ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LieSubalgebra.mem_inf
@[simp]
theorem mem_inf (x : L) : x ∈ K ⊓ K' ↔ x ∈ K ∧ x ∈ K' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K K' : LieSubalgebra R L) (x : L)
Goal: x ∈ K ⊓ K' ↔ x ∈ K ∧ x ∈ K'
-- LieSubalgebra.eq_bot_iff
theorem eq_bot_iff : K = ⊥ ↔ ∀ x : L, x ∈ K → x = 0 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L)
Goal: K = ⊥ ↔ ∀ x ∈ K, x = 0
-- LieSubalgebra.subsingleton_bot
theorem subsingleton_bot : Subsingleton (⊥ : LieSubalgebra R L) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: Subsingleton ↥⊥
-- LieSubalgebra.wellFounded_of_noetherian
theorem wellFounded_of_noetherian [IsNoetherian R L] :
    WellFounded ((· > ·) : LieSubalgebra R L → LieSubalgebra R L → Prop) 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] [IsNoetherian R L]
Goal: WellFounded fun x x_1 => x > x_1
-- LieSubalgebra.coe_inclusion
@[simp]
theorem coe_inclusion (x : K) : (inclusion h x : L) = x 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K K' : LieSubalgebra R L} (h : K ≤ K') (x : ↥K)
Goal: ↑((LieSubalgebra.inclusion h) x) = ↑x
-- LieSubalgebra.inclusion_apply
theorem inclusion_apply (x : K) : inclusion h x = ⟨x.1, h x.2⟩ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K K' : LieSubalgebra R L} (h : K ≤ K') (x : ↥K)
Goal: (LieSubalgebra.inclusion h) x = ⟨↑x, ⋯⟩
-- LieSubalgebra.inclusion_injective
theorem inclusion_injective : Function.Injective (inclusion h) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K K' : LieSubalgebra R L} (h : K ≤ K')
Goal: Function.Injective ⇑(LieSubalgebra.inclusion h)
-- LieSubalgebra.mem_ofLe
@[simp]
theorem mem_ofLe (x : K') : x ∈ ofLe h ↔ (x : L) ∈ K 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K K' : LieSubalgebra R L} (h : K ≤ K') (x : ↥K')
Goal: x ∈ LieSubalgebra.ofLe h ↔ ↑x ∈ K
-- LieSubalgebra.ofLe_eq_comap_incl
theorem ofLe_eq_comap_incl : ofLe h = K.comap K'.incl 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K K' : LieSubalgebra R L} (h : K ≤ K')
Goal: LieSubalgebra.ofLe h = LieSubalgebra.comap K'.incl K
-- LieSubalgebra.coe_ofLe
@[simp]
theorem coe_ofLe : (ofLe h : Submodule R K') = LinearMap.range (Submodule.inclusion h) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K K' : LieSubalgebra R L} (h : K ≤ K')
Goal: (LieSubalgebra.ofLe h).toSubmodule = LinearMap.range (Submodule.inclusion h)
-- LieSubalgebra.equivOfLe_apply
@[simp]
theorem equivOfLe_apply (x : K) : equivOfLe h x = ⟨inclusion h x, (inclusion h).mem_range_self x⟩ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K K' : LieSubalgebra R L} (h : K ≤ K') (x : ↥K)
Goal: (LieSubalgebra.equivOfLe h) x = ⟨(LieSubalgebra.inclusion h) x, ⋯⟩
-- LieSubalgebra.map_le_iff_le_comap
theorem map_le_iff_le_comap {K : LieSubalgebra R L} {K' : LieSubalgebra R L₂} :
    map f K ≤ K' ↔ K ≤ comap f K' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] {f : L →ₗ⁅R⁆ L₂} {K : LieSubalgebra R L} {K' : LieSubalgebra R L₂}
Goal: LieSubalgebra.map f K ≤ K' ↔ K ≤ LieSubalgebra.comap f K'
-- LieSubalgebra.gc_map_comap
theorem gc_map_comap : GaloisConnection (map f) (comap f) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] {f : L →ₗ⁅R⁆ L₂}
Goal: GaloisConnection (LieSubalgebra.map f) (LieSubalgebra.comap f)
-- LieSubalgebra.mem_lieSpan
theorem mem_lieSpan {x : L} : x ∈ lieSpan R L s ↔ ∀ K : LieSubalgebra R L, s ⊆ K → x ∈ K 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s : Set L} {x : L}
Goal: x ∈ LieSubalgebra.lieSpan R L s ↔ ∀ (K : LieSubalgebra R L), s ⊆ ↑K → x ∈ K
-- LieSubalgebra.subset_lieSpan
theorem subset_lieSpan : s ⊆ lieSpan R L s 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s : Set L}
Goal: s ⊆ ↑(LieSubalgebra.lieSpan R L s)
-- LieSubalgebra.submodule_span_le_lieSpan
