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)
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

Tactic:
have restrict : ∑ ij ∈ R2N ×ˢ R2N with ij.1 ≤ N ∧ ij.2 ≤ N,
      ((ij.1 ! : ℚ)⁻¹ * (ij.2 ! : ℚ)⁻¹) • (a ^ ij.1 * b ^ ij.2) =
        ∑ ij ∈ RN ×ˢ RN, ((ij.1 ! : ℚ)⁻¹ * (ij.2 ! : ℚ)⁻¹) • (a ^ ij.1 * b ^ ij.2) := by
    apply sum_congr
    · ext x
      simp only [mem_filter, mem_product, mem_range, hR2N, hRN]
      lia
    · tauto

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)
restrict :
  ∑ ij ∈ R2N ×ˢ R2N with ij.1 ≤ N ∧ ij.2 ≤ N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) =
    ∑ ij ∈ RN ×ˢ RN, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i
Module: Mathlib.Algebra.Lie.Solvable

Goal state before:
pp: case succ
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
h : derivedSeries R L 1 = ⊤
n✝ : ℕ
⊢ derivedSeries R L (n✝ + 1) = ⊤

Tactic:
rwa [derivedSeries_succ_eq_top_iff]

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
inst✝ : LieModule R L M
N : LieSubmodule R L M
k : ℕ
⊢ lowerCentralSeries R L (↥N) k = ⊥ ↔ LieSubmodule.lcs k N = ⊥

Tactic:
N.lowerCentralSeries_eq_lcs_comap k,

Goal state after:
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
N : LieSubmodule R L M
k : ℕ
⊢ LieSubmodule.comap N.incl (LieSubmodule.lcs k N) = ⊥ ↔ LieSubmodule.lcs k N = ⊥
Module: Mathlib.FieldTheory.Finiteness

Goal state before:
pp: case mpr
K : Type u
V : Type v
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
b : Basis (↑(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V
hbfinite : (Basis.ofVectorSpaceIndex K V).Finite
⊢ span K (Basis.ofVectorSpaceIndex K V) = span K (Set.range Subtype.val)

Tactic:
Subtype.range_coe

Goal state after:
case mpr
K : Type u
V : Type v
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
b : Basis (↑(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V
hbfinite : (Basis.ofVectorSpaceIndex K V).Finite
⊢ span K (Basis.ofVectorSpaceIndex K V) = span K (Basis.ofVectorSpaceIndex K V)
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
⊢ IsEngelian R L

Tactic:
by
  intro M _i1 _i2 _i3 _i4 h
  rw [← isNilpotent_range_toEnd_iff R]
  let L' := (toEnd R L M).range
  replace h : ∀ y : L', IsNilpotent (y : Module.End R M) := by
    rintro ⟨-, ⟨y, rfl⟩⟩
    simp [h]
  change LieModule.IsNilpotent L' M
  let s := {K : LieSubalgebra R L' | LieAlgebra.IsEngelian R K}
  have hs : s.Nonempty := ⟨⊥, LieAlgebra.isEngelian_of_subsingleton⟩
  suffices ⊤ ∈ s by
    rw [← isNilpotent_of_top_iff (R := R)]
    apply this M
    simp [LieSubalgebra.toEnd_eq, h]
  have : ∀ K ∈ s, K ≠ ⊤ → ∃ K' ∈ s, K < K' := by
    rintro K (hK₁ : LieAlgebra.IsEngelian R K) hK₂
    apply LieAlgebra.exists_engelian_lieSubalgebra_of_lt_normalizer hK₁
    apply lt_of_le_of_ne K.le_normalizer
    rw [Ne, eq_comm, K.normalizer_eq_self_iff, ← Ne, ←
      LieSubmodule.nontrivial_iff_ne_bot R K]
    have : Nontrivial (L' ⧸ K.toLieSubmodule) := Submodule.Quotient.nontrivial_iff.2 <| by simpa
    have : LieModule.IsNilpotent K (L' ⧸ K.toLieSubmodule) := by
      refine hK₁ _ fun x => ?_
      have hx := LieAlgebra.isNilpotent_ad_of_isNilpotent (h x)
      apply Module.End.IsNilpotent.mapQ ?_ hx
      intro X HX
      simp only [LieSubalgebra.coe_toLieSubmodule, LieSubalgebra.mem_toSubmodule] at HX
      simp only [LieSubalgebra.coe_toLieSubmodule, Submodule.mem_comap, ad_apply,
        LieSubalgebra.mem_toSubmodule]
      exact LieSubalgebra.lie_mem K x.prop HX
    exact nontrivial_max_triv_of_isNilpotent R K (L' ⧸ K.toLieSubmodule)
  have _i5 : IsNoetherian R L' := by
    refine isNoetherian_of_surjective (LieHom.rangeRestrict (toEnd R L M)).toLinearMap ?_
    simp only [LinearMap.range_eq_top]
    exact LieHom.surjective_rangeRestrict (toEnd R L M)
  obtain ⟨K, hK₁, hK₂⟩ := (LieSubalgebra.wellFoundedGT_of_noetherian R L').wf.has_min s hs
  obtain rfl : K = ⊤ := by grind
  exact hK₁

