjoin F ↑s)
this✝ : Algebra.EssFiniteType ↥(Algebra.adjoin F ↑s) ↥(adjoin F ↑s)
this : Algebra.EssFiniteType F ↥⊤
⊢ Algebra.EssFiniteType F E
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra

Goal state before:
pp: F : Type u_1
A : Type u_2
K : Type u_3
L : Type u_4
inst✝⁸ : Field F
inst✝⁷ : CommRing A
inst✝⁶ : Algebra F A
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : Field L
inst✝ : Algebra F L
g : A →ₐ[F] L
f : K →ₐ[F] L
h : (↑f).comp (algebraMap A K) = ↑g
⊢ f.fieldRange = IntermediateField.adjoin F ↑g.range

Tactic:
by
  apply IntermediateField.toSubfield_injective
  simp_rw [AlgHom.fieldRange_toSubfield, IntermediateField.adjoin_toSubfield]
  convert! ringHom_fieldRange_eq_of_comp_eq h using 2
  exact Set.union_eq_self_of_subset_left fun _ ⟨x, hx⟩ ↦ ⟨algebraMap F A x, by simp [← hx]⟩

Goal state after:
Module: Mathlib.RingTheory.Adjoin.Field

Goal state before:
pp: F : Type u_5
E : Type u_6
K : Type u_7
inst✝⁹ : CommSemiring F
inst✝⁸ : StrongRankCondition F
inst✝⁷ : CommSemiring E
inst✝⁶ : StrongRankCondition E
inst✝⁵ : Semiring K
inst✝⁴ : SMul F E
inst✝³ : Algebra E K
inst✝² : Algebra F K
inst✝¹ : IsScalarTower F E K
L : Subalgebra F K
inst✝ : Module.Free F ↥L
⊢ Module.rank E ↥(Algebra.adjoin E ↑L) ≤ Module.rank F ↥L

Tactic:
rw [← rank_toSubmodule, Module.Free.rank_eq_card_chooseBasisIndex F L,
    L.adjoin_eq_span_basis E (Module.Free.chooseBasis F L)]

Goal state after:
F : Type u_5
E : Type u_6
K : Type u_7
inst✝⁹ : CommSemiring F
inst✝⁸ : StrongRankCondition F
inst✝⁷ : CommSemiring E
inst✝⁶ : StrongRankCondition E
inst✝⁵ : Semiring K
inst✝⁴ : SMul F E
inst✝³ : Algebra E K
inst✝² : Algebra F K
inst✝¹ : IsScalarTower F E K
L : Subalgebra F K
inst✝ : Module.Free F ↥L
⊢ Module.rank E ↥(Submodule.span E (Set.range fun i ↦ ↑((Module.Free.chooseBasis F ↥L) i))) ≤
    Cardinal.mk (Module.Free.ChooseBasisIndex F ↥L)
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Defs

Goal state before:
pp: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
ι : Sort u_3
f : ι → IntermediateField F E
⊢ ⨆ i, f i = adjoin F (⋃ i, ↑(f i))

Tactic:
adjoin_iUnion,

Goal state after:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
ι : Sort u_3
f : ι → IntermediateField F E
⊢ ⨆ i, f i = ⨆ i, adjoin F ↑(f i)
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
ι : Type u_3
i : ι → E
x : E
⊢ (∃ r, (∃ x, (MvPolynomial.aeval i) x = r) ∧ ∃ s, (∃ x, (MvPolynomial.aeval i) x = s) ∧ x = r / s) ↔
    ∃ r s, x = (MvPolynomial.aeval i) r / (MvPolynomial.aeval i) s

Tactic:
exists_exists_eq_and

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
s : Set (IntermediateField F E)
hne : s.Nonempty
⊢ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s → x ∈ ⨆ a, ↑a → ∃ x_1, ∃ (_ : x_1 ∈ s), x ∈ x_1

Tactic:
hs

Goal state after:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
s : Set (IntermediateField F E)
hne : s.Nonempty
hs : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ x ∈ ⨆ a, ↑a → ∃ x_1, ∃ (_ : x_1 ∈ s), x ∈ x_1
Module: Mathlib.LinearAlgebra.Eigenspace.Pi

Goal state before:
pp: case ind.inr
ι : Type u_1
K : Type u_3
inst✝³ : Field K
n : ℕ
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : FiniteDimensional K M
f : ι → End K M
h : ∀ (i j : ι) (φ : K), MapsTo ⇑(f i) ↑((f j).maxGenEigenspace φ) ↑((f j).maxGenEigenspace φ)
h' : ∀ (i : ι), ⨆ μ, (f i).maxGenEigenspace μ = ⊤
h_dim : finrank K M = n
i : ι
hy : ∀ (φ : K), finrank K ↥((f i).maxGenEigenspace φ) < n
hi : ∀ (j : ι) (φ : K), MapsTo ⇑(f j) ↑((f i).maxGenEigenspace φ) ↑((f i).maxGenEigenspace φ)
ih : ∀ (φ : K), ⨆ χ, ⨆ (_ : χ i = φ), ⨅ j, maxGenEigenspace (LinearMap.restrict (f j) ⋯) (χ j) = ⊤
⊢ ⨆ χ, ⨅ i, (f i).maxGenEigenspace (χ i) = ⊤

Tactic:
replace ih (φ : K) :
        ⨆ (χ : ι → K) (_ : χ i = φ), ⨅ j, maxGenEigenspace (f j) (χ j) =
        maxGenEigenspace (f i) φ := by
      have (χ : ι → K) (hχ : χ i = φ) : ⨅ j, maxGenEigenspace (f j) (χ j) =
          (⨅ j, maxGenEigenspace ((f j).restrict (hi j φ)) (χ j)).map
            ((f i).maxGenEigenspace φ).subtype := by
        rw [← hχ, iInf_maxGenEigenspace_restrict_map_subtype_eq]
      simp_rw [biSup_congr this, ← Submodule.map_iSup, ih, Submodule.map_top,
        Submodule.range_subtype]

Goal state after:
case ind.inr
ι : Type u_1
K : Type u_3
inst✝³ : Field K
n : ℕ
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : FiniteDimensional K M
f : ι → End K M
h : ∀ (i j : ι) (φ : K), MapsTo ⇑(f i) ↑((f j).maxGenEigenspace φ) ↑((f j).maxGenEigenspace φ)
h' : ∀ (i : ι), ⨆ μ, (f i).maxGenEigenspace μ = ⊤
h_dim : finrank K M = n
i : ι
hy : ∀ (φ : K), finrank K ↥((f i).maxGenEigenspace φ) < n
hi : ∀ (j : ι) (φ : K), MapsTo ⇑(f j) ↑((f i).maxGenEigenspace φ) ↑((f i).maxGenEigenspace φ)
ih : ∀ (φ : K), ⨆ χ, ⨆ (_ : χ i = φ), ⨅ j, (f j).maxGenEigenspace (χ j) = (f i).maxGenEigenspace φ
⊢ ⨆ χ, ⨅ i, (f i).maxGenEigenspace (χ i) = ⊤