ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LieModule.isNilpotent_toEnd_sub_algebraMap
lemma isNilpotent_toEnd_sub_algebraMap [IsNoetherian R M] (χ : L → R) (x : L) :
    _root_.IsNilpotent <| toEnd R L (weightSpace M χ) x - algebraMap R _ (χ x) 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M] (χ : L → R) (x : L)
Goal: IsNilpotent
  ((LieModule.toEnd R L ↥↑(LieModule.weightSpace M χ)) x -
    (algebraMap R (Module.End R ↥↑(LieModule.weightSpace M χ))) (χ x))
-- LieModule.isNilpotent_toEnd_weightSpace_zero
theorem isNilpotent_toEnd_weightSpace_zero [IsNoetherian R M] (x : L) :
    _root_.IsNilpotent <| toEnd R L (weightSpace M (0 : L → R)) x 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M] (x : L)
Goal: IsNilpotent ((LieModule.toEnd R L ↥↑(LieModule.weightSpace M 0)) x)
-- LieModule.weightSpace_zero_normalizer_eq_self
@[simp]
lemma weightSpace_zero_normalizer_eq_self :
    (weightSpace M (0 : L → R)).normalizer = weightSpace M 0 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: (LieModule.weightSpace M 0).normalizer = LieModule.weightSpace M 0
-- LieModule.iSup_ucs_le_weightSpace_zero
lemma iSup_ucs_le_weightSpace_zero :
    ⨆ k, (⊥ : LieSubmodule R L M).ucs k ≤ weightSpace M (0 : L → R) 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: ⨆ k, LieSubmodule.ucs k ⊥ ≤ LieModule.weightSpace M 0
-- LieModule.iSup_ucs_eq_weightSpace_zero
lemma iSup_ucs_eq_weightSpace_zero [IsNoetherian R M] :
    ⨆ k, (⊥ : LieSubmodule R L M).ucs k = weightSpace M (0 : L → R) 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M]
Goal: ⨆ k, LieSubmodule.ucs k ⊥ = LieModule.weightSpace M 0
-- LieModule.mem_posFittingCompOf
lemma mem_posFittingCompOf (x : L) (m : M) :
    m ∈ posFittingCompOf R M x ↔ ∀ (k : ℕ), ∃ n, (toEnd R L M x ^ k) n = m 
Arguments: (R : Type u_2) {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L) (m : M)
Goal: m ∈ LieModule.posFittingCompOf R M x ↔ ∀ (k : ℕ), ∃ n, ((LieModule.toEnd R L M) x ^ k) n = m
-- LieModule.posFittingCompOf_le_lowerCentralSeries
@[simp] lemma posFittingCompOf_le_lowerCentralSeries (x : L) (k : ℕ) :
    posFittingCompOf R M x ≤ lowerCentralSeries R L M k 
Arguments: (R : Type u_2) {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L) (k : ℕ)
Goal: LieModule.posFittingCompOf R M x ≤ LieModule.lowerCentralSeries R L M k
-- LieModule.posFittingCompOf_eq_bot_of_isNilpotent
@[simp] lemma posFittingCompOf_eq_bot_of_isNilpotent
    [IsNilpotent R L M] (x : L) :
    posFittingCompOf R M x = ⊥ 
Arguments: (R : Type u_2) {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieModule.IsNilpotent R L M] (x : L)
Goal: LieModule.posFittingCompOf R M x = ⊥
-- LieModule.mem_posFittingComp
lemma mem_posFittingComp (m : M) :
    m ∈ posFittingComp R L M ↔ m ∈ ⨆ (x : L), posFittingCompOf R M x 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (m : M)
Goal: m ∈ LieModule.posFittingComp R L M ↔ m ∈ ⨆ x, LieModule.posFittingCompOf R M x
-- LieModule.posFittingCompOf_le_posFittingComp
lemma posFittingCompOf_le_posFittingComp (x : L) :