theorem submodule_span_le_lieSpan : Submodule.span R s ≤ lieSpan R L s 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s : Set L}
Goal: Submodule.span R s ≤ (LieSubalgebra.lieSpan R L s).toSubmodule
-- LieSubalgebra.lieSpan_le
theorem lieSpan_le {K} : lieSpan R L s ≤ K ↔ s ⊆ K 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s : Set L} {K : LieSubalgebra R L}
Goal: LieSubalgebra.lieSpan R L s ≤ K ↔ s ⊆ ↑K
-- LieSubalgebra.lieSpan_mono
theorem lieSpan_mono {t : Set L} (h : s ⊆ t) : lieSpan R L s ≤ lieSpan R L t 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s t : Set L} (h : s ⊆ t)
Goal: LieSubalgebra.lieSpan R L s ≤ LieSubalgebra.lieSpan R L t
-- LieSubalgebra.lieSpan_eq
theorem lieSpan_eq : lieSpan R L (K : Set L) = K 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L)
Goal: LieSubalgebra.lieSpan R L ↑K = K
-- LieSubalgebra.coe_lieSpan_submodule_eq_iff
theorem coe_lieSpan_submodule_eq_iff {p : Submodule R L} :
    (lieSpan R L (p : Set L) : Submodule R L) = p ↔ ∃ K : LieSubalgebra R L, ↑K = p 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {p : Submodule R L}
Goal: (LieSubalgebra.lieSpan R L ↑p).toSubmodule = p ↔ ∃ K, K.toSubmodule = p
-- LieSubalgebra.span_empty
@[simp]
theorem span_empty : lieSpan R L (∅ : Set L) = ⊥ 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: LieSubalgebra.lieSpan R L ∅ = ⊥
-- LieSubalgebra.span_univ
@[simp]
theorem span_univ : lieSpan R L (Set.univ : Set L) = ⊤ 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: LieSubalgebra.lieSpan R L Set.univ = ⊤
-- LieSubalgebra.span_union
theorem span_union (s t : Set L) : lieSpan R L (s ∪ t) = lieSpan R L s ⊔ lieSpan R L t 
Arguments: (R : Type u) {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (s t : Set L)
Goal: LieSubalgebra.lieSpan R L (s ∪ t) = LieSubalgebra.lieSpan R L s ⊔ LieSubalgebra.lieSpan R L t
-- LieSubalgebra.span_iUnion
theorem span_iUnion {ι} (s : ι → Set L) : lieSpan R L (⋃ i, s i) = ⨆ i, lieSpan R L (s i) 
Arguments: (R : Type u) {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {ι : Sort u_1} (s : ι → Set L)
Goal: LieSubalgebra.lieSpan R L (⋃ i, s i) = ⨆ i, LieSubalgebra.lieSpan R L (s i)
-- LieSubalgebra.lieSpan_induction
@[elab_as_elim]
theorem lieSpan_induction {p : L → Prop} {x : L} (h : x ∈ lieSpan R L s) (mem : ∀ x ∈ s, p x)
    (zero : p 0) (smul : ∀ (r : R), ∀ {x : L}, p x → p (r • x))
    (add : ∀ x y, p x → p y → p (x + y)) (lie : ∀ x y, p x → p y → p ⁅x, y⁆) : p x 
Arguments: (R : Type u) {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s : Set L} {p : L → Prop} {x : L} (h : x ∈ LieSubalgebra.lieSpan R L s) (mem : ∀ x ∈ s, p x) (zero : p 0) (smul : ∀ (r : R) {x : L}, p x → p (r • x)) (add : ∀ (x y : L), p x → p y → p (x + y)) (lie : ∀ (x y : L), p x → p y → p ⁅x, y⁆)
Goal: p x
-- LieEquiv.ofInjective_apply
@[simp]
theorem ofInjective_apply (f : L₁ →ₗ⁅R⁆ L₂) (h : Function.Injective f) (x : L₁) :
    ↑(ofInjective f h x) = f x 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (h : Function.Injective ⇑f) (x : L₁)
Goal: ↑((LieEquiv.ofInjective f h) x) = f x
-- LieEquiv.ofEq_apply
@[simp]
theorem ofEq_apply (L L' : LieSubalgebra R L₁) (h : (L : Set L₁) = L') (x : L) :
    (↑(ofEq L L' h x) : L₁) = x 
Arguments: {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] (L L' : LieSubalgebra R L₁) (h : ↑L = ↑L') (x : ↥L)
Goal: ↑((LieEquiv.ofEq L L' h) x) = ↑x
-- LieEquiv.lieSubalgebraMap_apply
@[simp]
theorem lieSubalgebraMap_apply (x : L₁'') : ↑(e.lieSubalgebraMap _ x) = e x 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (L₁'' : LieSubalgebra R L₁) (e : L₁ ≃ₗ⁅R⁆ L₂) (x : ↥L₁'')
Goal: ↑((LieEquiv.lieSubalgebraMap L₁'' e) x) = e ↑x
-- LieEquiv.ofSubalgebras_apply
@[simp]
