ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LieModule.lie_mem_weightSpaceChain_of_weightSpace_eq_bot_left
lemma lie_mem_weightSpaceChain_of_weightSpace_eq_bot_left [LieAlgebra.IsNilpotent R H]
    (hp : weightSpace M (p • α + χ) = ⊥)
    {x : L} (hx : x ∈ rootSpace H (-α))
    {y : M} (hy : y ∈ weightSpaceChain M α χ p q) :
    ⁅x, y⁆ ∈ weightSpaceChain M α χ p q 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {H : LieSubalgebra R L} (α χ : ↥H → R) (p q : ℤ) [LieAlgebra.IsNilpotent R ↥H] (hp : LieModule.weightSpace M (p • α + χ) = ⊥) {x : L} (hx : x ∈ LieAlgebra.rootSpace H (-α)) {y : M} (hy : y ∈ LieModule.weightSpaceChain M α χ p q)
Goal: ⁅x, y⁆ ∈ LieModule.weightSpaceChain M α χ p q
-- LieModule.trace_toEnd_weightSpaceChain_eq_zero
lemma trace_toEnd_weightSpaceChain_eq_zero
    (hp : weightSpace M (p • α + χ) = ⊥)
    (hq : weightSpace M (q • α + χ) = ⊥)
    {x : H} (hx : x ∈ corootSpace α) :
    LinearMap.trace R _ (toEnd R H (weightSpaceChain M α χ p q) x) = 0 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {H : LieSubalgebra R L} (α χ : ↥H → R) (p q : ℤ) [H.IsCartanSubalgebra] [IsNoetherian R L] (hp : LieModule.weightSpace M (p • α + χ) = ⊥) (hq : LieModule.weightSpace M (q • α + χ) = ⊥) {x : ↥H} (hx : x ∈ LieAlgebra.corootSpace α)
Goal: (LinearMap.trace R ↥↑(LieModule.weightSpaceChain M α χ p q))
    ((LieModule.toEnd R ↥H ↥↑(LieModule.weightSpaceChain M α χ p q)) x) =
  0
-- LieModule.exists_forall_mem_corootSpace_smul_add_eq_zero
lemma exists_forall_mem_corootSpace_smul_add_eq_zero
    [IsDomain R] [IsPrincipalIdealRing R] [CharZero R] [NoZeroSMulDivisors R M] [IsNoetherian R M]
    (hα : α ≠ 0) (hχ : weightSpace M χ ≠ ⊥) :
    ∃ a b : ℤ, 0 < b ∧ ∀ x ∈ corootSpace α, (a • α + b • χ) x = 0 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {H : LieSubalgebra R L} (α χ : ↥H → R) [H.IsCartanSubalgebra] [IsNoetherian R L] [IsDomain R] [IsPrincipalIdealRing R] [CharZero R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (hα : α ≠ 0) (hχ : LieModule.weightSpace M χ ≠ ⊥)
Goal: ∃ a b, 0 < b ∧ ∀ x ∈ LieAlgebra.corootSpace α, (a • α + b • χ) x = 0
-- LieModule.chainTopCoeff_neg
@[simp] lemma chainTopCoeff_neg : chainTopCoeff (-α) β = chainBotCoeff α β 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M)
Goal: LieModule.chainTopCoeff (-α) β = LieModule.chainBotCoeff α β
-- LieModule.chainBotCoeff_neg
@[simp] lemma chainBotCoeff_neg : chainBotCoeff (-α) β = chainTopCoeff α β 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M)
Goal: LieModule.chainBotCoeff (-α) β = LieModule.chainTopCoeff α β
-- LieModule.chainTopCoeff_zero
@[simp] lemma chainTopCoeff_zero : chainTopCoeff 0 β = 0 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (β : LieModule.Weight R L M)
Goal: LieModule.chainTopCoeff 0 β = 0
-- LieModule.chainBotCoeff_zero
@[simp] lemma chainBotCoeff_zero : chainBotCoeff 0 β = 0 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (β : LieModule.Weight R L M)
Goal: LieModule.chainBotCoeff 0 β = 0
-- LieModule.chainTopCoeff_add_one
lemma chainTopCoeff_add_one :
    letI := Classical.propDecidable
    chainTopCoeff α β + 1 =
      Nat.find (eventually_weightSpace_smul_add_eq_bot M α β hα).exists 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M) (hα : α ≠ 0)
Goal: LieModule.chainTopCoeff α β + 1 = Nat.find ⋯
-- LieModule.weightSpace_chainTopCoeff_add_one_nsmul_add