    posFittingCompOf R M x ≤ posFittingComp R L M 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L)
Goal: LieModule.posFittingCompOf R M x ≤ LieModule.posFittingComp R L M
-- LieModule.posFittingComp_le_iInf_lowerCentralSeries
lemma posFittingComp_le_iInf_lowerCentralSeries :
    posFittingComp R L M ≤ ⨅ k, lowerCentralSeries R L M k 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: LieModule.posFittingComp R L M ≤ ⨅ k, LieModule.lowerCentralSeries R L M k
-- LieModule.iInf_lowerCentralSeries_eq_posFittingComp
@[simp] lemma iInf_lowerCentralSeries_eq_posFittingComp
    [IsNoetherian R M] [IsArtinian R M] :
    ⨅ k, lowerCentralSeries R L M k = posFittingComp R L M 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M] [IsArtinian R M]
Goal: ⨅ k, LieModule.lowerCentralSeries R L M k = LieModule.posFittingComp R L M
-- LieModule.posFittingComp_eq_bot_of_isNilpotent
@[simp] lemma posFittingComp_eq_bot_of_isNilpotent
    [IsNilpotent R L M] :
    posFittingComp R L M = ⊥ 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieModule.IsNilpotent R L M]
Goal: LieModule.posFittingComp R L M = ⊥
-- LieModule.map_posFittingComp_le
lemma map_posFittingComp_le :
    (posFittingComp R L M).map f ≤ posFittingComp R L M₂ 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {M₂ : Type u_5} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] (f : M →ₗ⁅R,L⁆ M₂)
Goal: LieSubmodule.map f (LieModule.posFittingComp R L M) ≤ LieModule.posFittingComp R L M₂
-- LieModule.map_weightSpace_le
lemma map_weightSpace_le :
    (weightSpace M χ).map f ≤ weightSpace M₂ χ 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {M₂ : Type u_5} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] {χ : L → R} (f : M →ₗ⁅R,L⁆ M₂)
Goal: LieSubmodule.map f (LieModule.weightSpace M χ) ≤ LieModule.weightSpace M₂ χ
-- LieModule.comap_weightSpace_eq_of_injective
lemma comap_weightSpace_eq_of_injective (hf : Injective f) :
    (weightSpace M₂ χ).comap f = weightSpace M χ 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {M₂ : Type u_5} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] {χ : L → R} {f : M →ₗ⁅R,L⁆ M₂} (hf : Function.Injective ⇑f)
Goal: LieSubmodule.comap f (LieModule.weightSpace M₂ χ) = LieModule.weightSpace M χ
-- LieModule.map_weightSpace_eq_of_injective
lemma map_weightSpace_eq_of_injective (hf : Injective f) :
    (weightSpace M χ).map f = weightSpace M₂ χ ⊓ f.range 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {M₂ : Type u_5} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] {χ : L → R} {f : M →ₗ⁅R,L⁆ M₂} (hf : Function.Injective ⇑f)
Goal: LieSubmodule.map f (LieModule.weightSpace M χ) = LieModule.weightSpace M₂ χ ⊓ f.range
-- LieModule.map_weightSpace_eq
lemma map_weightSpace_eq (e : M ≃ₗ⁅R,L⁆ M₂) :
    (weightSpace M χ).map e = weightSpace M₂ χ 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {M₂ : Type u_5} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] {χ : L → R} (e : M ≃ₗ⁅R,L⁆ M₂)
Goal: LieSubmodule.map e.toLieModuleHom (LieModule.weightSpace M χ) = LieModule.weightSpace M₂ χ
