ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LieModule.trivial_iff_lower_central_eq_bot
theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries R L M 1 = ⊥ 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M]
Goal: LieModule.IsTrivial L M ↔ LieModule.lowerCentralSeries R L M 1 = ⊥
-- LieModule.iterate_toEnd_mem_lowerCentralSeries
theorem iterate_toEnd_mem_lowerCentralSeries (x : L) (m : M) (k : ℕ) :
    (toEnd R L M x)^[k] m ∈ lowerCentralSeries R L M k 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L) (m : M) (k : ℕ)
Goal: (⇑((LieModule.toEnd R L M) x))^[k] m ∈ LieModule.lowerCentralSeries R L M k
-- LieModule.iterate_toEnd_mem_lowerCentralSeries₂
theorem iterate_toEnd_mem_lowerCentralSeries₂ (x y : L) (m : M) (k : ℕ) :
    (toEnd R L M x ∘ₗ toEnd R L M y)^[k] m ∈
      lowerCentralSeries R L M (2 * k) 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x y : L) (m : M) (k : ℕ)
Goal: (⇑((LieModule.toEnd R L M) x ∘ₗ (LieModule.toEnd R L M) y))^[k] m ∈ LieModule.lowerCentralSeries R L M (2 * k)
-- LieModule.map_lowerCentralSeries_le
theorem map_lowerCentralSeries_le (f : M →ₗ⁅R,L⁆ M₂) :
    (lowerCentralSeries R L M k).map f ≤ lowerCentralSeries R L M₂ k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (k : ℕ) {M₂ : Type w₁} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] (f : M →ₗ⁅R,L⁆ M₂)
Goal: LieSubmodule.map f (LieModule.lowerCentralSeries R L M k) ≤ LieModule.lowerCentralSeries R L M₂ k
-- LieModule.map_lowerCentralSeries_eq
lemma map_lowerCentralSeries_eq {f : M →ₗ⁅R,L⁆ M₂} (hf : Function.Surjective f) :
    (lowerCentralSeries R L M k).map f = lowerCentralSeries R L M₂ k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (k : ℕ) {M₂ : Type w₁} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] {f : M →ₗ⁅R,L⁆ M₂} (hf : Function.Surjective ⇑f)
Goal: LieSubmodule.map f (LieModule.lowerCentralSeries R L M k) = LieModule.lowerCentralSeries R L M₂ k
-- LieModule.derivedSeries_le_lowerCentralSeries
theorem derivedSeries_le_lowerCentralSeries (k : ℕ) :
    derivedSeries R L k ≤ lowerCentralSeries R L L k 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] (k : ℕ)
Goal: LieAlgebra.derivedSeries R L k ≤ LieModule.lowerCentralSeries R L L k
-- LieModule.IsNilpotent.nilpotent
class IsNilpotent : Prop where
  nilpotent : ∃ k, lowerCentralSeries R L M k = ⊥
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [self : LieModule.IsNilpotent R L M]
Goal: ∃ k, LieModule.lowerCentralSeries R L M k = ⊥
-- LieModule.exists_lowerCentralSeries_eq_bot_of_isNilpotent
theorem exists_lowerCentralSeries_eq_bot_of_isNilpotent [IsNilpotent R L M] :
    ∃ k, lowerCentralSeries R L M k = ⊥ 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule.IsNilpotent R L M]
Goal: ∃ k, LieModule.lowerCentralSeries R L M k = ⊥
-- LieModule.iInf_lowerCentralSeries_eq_bot_of_isNilpotent
@[simp] lemma iInf_lowerCentralSeries_eq_bot_of_isNilpotent [IsNilpotent R L M] :
    ⨅ k, lowerCentralSeries R L M k = ⊥ 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule.IsNilpotent R L M]
Goal: ⨅ k, LieModule.lowerCentralSeries R L M k = ⊥
-- LieModule.isNilpotent_iff