lemma weightSpace_chainTopCoeff_add_one_nsmul_add :
    weightSpace M ((chainTopCoeff α β + 1) • α + β : L → R) = ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M) (hα : α ≠ 0)
Goal: LieModule.weightSpace M ((LieModule.chainTopCoeff α β + 1) • α + ⇑β) = ⊥
-- LieModule.weightSpace_chainTopCoeff_add_one_zsmul_add
lemma weightSpace_chainTopCoeff_add_one_zsmul_add :
    weightSpace M ((chainTopCoeff α β + 1 : ℤ) • α + β : L → R) = ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M) (hα : α ≠ 0)
Goal: LieModule.weightSpace M ((↑(LieModule.chainTopCoeff α β) + 1) • α + ⇑β) = ⊥
-- LieModule.weightSpace_nsmul_add_ne_bot_of_le
lemma weightSpace_nsmul_add_ne_bot_of_le {n} (hn : n ≤ chainTopCoeff α β) :
    weightSpace M (n • α + β : L → R) ≠ ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M) {n : ℕ} (hn : n ≤ LieModule.chainTopCoeff α β)
Goal: LieModule.weightSpace M (n • α + ⇑β) ≠ ⊥
-- LieModule.weightSpace_chainBotCoeff_sub_one_zsmul_sub
lemma weightSpace_chainBotCoeff_sub_one_zsmul_sub :
    weightSpace M ((-chainBotCoeff α β - 1 : ℤ) • α + β : L → R) = ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M) (hα : α ≠ 0)
Goal: LieModule.weightSpace M ((-↑(LieModule.chainBotCoeff α β) - 1) • α + ⇑β) = ⊥
-- LieModule.weightSpace_zsmul_add_ne_bot
lemma weightSpace_zsmul_add_ne_bot {n : ℤ}
    (hn : -chainBotCoeff α β ≤ n) (hn' : n ≤ chainTopCoeff α β) :
      weightSpace M (n • α + β : L → R) ≠ ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M) {n : ℤ} (hn : -↑(LieModule.chainBotCoeff α β) ≤ n) (hn' : n ≤ ↑(LieModule.chainTopCoeff α β))
Goal: LieModule.weightSpace M (n • α + ⇑β) ≠ ⊥
-- LieModule.weightSpace_neg_zsmul_add_ne_bot
lemma weightSpace_neg_zsmul_add_ne_bot {n : ℕ} (hn : n ≤ chainBotCoeff α β) :
    weightSpace M ((-n : ℤ) • α + β : L → R) ≠ ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M) {n : ℕ} (hn : n ≤ LieModule.chainBotCoeff α β)
Goal: LieModule.weightSpace M (-↑n • α + ⇑β) ≠ ⊥
-- LieModule.coe_chainTop'
lemma coe_chainTop' : (chainTop α β : L → R) = chainTopCoeff α β • α + β 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M)
Goal: ⇑(LieModule.chainTop α β) = LieModule.chainTopCoeff α β • α + ⇑β
-- LieModule.coe_chainTop
@[simp] lemma coe_chainTop : (chainTop α β : L → R) = (chainTopCoeff α β : ℤ) • α + β 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M)
Goal: ⇑(LieModule.chainTop α β) = ↑(LieModule.chainTopCoeff α β) • α + ⇑β
-- LieModule.coe_chainBot
@[simp] lemma coe_chainBot : (chainBot α β : L → R) = (-chainBotCoeff α β : ℤ) • α + β 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M)
Goal: ⇑(LieModule.chainBot α β) = -↑(LieModule.chainBotCoeff α β) • α + ⇑β
-- LieModule.chainTop_neg
@[simp] lemma chainTop_neg : chainTop (-α) β = chainBot α β 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M)
Goal: LieModule.chainTop (-α) β = LieModule.chainBot α β
-- LieModule.chainBot_neg
@[simp] lemma chainBot_neg : chainBot (-α) β = chainTop α β 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M)
Goal: LieModule.chainBot (-α) β = LieModule.chainTop α β
-- LieModule.chainTop_zero
@[simp] lemma chainTop_zero : chainTop 0 β = β 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (β : LieModule.Weight R L M)
Goal: LieModule.chainTop 0 β = β
-- LieModule.chainBot_zero
@[simp] lemma chainBot_zero : chainBot 0 β = β 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (β : LieModule.Weight R L M)