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

Goal state before:
pp: R : Type u₁
L : Type u₂
M : Type u₄
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M
⊢ IsNilpotent L M ↔ ∀ (x : L), _root_.IsNilpotent ((toEnd R L M) x)

Tactic:
rw [← isNilpotent_range_toEnd_iff (R := R), LieModule.isNilpotent_iff_forall (R := R)]; simp

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

Goal state before:
pp: case h
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
a✝ : Algebra.EssFiniteType F E
x : E
x✝ : x ∈ ⊤
⊢ x ∈ adjoin F ↑(Algebra.EssFiniteType.finset F E)

Tactic:
obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq (Algebra.EssFiniteType.submonoid F E) x

Goal state after:
case h
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
a✝ : Algebra.EssFiniteType F E
x : ↥(Algebra.EssFiniteType.subalgebra F E)
s : ↥(Algebra.EssFiniteType.submonoid F E)
x✝ : IsLocalization.mk' E x s ∈ ⊤
⊢ IsLocalization.mk' E x s ∈ adjoin F ↑(Algebra.EssFiniteType.finset F E)
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra

Goal state before:
pp: F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : EssFiniteType F E
inst✝ : Algebra.IsAlgebraic F E
s : Finset E
hs : adjoin F ↑s = ⊤
⊢ Module.Finite F E

Tactic:
have : Algebra.FiniteType F E := by
    use s
    rw [← adjoin_toSubalgebra_of_isAlgebraic fun x hx ↦ Algebra.IsAlgebraic.isAlgebraic x]
    simpa [← toSubalgebra_inj] using hs

Goal state after:
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : EssFiniteType F E
inst✝ : Algebra.IsAlgebraic F E
s : Finset E
hs : adjoin F ↑s = ⊤
this : FiniteType F E
⊢ Module.Finite 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:
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.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
S : Set E
x : E
⊢ x ∈ adjoin F S ↔ x ∈ adjoin F (Set.range Subtype.val)

Tactic:
Subtype.range_coe

Goal state after:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
x : E
⊢ x ∈ adjoin F S ↔ x ∈ adjoin F S
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

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 : IsAlgClosed K
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
⊢ finrank K V = n → ⨆ μ, f.maxGenEigenspace μ = ⊤

Tactic:
h_dim

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 : IsAlgClosed K
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
h_dim : finrank K V = n
⊢ ⨆ μ, f.maxGenEigenspace μ = ⊤
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
⊢ 0 < finrank ?m.80 V

Tactic:
h_dim

Goal state after:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
⊢ 0 < n + 1
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
this : Nontrivial V
μ₀ : K
hμ₀ : f.HasEigenvalue μ₀
ES : Submodule K V := (f.genEigenspace μ₀) ↑(finrank K V)
ER : Submodule K V := f.genEigenrange μ₀ ↑(finrank K V)
h_f_ER : ∀ x ∈ ER, f x ∈ ER
f' : End K ↥ER := LinearMap.restrict f h_f_ER
⊢ 0 < finrank K ↥((f.genEigenspace μ₀) ↑(finrank K V))