theorem isNilpotent_iff : IsNilpotent R L M ↔ ∃ k, lowerCentralSeries R L M k = ⊥ 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M]
Goal: LieModule.IsNilpotent R L M ↔ ∃ k, LieModule.lowerCentralSeries R L M k = ⊥
-- LieSubmodule.isNilpotent_iff_exists_lcs_eq_bot
theorem _root_.LieSubmodule.isNilpotent_iff_exists_lcs_eq_bot (N : LieSubmodule R L M) :
    LieModule.IsNilpotent R L N ↔ ∃ k, N.lcs k = ⊥ 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M)
Goal: LieModule.IsNilpotent R L ↥↑N ↔ ∃ k, LieSubmodule.lcs k N = ⊥
-- LieModule.exists_forall_pow_toEnd_eq_zero
theorem exists_forall_pow_toEnd_eq_zero [hM : IsNilpotent R L M] :
    ∃ k : ℕ, ∀ x : L, toEnd R L M x ^ k = 0 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [hM : LieModule.IsNilpotent R L M]
Goal: ∃ k, ∀ (x : L), (LieModule.toEnd R L M) x ^ k = 0
-- LieModule.isNilpotent_toEnd_of_isNilpotent
theorem isNilpotent_toEnd_of_isNilpotent [IsNilpotent R L M] (x : L) :
    _root_.IsNilpotent (toEnd R L M x) 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieModule.IsNilpotent R L M] (x : L)
Goal: IsNilpotent ((LieModule.toEnd R L M) x)
-- LieModule.isNilpotent_toEnd_of_isNilpotent₂
theorem isNilpotent_toEnd_of_isNilpotent₂ [IsNilpotent R L M] (x y : L) :
    _root_.IsNilpotent (toEnd R L M x ∘ₗ toEnd R L M y) 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieModule.IsNilpotent R L M] (x y : L)
Goal: IsNilpotent ((LieModule.toEnd R L M) x ∘ₗ (LieModule.toEnd R L M) y)
-- LieModule.maxGenEigenSpace_toEnd_eq_top
@[simp] lemma maxGenEigenSpace_toEnd_eq_top [IsNilpotent R L M] (x : L) :
    ((toEnd R L M x).maxGenEigenspace 0) = ⊤ 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieModule.IsNilpotent R L M] (x : L)
Goal: ((LieModule.toEnd R L M) x).maxGenEigenspace 0 = ⊤
-- LieModule.nilpotentOfNilpotentQuotient
theorem nilpotentOfNilpotentQuotient {N : LieSubmodule R L M} (h₁ : N ≤ maxTrivSubmodule R L M)
    (h₂ : IsNilpotent R L (M ⧸ N)) : IsNilpotent R L M 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {N : LieSubmodule R L M} (h₁ : N ≤ LieModule.maxTrivSubmodule R L M) (h₂ : LieModule.IsNilpotent R L (M ⧸ N))
Goal: LieModule.IsNilpotent R L M
-- LieModule.isNilpotent_quotient_iff
theorem isNilpotent_quotient_iff :
    IsNilpotent R L (M ⧸ N) ↔ ∃ k, lowerCentralSeries R L M k ≤ N 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M)
Goal: LieModule.IsNilpotent R L (M ⧸ N) ↔ ∃ k, LieModule.lowerCentralSeries R L M k ≤ N
-- LieModule.iInf_lcs_le_of_isNilpotent_quot
theorem iInf_lcs_le_of_isNilpotent_quot (h : IsNilpotent R L (M ⧸ N)) :
    ⨅ k, lowerCentralSeries R L M k ≤ N 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (h : LieModule.IsNilpotent R L (M ⧸ N))
Goal: ⨅ k, LieModule.lowerCentralSeries R L M k ≤ N
-- LieModule.nilpotencyLength_eq_zero_iff
@[simp]
theorem nilpotencyLength_eq_zero_iff [IsNilpotent R L M] :
    nilpotencyLength R L M = 0 ↔ Subsingleton M 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule.IsNilpotent R L M]
Goal: LieModule.nilpotencyLength R L M = 0 ↔ Subsingleton M
-- LieModule.nilpotencyLength_eq_succ_iff