Goal: LieModule.chainBot 0 β = β
-- LieModule.weightSpace_add_chainTop
lemma weightSpace_add_chainTop :
    weightSpace M (α + chainTop α β : L → R) = ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M) (hα : α ≠ 0)
Goal: LieModule.weightSpace M (α + ⇑(LieModule.chainTop α β)) = ⊥
-- LieModule.weightSpace_neg_add_chainBot
lemma weightSpace_neg_add_chainBot :
    weightSpace M (-α + chainBot α β : L → R) = ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M) (hα : α ≠ 0)
Goal: LieModule.weightSpace M (-α + ⇑(LieModule.chainBot α β)) = ⊥
-- LieModule.chainTop_isNonZero'
lemma chainTop_isNonZero' (hα' : weightSpace M α ≠ ⊥) :
    (chainTop α β).IsNonZero 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α : L → R) (β : LieModule.Weight R L M) (hα : α ≠ 0) (hα' : LieModule.weightSpace M α ≠ ⊥)
Goal: (LieModule.chainTop α β).IsNonZero
-- LieModule.chainTop_isNonZero
lemma chainTop_isNonZero (α β : Weight R L M) (hα : α.IsNonZero) :
    (chainTop α β).IsNonZero 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (α β : LieModule.Weight R L M) (hα : α.IsNonZero)
Goal: (LieModule.chainTop (⇑α) β).IsNonZero
-- LieModule.traceForm_apply_apply
lemma traceForm_apply_apply (x y : L) :
    traceForm R L M x y = trace R _ (φ x ∘ₗ φ y) 
Arguments: (R : Type u_1) (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] (x y : L)
Goal: ((LieModule.traceForm R L M) x) y = (LinearMap.trace R M) (φ x ∘ₗ φ y)
-- LieModule.traceForm_comm
lemma traceForm_comm (x y : L) : traceForm R L M x y = traceForm R L M y x 
Arguments: (R : Type u_1) (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] (x y : L)
Goal: ((LieModule.traceForm R L M) x) y = ((LieModule.traceForm R L M) y) x
-- LieModule.traceForm_isSymm
lemma traceForm_isSymm : LinearMap.IsSymm (traceForm R L M) 
Arguments: (R : Type u_1) (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]
Goal: LinearMap.IsSymm (LieModule.traceForm R L M)
-- LieModule.traceForm_flip
@[simp] lemma traceForm_flip : LinearMap.flip (traceForm R L M) = traceForm R L M 
Arguments: (R : Type u_1) (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]
Goal: LinearMap.flip (LieModule.traceForm R L M) = LieModule.traceForm R L M
-- LieModule.traceForm_apply_lie_apply
lemma traceForm_apply_lie_apply (x y z : L) :
    traceForm R L M ⁅x, y⁆ z = traceForm R L M x ⁅y, z⁆ 
Arguments: (R : Type u_1) (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] (x y z : L)
Goal: ((LieModule.traceForm R L M) ⁅x, y⁆) z = ((LieModule.traceForm R L M) x) ⁅y, z⁆
-- LieModule.traceForm_apply_lie_apply'
lemma traceForm_apply_lie_apply' (x y z : L) :
    traceForm R L M ⁅x, y⁆ z = - traceForm R L M y ⁅x, z⁆ 
Arguments: (R : Type u_1) (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] (x y z : L)
Goal: ((LieModule.traceForm R L M) ⁅x, y⁆) z = -((LieModule.traceForm R L M) y) ⁅x, z⁆
-- LieModule.traceForm_lieInvariant
lemma traceForm_lieInvariant : (traceForm R L M).lieInvariant L 
Arguments: (R : Type u_1) (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]
Goal: LinearMap.BilinForm.lieInvariant L (LieModule.traceForm R L M)
-- LieModule.lie_traceForm_eq_zero
@[simp] lemma lie_traceForm_eq_zero (x : L) : ⁅x, traceForm R L M⁆ = 0 
Arguments: (R : Type u_1) (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] (x : L)
Goal: ⁅x, LieModule.traceForm R L M⁆ = 0