-- LieModule.map_posFittingComp_eq
lemma map_posFittingComp_eq (e : M ≃ₗ⁅R,L⁆ M₂) :
    (posFittingComp R L M).map e = posFittingComp R L M₂ 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {M₂ : Type u_5} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] (e : M ≃ₗ⁅R,L⁆ M₂)
Goal: LieSubmodule.map e.toLieModuleHom (LieModule.posFittingComp R L M) = LieModule.posFittingComp R L M₂
-- LieModule.posFittingComp_map_incl_sup_of_codisjoint
lemma posFittingComp_map_incl_sup_of_codisjoint [IsNoetherian R M] [IsArtinian R M]
    {N₁ N₂ : LieSubmodule R L M} (h : Codisjoint N₁ N₂) :
    (posFittingComp R L N₁).map N₁.incl ⊔ (posFittingComp R L N₂).map N₂.incl =
    posFittingComp R L M 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M] [IsArtinian R M] {N₁ N₂ : LieSubmodule R L M} (h : Codisjoint N₁ N₂)
Goal: LieSubmodule.map N₁.incl (LieModule.posFittingComp R L ↥↑N₁) ⊔
    LieSubmodule.map N₂.incl (LieModule.posFittingComp R L ↥↑N₂) =
  LieModule.posFittingComp R L M
-- LieModule.weightSpace_weightSpaceOf_map_incl
lemma weightSpace_weightSpaceOf_map_incl (x : L) (χ : L → R) :
    (weightSpace (weightSpaceOf M (χ x) x) χ).map (weightSpaceOf M (χ x) x).incl =
    weightSpace M χ 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L) (χ : L → R)
Goal: LieSubmodule.map (LieModule.weightSpaceOf M (χ x) x).incl
    (LieModule.weightSpace (↥↑(LieModule.weightSpaceOf M (χ x) x)) χ) =
  LieModule.weightSpace M χ
-- LieModule.isCompl_weightSpaceOf_zero_posFittingCompOf
lemma isCompl_weightSpaceOf_zero_posFittingCompOf (x : L) :
    IsCompl (weightSpaceOf M 0 x) (posFittingCompOf R M x) 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M] [IsArtinian R M] (x : L)
Goal: IsCompl (LieModule.weightSpaceOf M 0 x) (LieModule.posFittingCompOf R M x)
-- _private.Mathlib.Algebra.Lie.Weights.Basic.0.LieModule.isCompl_weightSpace_zero_posFittingComp_aux
private lemma isCompl_weightSpace_zero_posFittingComp_aux
    (h : ∀ N < (⊤ : LieSubmodule R L M), IsCompl (weightSpace N 0) (posFittingComp R L N)) :
    IsCompl (weightSpace M 0) (posFittingComp R L M) 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M] [IsArtinian R M] (h : ∀ N < ⊤, IsCompl (LieModule.weightSpace (↥↑N) 0) (LieModule.posFittingComp R L ↥↑N))
Goal: IsCompl (LieModule.weightSpace M 0) (LieModule.posFittingComp R L M)
-- LieModule.isCompl_weightSpace_zero_posFittingComp
lemma isCompl_weightSpace_zero_posFittingComp :
    IsCompl (weightSpace M 0) (posFittingComp R L M) 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M] [IsArtinian R M]
Goal: IsCompl (LieModule.weightSpace M 0) (LieModule.posFittingComp R L M)
-- LieModule.disjoint_weightSpaceOf
lemma disjoint_weightSpaceOf [NoZeroSMulDivisors R M] {x : L} {φ₁ φ₂ : R} (h : φ₁ ≠ φ₂) :
    Disjoint (weightSpaceOf M φ₁ x) (weightSpaceOf M φ₂ x) 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [NoZeroSMulDivisors R M] {x : L} {φ₁ φ₂ : R} (h : φ₁ ≠ φ₂)
Goal: Disjoint (LieModule.weightSpaceOf M φ₁ x) (LieModule.weightSpaceOf M φ₂ x)
-- LieModule.disjoint_weightSpace