theorem nilpotencyLength_eq_succ_iff (k : ℕ) :
    nilpotencyLength R L M = k + 1 ↔
      lowerCentralSeries R L M (k + 1) = ⊥ ∧ lowerCentralSeries R L M k ≠ ⊥ 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] (k : ℕ)
Goal: LieModule.nilpotencyLength R L M = k + 1 ↔
  LieModule.lowerCentralSeries R L M (k + 1) = ⊥ ∧ LieModule.lowerCentralSeries R L M k ≠ ⊥
-- LieModule.nilpotencyLength_eq_one_iff
@[simp]
theorem nilpotencyLength_eq_one_iff [Nontrivial M] :
    nilpotencyLength R L M = 1 ↔ IsTrivial L M 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [Nontrivial M]
Goal: LieModule.nilpotencyLength R L M = 1 ↔ LieModule.IsTrivial L M
-- LieModule.isTrivial_of_nilpotencyLength_le_one
theorem isTrivial_of_nilpotencyLength_le_one [IsNilpotent R L M] (h : nilpotencyLength R L M ≤ 1) :
    IsTrivial L M 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule.IsNilpotent R L M] (h : LieModule.nilpotencyLength R L M ≤ 1)
Goal: LieModule.IsTrivial L M
-- LieModule.lowerCentralSeriesLast_le_max_triv
theorem lowerCentralSeriesLast_le_max_triv :
    lowerCentralSeriesLast R L M ≤ maxTrivSubmodule R L M 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: LieModule.lowerCentralSeriesLast R L M ≤ LieModule.maxTrivSubmodule R L M
-- LieModule.nontrivial_lowerCentralSeriesLast
theorem nontrivial_lowerCentralSeriesLast [Nontrivial M] [IsNilpotent R L M] :
    Nontrivial (lowerCentralSeriesLast R L M) 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [Nontrivial M] [LieModule.IsNilpotent R L M]
Goal: Nontrivial ↥↑(LieModule.lowerCentralSeriesLast R L M)
-- LieModule.lowerCentralSeriesLast_le_of_not_isTrivial
theorem lowerCentralSeriesLast_le_of_not_isTrivial [IsNilpotent R L M] (h : ¬ IsTrivial L M) :
    lowerCentralSeriesLast R L M ≤ lowerCentralSeries R L M 1 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule.IsNilpotent R L M] (h : ¬LieModule.IsTrivial L M)
Goal: LieModule.lowerCentralSeriesLast R L M ≤ LieModule.lowerCentralSeries R L M 1
-- LieModule.disjoint_lowerCentralSeries_maxTrivSubmodule_iff
lemma disjoint_lowerCentralSeries_maxTrivSubmodule_iff [IsNilpotent R L M] :
    Disjoint (lowerCentralSeries R L M 1) (maxTrivSubmodule R L M) ↔ IsTrivial L M 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieModule.IsNilpotent R L M]
Goal: Disjoint (LieModule.lowerCentralSeries R L M 1) (LieModule.maxTrivSubmodule R L M) ↔ LieModule.IsTrivial L M
-- LieModule.nontrivial_max_triv_of_isNilpotent
theorem nontrivial_max_triv_of_isNilpotent [Nontrivial M] [IsNilpotent R L M] :
    Nontrivial (maxTrivSubmodule R L M) 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Nontrivial M] [LieModule.IsNilpotent R L M]
Goal: Nontrivial ↥↑(LieModule.maxTrivSubmodule R L M)
-- LieModule.coe_lcs_range_toEnd_eq
@[simp]
theorem coe_lcs_range_toEnd_eq (k : ℕ) :
    (lowerCentralSeries R (toEnd R L M).range M k : Submodule R M) =
      lowerCentralSeries R L M k 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (k : ℕ)
Goal: ↑(LieModule.lowerCentralSeries R (↥(LieModule.toEnd R L M).range) M k) = ↑(LieModule.lowerCentralSeries R L M k)
-- LieModule.isNilpotent_range_toEnd_iff