-- LieModule.traceForm_eq_zero_of_isNilpotent
@[simp] lemma traceForm_eq_zero_of_isNilpotent [IsReduced R] [IsNilpotent R L M] :
    traceForm R L M = 0 
Arguments: (R : Type u_1) (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] [Module.Free R M] [Module.Finite R M] [IsReduced R] [LieModule.IsNilpotent R L M]
Goal: LieModule.traceForm R L M = 0
-- LieModule.traceForm_weightSpace_eq
@[simp]
lemma traceForm_weightSpace_eq [IsDomain R] [IsPrincipalIdealRing R]
    [LieAlgebra.IsNilpotent R L] [IsNoetherian R M] [LinearWeights R L M] (χ : L → R) (x y : L) :
    traceForm R L (weightSpace M χ) x y = finrank R (weightSpace M χ) • (χ x * χ y) 
Arguments: (R : Type u_1) (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] [Module.Free R M] [Module.Finite R M] [IsDomain R] [IsPrincipalIdealRing R] [LieAlgebra.IsNilpotent R L] [IsNoetherian R M] [LieModule.LinearWeights R L M] (χ : L → R) (x y : L)
Goal: ((LieModule.traceForm R L ↥↑(LieModule.weightSpace M χ)) x) y =
  FiniteDimensional.finrank R ↥↑(LieModule.weightSpace M χ) • (χ x * χ y)
-- LieModule.traceForm_eq_zero_if_mem_lcs_of_mem_ucs
lemma traceForm_eq_zero_if_mem_lcs_of_mem_ucs {x y : L} (k : ℕ)
    (hx : x ∈ (⊤ : LieIdeal R L).lcs L k) (hy : y ∈ (⊥ : LieIdeal R L).ucs k) :
    traceForm R L M x y = 0 
Arguments: (R : Type u_1) (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] {x y : L} (k : ℕ) (hx : x ∈ ⊤.lcs L k) (hy : y ∈ LieSubmodule.ucs k ⊥)
Goal: ((LieModule.traceForm R L M) x) y = 0
-- LieModule.traceForm_apply_eq_zero_of_mem_lcs_of_mem_center
lemma traceForm_apply_eq_zero_of_mem_lcs_of_mem_center {x y : L}
    (hx : x ∈ lowerCentralSeries R L L 1) (hy : y ∈ LieAlgebra.center R L) :
    traceForm R L M x y = 0 
Arguments: (R : Type u_1) (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] {x y : L} (hx : x ∈ LieModule.lowerCentralSeries R L L 1) (hy : y ∈ LieAlgebra.center R L)
Goal: ((LieModule.traceForm R L M) x) y = 0
-- LieModule.traceForm_eq_zero_of_isTrivial
@[simp] lemma traceForm_eq_zero_of_isTrivial [IsTrivial L M] :
    traceForm R L M = 0 
Arguments: (R : Type u_1) (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] [LieModule.IsTrivial L M]
Goal: LieModule.traceForm R L M = 0
-- LieModule.eq_zero_of_mem_weightSpace_mem_posFitting
lemma eq_zero_of_mem_weightSpace_mem_posFitting [LieAlgebra.IsNilpotent R L]
    {B : LinearMap.BilinForm R M} (hB : ∀ (x : L) (m n : M), B ⁅x, m⁆ n = - B m ⁅x, n⁆)
    {m₀ m₁ : M} (hm₀ : m₀ ∈ weightSpace M (0 : L → R)) (hm₁ : m₁ ∈ posFittingComp R L M) :
    B m₀ m₁ = 0 
Arguments: (R : Type u_1) (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] [LieAlgebra.IsNilpotent R L] {B : LinearMap.BilinForm R M} (hB : ∀ (x : L) (m n : M), (B ⁅x, m⁆) n = -(B m) ⁅x, n⁆) {m₀ m₁ : M} (hm₀ : m₀ ∈ LieModule.weightSpace M 0) (hm₁ : m₁ ∈ LieModule.posFittingComp R L M)
Goal: (B m₀) m₁ = 0
-- LieModule.trace_toEnd_eq_zero_of_mem_lcs
lemma trace_toEnd_eq_zero_of_mem_lcs
    {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
    trace R _ (toEnd R L M x) = 0 
Arguments: (R : Type u_1) (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] {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ LieModule.lowerCentralSeries R L L k)
Goal: (LinearMap.trace R M) (φ x) = 0
-- LieModule.traceForm_lieSubalgebra_mk_left
@[simp]
lemma traceForm_lieSubalgebra_mk_left (L' : LieSubalgebra R L) {x : L} (hx : x ∈ L') (y : L') :