lemma disjoint_weightSpace [NoZeroSMulDivisors R M] {χ₁ χ₂ : L → R} (h : χ₁ ≠ χ₂) :
    Disjoint (weightSpace M χ₁) (weightSpace M χ₂) 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [NoZeroSMulDivisors R M] {χ₁ χ₂ : L → R} (h : χ₁ ≠ χ₂)
Goal: Disjoint (LieModule.weightSpace M χ₁) (LieModule.weightSpace M χ₂)
-- LieModule.injOn_weightSpace
lemma injOn_weightSpace [NoZeroSMulDivisors R M] :
    InjOn (fun (χ : L → R) ↦ weightSpace M χ) {χ | weightSpace M χ ≠ ⊥} 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [NoZeroSMulDivisors R M]
Goal: Set.InjOn (fun χ => LieModule.weightSpace M χ) {χ | LieModule.weightSpace M χ ≠ ⊥}
-- LieModule.independent_weightSpace
lemma independent_weightSpace [NoZeroSMulDivisors R M] :
    CompleteLattice.Independent fun (χ : L → R) ↦ weightSpace M χ 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [NoZeroSMulDivisors R M]
Goal: CompleteLattice.Independent fun χ => LieModule.weightSpace M χ
-- LieModule.independent_weightSpace'
lemma independent_weightSpace' [NoZeroSMulDivisors R M] :
    CompleteLattice.Independent fun χ : Weight R L M ↦ weightSpace M χ 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [NoZeroSMulDivisors R M]
Goal: CompleteLattice.Independent fun χ => LieModule.weightSpace M ⇑χ
-- LieModule.independent_weightSpaceOf
lemma independent_weightSpaceOf [NoZeroSMulDivisors R M] (x : L) :
    CompleteLattice.Independent fun (χ : R) ↦ weightSpaceOf M χ x 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [NoZeroSMulDivisors R M] (x : L)
Goal: CompleteLattice.Independent fun χ => LieModule.weightSpaceOf M χ x
-- LieModule.finite_weightSpaceOf_ne_bot
lemma finite_weightSpaceOf_ne_bot [NoZeroSMulDivisors R M] [IsNoetherian R M] (x : L) :
    {χ : R | weightSpaceOf M χ x ≠ ⊥}.Finite 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [NoZeroSMulDivisors R M] [IsNoetherian R M] (x : L)
Goal: {χ | LieModule.weightSpaceOf M χ x ≠ ⊥}.Finite
-- LieModule.finite_weightSpace_ne_bot
lemma finite_weightSpace_ne_bot [NoZeroSMulDivisors R M] [IsNoetherian R M] :
    {χ : L → R | weightSpace M χ ≠ ⊥}.Finite 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [NoZeroSMulDivisors R M] [IsNoetherian R M]
Goal: {χ | LieModule.weightSpace M χ ≠ ⊥}.Finite
-- LieModule.IsTriangularizable.iSup_eq_top
class IsTriangularizable : Prop :=
  iSup_eq_top : ∀ x, ⨆ φ, ⨆ k, (toEnd R L M x).genEigenspace φ k = ⊤
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [self : LieModule.IsTriangularizable R L M] (x : L)
Goal: ⨆ φ, ⨆ k, (((LieModule.toEnd R L M) x).genEigenspace φ) k = ⊤
-- LieModule.iSup_weightSpaceOf_eq_top
@[simp]
lemma iSup_weightSpaceOf_eq_top [IsTriangularizable R L M] (x : L) :
    ⨆ (φ : R), weightSpaceOf M φ x = ⊤ 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieModule.IsTriangularizable R L M] (x : L)
Goal: ⨆ φ, LieModule.weightSpaceOf M φ x = ⊤
-- LieModule.trace_toEnd_weightSpace
@[simp]
lemma trace_toEnd_weightSpace [IsDomain R] [IsPrincipalIdealRing R]
    [Module.Free R M] [Module.Finite R M] (χ : L → R) (x : L) :
    trace R _ (toEnd R L (weightSpace M χ) x) = finrank R (weightSpace M χ) • χ x 