@[simp]
theorem isNilpotent_range_toEnd_iff :
    IsNilpotent R (toEnd R L M).range M ↔ IsNilpotent R L M 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: LieModule.IsNilpotent R (↥(LieModule.toEnd R L M).range) M ↔ LieModule.IsNilpotent R L M
-- LieSubmodule.ucs_zero
@[simp]
theorem ucs_zero : N.ucs 0 = N 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M)
Goal: LieSubmodule.ucs 0 N = N
-- LieSubmodule.ucs_succ
@[simp]
theorem ucs_succ (k : ℕ) : N.ucs (k + 1) = (N.ucs k).normalizer 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (k : ℕ)
Goal: LieSubmodule.ucs (k + 1) N = (LieSubmodule.ucs k N).normalizer
-- LieSubmodule.ucs_add
theorem ucs_add (k l : ℕ) : N.ucs (k + l) = (N.ucs l).ucs k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (k l : ℕ)
Goal: LieSubmodule.ucs (k + l) N = LieSubmodule.ucs k (LieSubmodule.ucs l N)
-- LieSubmodule.ucs_mono
@[gcongr, mono]
theorem ucs_mono (k : ℕ) (h : N₁ ≤ N₂) : N₁.ucs k ≤ N₂.ucs k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {N₁ N₂ : LieSubmodule R L M} (k : ℕ) (h : N₁ ≤ N₂)
Goal: LieSubmodule.ucs k N₁ ≤ LieSubmodule.ucs k N₂
-- LieSubmodule.ucs_eq_self_of_normalizer_eq_self
theorem ucs_eq_self_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) : N₁.ucs k = N₁ 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {N₁ : LieSubmodule R L M} (h : N₁.normalizer = N₁) (k : ℕ)
Goal: LieSubmodule.ucs k N₁ = N₁
-- LieSubmodule.ucs_le_of_normalizer_eq_self
theorem ucs_le_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
    (⊥ : LieSubmodule R L M).ucs k ≤ N₁ 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {N₁ : LieSubmodule R L M} (h : N₁.normalizer = N₁) (k : ℕ)
Goal: LieSubmodule.ucs k ⊥ ≤ N₁
-- LieSubmodule.lcs_add_le_iff
theorem lcs_add_le_iff (l k : ℕ) : N₁.lcs (l + k) ≤ N₂ ↔ N₁.lcs l ≤ N₂.ucs k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {N₁ N₂ : LieSubmodule R L M} (l k : ℕ)
Goal: LieSubmodule.lcs (l + k) N₁ ≤ N₂ ↔ LieSubmodule.lcs l N₁ ≤ LieSubmodule.ucs k N₂
-- LieSubmodule.lcs_le_iff
theorem lcs_le_iff (k : ℕ) : N₁.lcs k ≤ N₂ ↔ N₁ ≤ N₂.ucs k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {N₁ N₂ : LieSubmodule R L M} (k : ℕ)
Goal: LieSubmodule.lcs k N₁ ≤ N₂ ↔ N₁ ≤ LieSubmodule.ucs k N₂
-- LieSubmodule.gc_lcs_ucs
theorem gc_lcs_ucs (k : ℕ) :
    GaloisConnection (fun N : LieSubmodule R L M => N.lcs k) fun N : LieSubmodule R L M =>
      N.ucs k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (k : ℕ)
Goal: GaloisConnection (fun N => LieSubmodule.lcs k N) fun N => LieSubmodule.ucs k N
-- LieSubmodule.ucs_eq_top_iff
theorem ucs_eq_top_iff (k : ℕ) : N.ucs k = ⊤ ↔ LieModule.lowerCentralSeries R L M k ≤ N 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (k : ℕ)
Goal: LieSubmodule.ucs k N = ⊤ ↔ LieModule.lowerCentralSeries R L M k ≤ N
-- LieModule.isNilpotent_iff_exists_ucs_eq_top
theorem _root_.LieModule.isNilpotent_iff_exists_ucs_eq_top :
    LieModule.IsNilpotent R L M ↔ ∃ k, (⊥ : LieSubmodule R L M).ucs k = ⊤ 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: LieModule.IsNilpotent R L M ↔ ∃ k, LieSubmodule.ucs k ⊥ = ⊤