Tactic:
h_dim

Goal state after:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
this : Nontrivial V
μ₀ : K
hμ₀ : f.HasEigenvalue μ₀
ES : Submodule K V := (f.genEigenspace μ₀) ↑(finrank K V)
ER : Submodule K V := f.genEigenrange μ₀ ↑(finrank K V)
h_f_ER : ∀ x ∈ ER, f x ∈ ER
f' : End K ↥ER := LinearMap.restrict f h_f_ER
⊢ 0 < finrank K ↥((f.genEigenspace μ₀) ↑(n + 1))
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
S : Finset E
⊢ IsCompactElement (adjoin F ↑S)

Tactic:
rw [← biSup_adjoin_simple]
  simp_rw [Finset.mem_coe, ← Finset.sup_eq_iSup]
  exact isCompactElement_finsetSup S fun x _ => adjoin_simple_isCompactElement x

Goal state after:
Module: Mathlib.FieldTheory.Extension

Goal state before:
pp: case refine_2
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Field K
inst✝⁵ : Algebra F E
inst✝⁴ : Algebra F K
S : Set E
L : Type u_4
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
f : L →ₐ[F] K
hK : ∀ s ∈ S, IsIntegral L s ∧ (Polynomial.map f.toRingHom (minpoly L s)).Splits
L' : IntermediateField F E := (IsScalarTower.toAlgHom F L E).fieldRange
f' : ↥L' →ₐ[F] K := f.comp ↑(AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F L E)).symm
this : ∃ φ, AlgHom.domRestrict (↥L') φ = f'
⊢ ∃ φ, AlgHom.domRestrict L φ = f

Tactic:
obtain ⟨φ, hφ⟩ := this

Goal state after:
case refine_2
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Field K
inst✝⁵ : Algebra F E
inst✝⁴ : Algebra F K
S : Set E
L : Type u_4
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
f : L →ₐ[F] K
hK : ∀ s ∈ S, IsIntegral L s ∧ (Polynomial.map f.toRingHom (minpoly L s)).Splits
L' : IntermediateField F E := (IsScalarTower.toAlgHom F L E).fieldRange
f' : ↥L' →ₐ[F] K := f.comp ↑(AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F L E)).symm
φ : ↥(adjoin (↥L') S) →ₐ[F] K
hφ : AlgHom.domRestrict (↥L') φ = f'
⊢ ∃ φ, AlgHom.domRestrict L φ = f
Module: Mathlib.Algebra.Lie.Weights.Cartan

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H : LieSubalgebra R L
inst✝ : LieRing.IsNilpotent ↥H
χ₁ χ₂ χ₃ : ↥H → R
hχ : χ₁ + χ₂ = χ₃
x : ↥(rootSpace H χ₁)
y : ↥(rootSpace H χ₂)
⊢ ↑((rootSpaceProduct R L H χ₁ χ₂ χ₃ hχ) (x ⊗ₜ[R] y)) = ⁅↑x, ↑y⁆

Tactic:
by
  simp only [rootSpaceProduct_def, coe_rootSpaceWeightSpaceProduct_tmul]

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Basic

Goal state before:
pp: R : 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 : LieRing.IsNilpotent L
inst✝ : IsNoetherian R M
χ : L → R
x : L
⊢ ∃ k, ↑(genWeightSpace M χ) ≤ LinearMap.ker (((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ^ k)

Tactic:
use (toEnd R L M x).maxGenEigenspaceIndex (χ x)
  intro m hm
  replace hm : m ∈ (toEnd R L M x).maxGenEigenspace (χ x) :=
    genWeightSpace_le_genWeightSpaceOf M x χ hm
  rwa [Module.End.maxGenEigenspace_eq, Module.End.genEigenspace_nat] at hm

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Basic

Goal state before:
pp: R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : LieRing.IsNilpotent L
x : L
m : M
⊢ m ∈ posFittingCompOf R M x ↔ ∀ (k : ℕ), ∃ n, ((toEnd R L M) x ^ k) n = m

Tactic:
simp [posFittingCompOf]