Arguments: (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsDomain R] [IsPrincipalIdealRing R] [Module.Free R M] [Module.Finite R M] (χ : L → R) (x : L)
Goal: (LinearMap.trace R ↥↑(LieModule.weightSpace M χ)) ((LieModule.toEnd R L ↥↑(LieModule.weightSpace M χ)) x) =
  FiniteDimensional.finrank R ↥↑(LieModule.weightSpace M χ) • χ x
-- LieModule.iSup_weightSpace_eq_top
lemma iSup_weightSpace_eq_top [IsTriangularizable K L M] :
    ⨆ χ : L → K, weightSpace M χ = ⊤ 
Arguments: (K : Type u_1) (L : Type u_3) (M : Type u_4) [LieRing L] [AddCommGroup M] [LieRingModule L M] [Field K] [LieAlgebra K L] [Module K M] [LieModule K L M] [LieAlgebra.IsNilpotent K L] [FiniteDimensional K M] [LieModule.IsTriangularizable K L M]
Goal: ⨆ χ, LieModule.weightSpace M χ = ⊤
-- LieModule.iSup_weightSpace_eq_top'
lemma iSup_weightSpace_eq_top' [IsTriangularizable K L M] :
    ⨆ χ : Weight K L M, weightSpace M χ = ⊤ 
Arguments: (K : Type u_1) (L : Type u_3) (M : Type u_4) [LieRing L] [AddCommGroup M] [LieRingModule L M] [Field K] [LieAlgebra K L] [Module K M] [LieModule K L M] [LieAlgebra.IsNilpotent K L] [FiniteDimensional K M] [LieModule.IsTriangularizable K L M]
Goal: ⨆ χ, LieModule.weightSpace M ⇑χ = ⊤
-- MvPolynomial.polynomial_eval_eval₂
theorem polynomial_eval_eval₂ [CommSemiring R] [CommSemiring S]
    {x : S} (f : R →+* Polynomial S) (g : σ → Polynomial S) (p : MvPolynomial σ R) :
    Polynomial.eval x (eval₂ f g p) =
      eval₂ ((Polynomial.evalRingHom x).comp f) (fun s => Polynomial.eval x (g s)) p 
Arguments: {R : Type u_1} {S : Type u_2} {σ : Type u_3} [CommSemiring R] [CommSemiring S] {x : S} (f : R →+* Polynomial S) (g : σ → Polynomial S) (p : MvPolynomial σ R)
Goal: Polynomial.eval x (MvPolynomial.eval₂ f g p) =
  MvPolynomial.eval₂ ((Polynomial.evalRingHom x).comp f) (fun s => Polynomial.eval x (g s)) p
-- MvPolynomial.eval_polynomial_eval_finSuccEquiv
theorem eval_polynomial_eval_finSuccEquiv {n : ℕ} {x : Fin n → R}
    [CommSemiring R] (f : MvPolynomial (Fin (n + 1)) R) (q : MvPolynomial (Fin n) R) :
    (eval x) (Polynomial.eval q (finSuccEquiv R n f)) = eval (Fin.cases (eval x q) x) f 
Arguments: {R : Type u_1} {n : ℕ} {x : Fin n → R} [CommSemiring R] (f : MvPolynomial (Fin (n + 1)) R) (q : MvPolynomial (Fin n) R)
Goal: (MvPolynomial.eval x) (Polynomial.eval q ((MvPolynomial.finSuccEquiv R n) f)) =
  (MvPolynomial.eval fun i => Fin.cases ((MvPolynomial.eval x) q) x i) f
-- Submodule.mem_localized'
lemma Submodule.mem_localized' (x : N) :
    x ∈ Submodule.localized' S p f M' ↔ ∃ m ∈ M', ∃ s : p, IsLocalizedModule.mk' f m s = x 
Arguments: {R : Type u} (S : Type u') {M : Type v} {N : Type v'} [CommRing R] [CommRing S] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Algebra R S] [Module S N] [IsScalarTower R S N] (p : Submonoid R) [IsLocalization p S] (f : M →ₗ[R] N) [IsLocalizedModule p f] (M' : Submodule R M) (x : N)
Goal: x ∈ Submodule.localized' S p f M' ↔ ∃ m ∈ M', ∃ s, IsLocalizedModule.mk' f m s = x
-- Submodule.toLocalized'_apply_coe
@[simps!]