    traceForm R L' M ⟨x, hx⟩ y = traceForm R L M x y 
Arguments: (R : Type u_1) (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] (L' : LieSubalgebra R L) {x : L} (hx : x ∈ L') (y : ↥L')
Goal: ((LieModule.traceForm R (↥L') M) ⟨x, hx⟩) y = ((LieModule.traceForm R L M) x) ↑y
-- LieModule.traceForm_lieSubalgebra_mk_right
@[simp]
lemma traceForm_lieSubalgebra_mk_right (L' : LieSubalgebra R L) {x : L'} {y : L} (hy : y ∈ L') :
    traceForm R L' M x ⟨y, hy⟩ = traceForm R L M x y 
Arguments: (R : Type u_1) (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] (L' : LieSubalgebra R L) {x : ↥L'} {y : L} (hy : y ∈ L')
Goal: ((LieModule.traceForm R (↥L') M) x) ⟨y, hy⟩ = ((LieModule.traceForm R L M) ↑x) y
-- LieModule.traceForm_eq_sum_weightSpaceOf
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ ∈ (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) 
Arguments: (R : Type u_1) (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] [Module.Free R M] [Module.Finite R M] [LieAlgebra.IsNilpotent R L] [IsDomain R] [IsPrincipalIdealRing R] [LieModule.IsTriangularizable R L M] (z : L)
Goal: LieModule.traceForm R L M = ∑ χ ∈ ⋯.toFinset, LieModule.traceForm R L ↥↑(LieModule.weightSpaceOf M χ z)
-- LieModule.lowerCentralSeries_one_inf_center_le_ker_traceForm
lemma lowerCentralSeries_one_inf_center_le_ker_traceForm :
    lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L ≤ LinearMap.ker (traceForm R L M) 
Arguments: (R : Type u_1) (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] [Module.Free R M] [Module.Finite R M] [LieAlgebra.IsNilpotent R L] [IsDomain R] [IsPrincipalIdealRing R]
Goal: (lieIdealSubalgebra R L (LieModule.lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L)).toSubmodule ≤
  LinearMap.ker (LieModule.traceForm R L M)
-- LieModule.isLieAbelian_of_ker_traceForm_eq_bot
lemma isLieAbelian_of_ker_traceForm_eq_bot (h : LinearMap.ker (traceForm R L M) = ⊥) :
    IsLieAbelian L 
Arguments: (R : Type u_1) (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] [Module.Free R M] [Module.Finite R M] [LieAlgebra.IsNilpotent R L] [IsDomain R] [IsPrincipalIdealRing R] (h : LinearMap.ker (LieModule.traceForm R L M) = ⊥)
Goal: IsLieAbelian L
-- LieSubmodule.trace_eq_trace_restrict_of_le_idealizer
lemma trace_eq_trace_restrict_of_le_idealizer
    (hy' : ∀ m ∈ N, (φ x ∘ₗ φ y) m ∈ N := fun m _ ↦ N.lie_mem (N.mem_idealizer.mp (h hy) m)) :
    trace R M (φ x ∘ₗ φ y) = trace R N ((φ x ∘ₗ φ y).restrict hy') 
Arguments: {R : Type u_1} {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] [Module.Free R M] [Module.Finite R M] [IsDomain R] [IsPrincipalIdealRing R] (N : LieSubmodule R L M) (I : LieIdeal R L) (h : I ≤ N.idealizer) (x : L) {y : L} (hy : y ∈ I) (hy' : ∀ m ∈ N, (φ x ∘ₗ φ y) m ∈ N := ⋯)
Goal: (LinearMap.trace R M) (φ x ∘ₗ φ y) = (LinearMap.trace R ↥↑N) ((φ x ∘ₗ φ y).restrict hy')
-- LieSubmodule.traceForm_eq_of_le_idealizer
lemma traceForm_eq_of_le_idealizer :
    traceForm R I N = (traceForm R L M).restrict I 
Arguments: {R : Type u_1} {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] [Module.Free R M] [Module.Finite R M] [IsDomain R] [IsPrincipalIdealRing R] (N : LieSubmodule R L M) (I : LieIdeal R L) (h : I ≤ N.idealizer)
Goal: LieModule.traceForm R ↥↑I ↥↑N = (LieModule.traceForm R L M).restrict (lieIdealSubalgebra R L I).toSubmodule