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Basic

Goal state before:
pp: case inr.h
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : LieRingModule L M
inst✝³ : LieModule R L M
inst✝² : LieRing.IsNilpotent L
inst✝¹ : IsNoetherian R M
inst✝ : IsArtinian R M
h : ∀ N < ⊤, IsCompl (genWeightSpace (↥N) 0) (posFittingComp R L ↥N)
M₀ : LieSubmodule R L M := ⋯
M₁ : LieSubmodule R L M := ⋯
x : L
M₀ₓ : LieSubmodule R L M := ⋯
hx : M₀ₓ ≠ ⊤
M₁ₓ : LieSubmodule R L M := ⋯
M₀ₓ₀ : LieSubmodule R L ↥M₀ₓ := ⋯
M₀ₓ₁ : LieSubmodule R L ↥M₀ₓ := ⋯
h₁ : IsCompl M₀ₓ M₁ₓ
h₂ : IsCompl M₀ₓ₀ M₀ₓ₁
h₃ : LieSubmodule.map M₀ₓ.incl M₀ₓ₀ = M₀
h₄ : LieSubmodule.map M₀ₓ.incl M₀ₓ₁ ⊔ M₁ₓ = M₁
⊢ Disjoint (LieSubmodule.map M₀ₓ.incl M₀ₓ₀) (LieSubmodule.map M₀ₓ.incl M₀ₓ₁)

Tactic:
rw [disjoint_iff, ← LieSubmodule.map_inf M₀ₓ.injective_incl, h₂.inf_eq_bot,
        LieSubmodule.map_bot]

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Basic

Goal state before:
pp: K : Type u_1
R : Type u_2
L : 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 : LieRing.IsNilpotent L
inst✝² : IsDomain R
inst✝¹ : Module.IsTorsionFree R M
inst✝ : IsNoetherian R M
⊢ Finite (Weight R L M)

Tactic:
have : Finite {χ : L → R | genWeightSpace M χ ≠ ⊥} := finite_genWeightSpace_ne_bot R L M

Goal state after:
K : Type u_1
R : Type u_2
L : 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 : LieRing.IsNilpotent L
inst✝² : IsDomain R
inst✝¹ : Module.IsTorsionFree R M
inst✝ : IsNoetherian R M
this : Finite ↑{χ | genWeightSpace M χ ≠ ⊥}
⊢ Finite (Weight R L M)
Module: Mathlib.Algebra.Lie.Weights.Basic

Goal state before:
pp: R : 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 : Nontrivial M
inst✝ : IsTriangularizable R L M
x : L
this : ∃ φ, ((toEnd R L M) x).maxGenEigenspace φ ≠ ⊥
⊢ ∃ φ, ((toEnd R L M) x).HasEigenvalue φ

Tactic:
obtain ⟨φ, hφ⟩ := this

Goal state after:
R : 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 : Nontrivial M
inst✝ : IsTriangularizable R L M
x : L
φ : R
hφ : ((toEnd R L M) x).maxGenEigenspace φ ≠ ⊥
⊢ ∃ φ, ((toEnd R L M) x).HasEigenvalue φ
Module: Mathlib.Algebra.Lie.Semisimple.Basic

Goal state before:
pp: case a
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
I : LieIdeal R L
hI : IsAtom I
J : LieIdeal R ↥I
y : ↥I
hy : y ∈ ↑↑J
a : L
ha : a ∈ I
b : L
hb : b ∈ sSup ({I' | IsAtom I'} \ {I})
⊢ ⁅b, ↑y⁆ = 0

Tactic:
rw [← LieSubmodule.mem_bot (R := R) (L := L),
              ← (IsSemisimple.sSupIndep_isAtom hI).eq_bot]

Goal state after:
case a
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
I : LieIdeal R L
hI : IsAtom I
J : LieIdeal R ↥I
y : ↥I
hy : y ∈ ↑↑J
a : L
ha : a ∈ I
b : L
hb : b ∈ sSup ({I' | IsAtom I'} \ {I})
⊢ ⁅b, ↑y⁆ ∈ I ⊓ sSup ({I | IsAtom I} \ {I})
Module: Mathlib.LinearAlgebra.Eigenspace.Semisimple

Goal state before:
pp: case mpr.inr
K : Type u_3
V : Type u_4
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
hf : f.IsSemisimple
h : ∀ (μ : K), f.HasEigenvalue μ → μ = 0
μ : K
hμ : μ ≠ 0
⊢ f.eigenspace μ ≤ f.eigenspace 0

Tactic:
have : f.eigenspace μ = ⊥ := not_not.mp (hasEigenvalue_iff.not.mp fun he ↦ hμ (h μ he))
      simp [this]