def Submodule.toLocalized' : M' →ₗ[R] M'.localized' S p f 
Arguments: {R : Type u} (S : Type u') {M : Type v} {N : Type v'} [CommRing R] [CommRing S] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Algebra R S] [Module S N] [IsScalarTower R S N] (p : Submonoid R) [IsLocalization p S] (f : M →ₗ[R] N) [IsLocalizedModule p f] (M' : Submodule R M) (c : ↥M')
Goal: ↑((Submodule.toLocalized' S p f M') c) = f ↑c
-- Submodule.toLocalizedQuotient'_mk
@[simp]
lemma Submodule.toLocalizedQuotient'_mk (x : M) :
    M'.toLocalizedQuotient' S p f (Submodule.Quotient.mk x) = Submodule.Quotient.mk (f x) 
Arguments: {R : Type u} (S : Type u') {M : Type v} {N : Type v'} [CommRing R] [CommRing S] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Algebra R S] [Module S N] [IsScalarTower R S N] (p : Submonoid R) [IsLocalization p S] (f : M →ₗ[R] N) [IsLocalizedModule p f] (M' : Submodule R M) (x : M)
Goal: (Submodule.toLocalizedQuotient' S p f M') (Submodule.Quotient.mk x) = Submodule.Quotient.mk (f x)
-- LinearMap.localized'_ker_eq_ker_localizedMap
lemma LinearMap.localized'_ker_eq_ker_localizedMap (g : M →ₗ[R] P) :
    Submodule.localized' S p f (LinearMap.ker g) =
      LinearMap.ker ((IsLocalizedModule.map p f f' g).extendScalarsOfIsLocalization p S) 
Arguments: {R : Type u} (S : Type u') {M : Type v} {N : Type v'} [CommRing R] [CommRing S] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Algebra R S] [Module S N] [IsScalarTower R S N] (p : Submonoid R) [IsLocalization p S] (f : M →ₗ[R] N) [IsLocalizedModule p f] {P : Type w} [AddCommGroup P] [Module R P] {Q : Type w'} [AddCommGroup Q] [Module R Q] [Module S Q] [IsScalarTower R S Q] (f' : P →ₗ[R] Q) [IsLocalizedModule p f'] (g : M →ₗ[R] P)
Goal: Submodule.localized' S p f (LinearMap.ker g) =
  LinearMap.ker (LinearMap.extendScalarsOfIsLocalization p S ((IsLocalizedModule.map p f f') g))
-- LinearMap.ker_localizedMap_eq_localized'_ker
lemma LinearMap.ker_localizedMap_eq_localized'_ker (g : M →ₗ[R] P) :
    LinearMap.ker (IsLocalizedModule.map p f f' g) =
      ((LinearMap.ker g).localized' S p f).restrictScalars _ 
Arguments: {R : Type u} (S : Type u') {M : Type v} {N : Type v'} [CommRing R] [CommRing S] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Algebra R S] [Module S N] [IsScalarTower R S N] (p : Submonoid R) [IsLocalization p S] (f : M →ₗ[R] N) [IsLocalizedModule p f] {P : Type w} [AddCommGroup P] [Module R P] {Q : Type w'} [AddCommGroup Q] [Module R Q] [Module S Q] [IsScalarTower R S Q] (f' : P →ₗ[R] Q) [IsLocalizedModule p f'] (g : M →ₗ[R] P)
Goal: LinearMap.ker ((IsLocalizedModule.map p f f') g) =
  Submodule.restrictScalars R (Submodule.localized' S p f (LinearMap.ker g))
-- LinearMap.toKerIsLocalized_apply_coe
@[simps!]
noncomputable def LinearMap.toKerIsLocalized (g : M →ₗ[R] P) :
    ker g →ₗ[R] ker (IsLocalizedModule.map p f f' g) 
Arguments: {R : Type u} {M : Type v} {N : Type v'} [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] (p : Submonoid R) (f : M →ₗ[R] N) [IsLocalizedModule p f] {P : Type w} [AddCommGroup P] [Module R P] {Q : Type w'} [AddCommGroup Q] [Module R Q] (f' : P →ₗ[R] Q) [IsLocalizedModule p f'] (g : M →ₗ[R] P) (c : ↥(LinearMap.ker g))
Goal: ↑((LinearMap.toKerIsLocalized p f f' g) c) = f ↑c
-- LinearMap.toKerLocalized_isLocalizedModule
lemma LinearMap.toKerLocalized_isLocalizedModule (g : M →ₗ[R] P) :
    IsLocalizedModule p (toKerIsLocalized p f f' g) 
Arguments: {R : Type u} (S : Type u') {M : Type v} {N : Type v'} [CommRing R] [CommRing S] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Algebra R S] [Module S N] [IsScalarTower R S N] (p : Submonoid R) [IsLocalization p S] (f : M →ₗ[R] N) [IsLocalizedModule p f] {P : Type w} [AddCommGroup P] [Module R P] {Q : Type w'} [AddCommGroup Q] [Module R Q] [Module S Q] [IsScalarTower R S Q] (f' : P →ₗ[R] Q) [IsLocalizedModule p f'] (g : M →ₗ[R] P)
Goal: IsLocalizedModule p (LinearMap.toKerIsLocalized p f f' g)