-- LieSubmodule.ucs_comap_incl
theorem ucs_comap_incl (k : ℕ) :
    ((⊥ : LieSubmodule R L M).ucs k).comap N.incl = (⊥ : LieSubmodule R L N).ucs k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (k : ℕ)
Goal: LieSubmodule.comap N.incl (LieSubmodule.ucs k ⊥) = LieSubmodule.ucs k ⊥
-- LieSubmodule.isNilpotent_iff_exists_self_le_ucs
theorem isNilpotent_iff_exists_self_le_ucs :
    LieModule.IsNilpotent R L N ↔ ∃ k, N ≤ (⊥ : LieSubmodule R L M).ucs k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M)
Goal: LieModule.IsNilpotent R L ↥↑N ↔ ∃ k, N ≤ LieSubmodule.ucs k ⊥
-- LieSubmodule.ucs_bot_one
theorem ucs_bot_one : (⊥ : LieSubmodule R L M).ucs 1 = LieModule.maxTrivSubmodule R L M 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: LieSubmodule.ucs 1 ⊥ = LieModule.maxTrivSubmodule R L M
-- Function.Surjective.lieModule_lcs_map_eq
theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
    (lowerCentralSeries R L M k : Submodule R M).map g = lowerCentralSeries R L₂ M₂ k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {L₂ : Type u_1} {M₂ : Type u_2} [LieRing L₂] [LieAlgebra R L₂] [AddCommGroup M₂] [Module R M₂] [LieRingModule L₂ M₂] [LieModule R L₂ M₂] {f : L →ₗ⁅R⁆ L₂} {g : M →ₗ[R] M₂} (hf : Function.Surjective ⇑f) (hg : Function.Surjective ⇑g) (hfg : ∀ (x : L) (m : M), ⁅f x, g m⁆ = g ⁅x, m⁆) (k : ℕ)
Goal: Submodule.map g ↑(LieModule.lowerCentralSeries R L M k) = ↑(LieModule.lowerCentralSeries R L₂ M₂ k)
-- Function.Surjective.lieModuleIsNilpotent
theorem Function.Surjective.lieModuleIsNilpotent [IsNilpotent R L M] : IsNilpotent R L₂ M₂ 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {L₂ : Type u_1} {M₂ : Type u_2} [LieRing L₂] [LieAlgebra R L₂] [AddCommGroup M₂] [Module R M₂] [LieRingModule L₂ M₂] [LieModule R L₂ M₂] {f : L →ₗ⁅R⁆ L₂} {g : M →ₗ[R] M₂} (hf : Function.Surjective ⇑f) (hg : Function.Surjective ⇑g) (hfg : ∀ (x : L) (m : M), ⁅f x, g m⁆ = g ⁅x, m⁆) [LieModule.IsNilpotent R L M]
Goal: LieModule.IsNilpotent R L₂ M₂
-- Equiv.lieModule_isNilpotent_iff
theorem Equiv.lieModule_isNilpotent_iff (f : L ≃ₗ⁅R⁆ L₂) (g : M ≃ₗ[R] M₂)
    (hfg : ∀ x m, ⁅f x, g m⁆ = g ⁅x, m⁆) : IsNilpotent R L M ↔ IsNilpotent R L₂ M₂ 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {L₂ : Type u_1} {M₂ : Type u_2} [LieRing L₂] [LieAlgebra R L₂] [AddCommGroup M₂] [Module R M₂] [LieRingModule L₂ M₂] [LieModule R L₂ M₂] (f : L ≃ₗ⁅R⁆ L₂) (g : M ≃ₗ[R] M₂) (hfg : ∀ (x : L) (m : M), ⁅f x, g m⁆ = g ⁅x, m⁆)
Goal: LieModule.IsNilpotent R L M ↔ LieModule.IsNilpotent R L₂ M₂
-- LieModule.isNilpotent_of_top_iff
@[simp]
theorem LieModule.isNilpotent_of_top_iff :
    IsNilpotent R (⊤ : LieSubalgebra R L) M ↔ IsNilpotent R L M 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: LieModule.IsNilpotent R (↥⊤) M ↔ LieModule.IsNilpotent R L M
-- LieModule.isNilpotent_of_top_iff'
@[simp] lemma LieModule.isNilpotent_of_top_iff' :
    IsNilpotent R L {x // x ∈ (⊤ : LieSubmodule R L M)} ↔ IsNilpotent R L M 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: LieModule.IsNilpotent R L ↥⊤ ↔ LieModule.IsNilpotent R L M
-- LieAlgebra.nilpotent_ad_of_nilpotent_algebra