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Chain

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
M : Type u_3
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : LieRing.IsNilpotent L
inst✝³ : IsAddTorsionFree R
inst✝² : IsDomain R
inst✝¹ : IsTorsionFree R M
inst✝ : IsNoetherian R M
α : L → R
β : Weight R L M
hα : α ≠ 0
hα' : genWeightSpace M α ≠ ⊥
e : (chainTop α β).IsZero
⊢ genWeightSpace M (α + 0) = ⊥

Tactic:
← e,

Goal state after:
R : Type u_1
L : Type u_2
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
M : Type u_3
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : LieRing.IsNilpotent L
inst✝³ : IsAddTorsionFree R
inst✝² : IsDomain R
inst✝¹ : IsTorsionFree R M
inst✝ : IsNoetherian R M
α : L → R
β : Weight R L M
hα : α ≠ 0
hα' : genWeightSpace M α ≠ ⊥
e : (chainTop α β).IsZero
⊢ genWeightSpace M (α + ⇑(chainTop α β)) = ⊥
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: R : Type u_1
L : Type u_3
M : Type u_4
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
x y : L
k : ℕ
hx : x ∈ ⊤.lcs L k
hy : y ∈ LieSubmodule.ucs k ⊥
⊢ ((traceForm R L M) x) y = 0

Tactic:
induction k generalizing x y with
  | zero =>
    replace hy : y = 0 := by simpa using hy
    simp [hy]
  | succ k ih =>
    rw [LieSubmodule.ucs_succ, LieSubmodule.mem_normalizer] at hy
    simp_rw [LieIdeal.lcs_succ, ← LieSubmodule.mem_toSubmodule,
      LieSubmodule.lieIdeal_oper_eq_linear_span', LieSubmodule.mem_top, true_and] at hx
    refine Submodule.span_induction ?_ ?_ (fun z w _ _ hz hw ↦ ?_) (fun t z _ hz ↦ ?_) hx
    · rintro - ⟨z, w, hw, rfl⟩
      rw [← lie_skew, map_neg, LinearMap.neg_apply, neg_eq_zero, traceForm_apply_lie_apply]
      exact ih hw (hy _)
    · simp
    · simp [hz, hw]
    · simp [hz]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Sequence

Goal state before:
pp: case h₁
R : Type u_1
inst✝ : Ring R
S : Sequence R
m : ℕ
hCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff
i : ℕ
hi : i ∈ Set.Iio m
hP : ↑S i ∈ ↑S '' Set.Iio m
⊢ ↑S i ∈ ↑(degreeLT R m)

Tactic:
SetLike.mem_coe,

Goal state after:
case h₁
R : Type u_1
inst✝ : Ring R
S : Sequence R
m : ℕ
hCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff
i : ℕ
hi : i ∈ Set.Iio m
hP : ↑S i ∈ ↑S '' Set.Iio m
⊢ ↑S i ∈ degreeLT R m
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : LieRing.IsNilpotent L
B : LinearMap.BilinForm R M
m₀ m₁ : M
hm₀ : m₀ ∈ genWeightSpace M 0
hm₁ : m₁ ∈ posFittingComp R L M
hB : ∀ (x : L) (k : ℕ) (m n : M), (B m) ((φ x ^ k) n) = (-1) ^ k • (B ((φ x ^ k) m)) n
this : ∀ (x : L), ∀ m ∈ posFittingCompOf R M x, (B m₀) m = 0
⊢ (B m₀) m₁ = 0

Tactic:
by
    refine LieSubmodule.iSup_induction (motive := fun m ↦ (B m₀) m = 0) _ hm₁ this (map_zero _) ?_
    simp_all

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Killing

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
e : L
heα : e ∈ rootSpace H ⇑α
f : L
hfα : f ∈ rootSpace H (-⇑α)
⊢ ((killingForm K L) e) f • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ∈
    ↑(K ∙ ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)))

Tactic:
SetLike.mem_coe,

Goal state after:
K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
e : L
heα : e ∈ rootSpace H ⇑α
f : L
hfα : f ∈ rootSpace H (-⇑α)
⊢ ((killingForm K L) e) f • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ∈
    K ∙ ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))