theorem LieAlgebra.nilpotent_ad_of_nilpotent_algebra [IsNilpotent R L] :
    ∃ k : ℕ, ∀ x : L, ad R L x ^ k = 0 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L]
Goal: ∃ k, ∀ (x : L), (LieAlgebra.ad R L) x ^ k = 0
-- coe_lowerCentralSeries_ideal_quot_eq
-- Porting note: added `LieSubmodule.toSubmodule` in the statement
theorem coe_lowerCentralSeries_ideal_quot_eq {I : LieIdeal R L} (k : ℕ) :
    LieSubmodule.toSubmodule (lowerCentralSeries R L (L ⧸ I) k) =
      LieSubmodule.toSubmodule (lowerCentralSeries R (L ⧸ I) (L ⧸ I) k) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {I : LieIdeal R L} (k : ℕ)
Goal: ↑(LieModule.lowerCentralSeries R L (L ⧸ I) k) = ↑(LieModule.lowerCentralSeries R (L ⧸ I) (L ⧸ I) k)
-- LieModule.coe_lowerCentralSeries_ideal_le
-- Porting note: added `LieSubmodule.toSubmodule` in the statement
theorem LieModule.coe_lowerCentralSeries_ideal_le {I : LieIdeal R L} (k : ℕ) :
    LieSubmodule.toSubmodule (lowerCentralSeries R I I k) ≤ lowerCentralSeries R L I k 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {I : LieIdeal R L} (k : ℕ)
Goal: ↑(LieModule.lowerCentralSeries R (↥↑I) (↥↑I) k) ≤ ↑(LieModule.lowerCentralSeries R L (↥↑I) k)
-- LieAlgebra.nilpotent_of_nilpotent_quotient
theorem LieAlgebra.nilpotent_of_nilpotent_quotient {I : LieIdeal R L} (h₁ : I ≤ center R L)
    (h₂ : IsNilpotent R (L ⧸ I)) : IsNilpotent R L 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {I : LieIdeal R L} (h₁ : I ≤ LieAlgebra.center R L) (h₂ : LieAlgebra.IsNilpotent R (L ⧸ I))
Goal: LieAlgebra.IsNilpotent R L
-- LieAlgebra.non_trivial_center_of_isNilpotent
theorem LieAlgebra.non_trivial_center_of_isNilpotent [Nontrivial L] [IsNilpotent R L] :
    Nontrivial <| center R L 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] [Nontrivial L] [LieAlgebra.IsNilpotent R L]
Goal: Nontrivial ↥↑(LieAlgebra.center R L)
-- LieIdeal.map_lowerCentralSeries_le
theorem LieIdeal.map_lowerCentralSeries_le (k : ℕ) {f : L →ₗ⁅R⁆ L'} :
    LieIdeal.map f (lowerCentralSeries R L L k) ≤ lowerCentralSeries R L' L' k 
Arguments: {R : Type u} {L : Type v} {L' : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (k : ℕ) {f : L →ₗ⁅R⁆ L'}
Goal: LieIdeal.map f (LieModule.lowerCentralSeries R L L k) ≤ LieModule.lowerCentralSeries R L' L' k
-- LieIdeal.lowerCentralSeries_map_eq
theorem LieIdeal.lowerCentralSeries_map_eq (k : ℕ) {f : L →ₗ⁅R⁆ L'} (h : Function.Surjective f) :
    LieIdeal.map f (lowerCentralSeries R L L k) = lowerCentralSeries R L' L' k 
Arguments: {R : Type u} {L : Type v} {L' : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (k : ℕ) {f : L →ₗ⁅R⁆ L'} (h : Function.Surjective ⇑f)
Goal: LieIdeal.map f (LieModule.lowerCentralSeries R L L k) = LieModule.lowerCentralSeries R L' L' k
-- Function.Injective.lieAlgebra_isNilpotent
theorem Function.Injective.lieAlgebra_isNilpotent [h₁ : IsNilpotent R L'] {f : L →ₗ⁅R⁆ L'}
    (h₂ : Function.Injective f) : IsNilpotent R L 
Arguments: {R : Type u} {L : Type v} {L' : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] [h₁ : LieAlgebra.IsNilpotent R L'] {f : L →ₗ⁅R⁆ L'} (h₂ : Function.Injective ⇑f)
Goal: LieAlgebra.IsNilpotent R L