theorem ofSubalgebras_apply (h : L₁'.map ↑e = L₂') (x : L₁') : ↑(e.ofSubalgebras _ _ h x) = e x 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (L₁' : LieSubalgebra R L₁) (L₂' : LieSubalgebra R L₂) (e : L₁ ≃ₗ⁅R⁆ L₂) (h : LieSubalgebra.map e.toLieHom L₁' = L₂') (x : ↥L₁')
Goal: ↑((LieEquiv.ofSubalgebras L₁' L₂' e h) x) = e ↑x
-- LieEquiv.ofSubalgebras_symm_apply
@[simp]
theorem ofSubalgebras_symm_apply (h : L₁'.map ↑e = L₂') (x : L₂') :
    ↑((e.ofSubalgebras _ _ h).symm x) = e.symm x 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (L₁' : LieSubalgebra R L₁) (L₂' : LieSubalgebra R L₂) (e : L₁ ≃ₗ⁅R⁆ L₂) (h : LieSubalgebra.map e.toLieHom L₁' = L₂') (x : ↥L₂')
Goal: ↑((LieEquiv.ofSubalgebras L₁' L₂' e h).symm x) = e.symm ↑x
-- LieDerivation.leibniz'
structure LieDerivation (R L M : Type*) [CommRing R] [LieRing L] [LieAlgebra R L]
    [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
    extends L →ₗ[R] M where
  protected leibniz' (a b : L) : toLinearMap ⁅a, b⁆ = ⁅a, toLinearMap b⁆ - ⁅b, toLinearMap a⁆
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (self : LieDerivation R L M) (a b : L)
Goal: ↑self ⁅a, b⁆ = ⁅a, ↑self b⁆ - ⁅b, ↑self a⁆
-- LieDerivation.toFun_eq_coe
theorem toFun_eq_coe : D.toFun = ⇑D 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M)
Goal: (↑D).toFun = ⇑D
-- LieDerivation.mk_coe
@[simp]
theorem mk_coe (f : L →ₗ[R] M) (h₁) : ((⟨f, h₁⟩ : LieDerivation R L M) : L → M) = f 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (f : L →ₗ[R] M) (h₁ : ∀ (a b : L), f ⁅a, b⁆ = ⁅a, f b⁆ - ⁅b, f a⁆)
Goal: ⇑{ toLinearMap := f, leibniz' := h₁ } = ⇑f
-- LieDerivation.coeFn_coe
@[simp, norm_cast]
theorem coeFn_coe (f : LieDerivation R L M) : ⇑(f : L →ₗ[R] M) = f 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (f : LieDerivation R L M)
Goal: ⇑↑f = ⇑f
-- LieDerivation.coe_injective
theorem coe_injective : @Function.Injective (LieDerivation R L M) (L → M) DFunLike.coe 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: Function.Injective DFunLike.coe
-- LieDerivation.ext
@[ext]
theorem ext (H : ∀ a, D1 a = D2 a) : D1 = D2 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 D2 : LieDerivation R L M} (H : ∀ (a : L), D1 a = D2 a)
Goal: D1 = D2
-- LieDerivation.congr_fun
theorem congr_fun (h : D1 = D2) (a : L) : D1 a = D2 a 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 D2 : LieDerivation R L M} (h : D1 = D2) (a : L)
Goal: D1 a = D2 a
-- LieDerivation.apply_lie_eq_sub
@[simp]
lemma apply_lie_eq_sub (D : LieDerivation R L M) (a b : L) :
    D ⁅a, b⁆ = ⁅a, D b⁆ - ⁅b, D a⁆ 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) (a b : L)
Goal: D ⁅a, b⁆ = ⁅a, D b⁆ - ⁅b, D a⁆
-- LieDerivation.apply_lie_eq_add
lemma apply_lie_eq_add (D : LieDerivation R L L) (a b : L) :
    D ⁅a, b⁆ = ⁅a, D b⁆ + ⁅D a, b⁆ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (D : LieDerivation R L L) (a b : L)
Goal: D ⁅a, b⁆ = ⁅a, D b⁆ + ⁅D a, b⁆
-- LieDerivation.eqOn_lieSpan
theorem eqOn_lieSpan {s : Set L} (h : Set.EqOn D1 D2 s) :
    Set.EqOn D1 D2 (LieSubalgebra.lieSpan R L s) 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 D2 : LieDerivation R L M} {s : Set L} (h : Set.EqOn (⇑D1) (⇑D2) s)
Goal: Set.EqOn ⇑D1 ⇑D2 ↑(LieSubalgebra.lieSpan R L s)
-- LieDerivation.ext_of_lieSpan_eq_top
theorem ext_of_lieSpan_eq_top (s : Set L) (hs : LieSubalgebra.lieSpan R L s = ⊤)
    (h : Set.EqOn D1 D2 s) : D1 = D2 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 D2 : LieDerivation R L M} (s : Set L) (hs : LieSubalgebra.lieSpan R L s = ⊤) (h : Set.EqOn (⇑D1) (⇑D2) s)
Goal: D1 = D2
-- LieDerivation.coe_zero
@[simp]
theorem coe_zero : ⇑(0 : LieDerivation R L M) = 0 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: ⇑0 = 0
-- LieDerivation.coe_zero_linearMap
@[simp]
theorem coe_zero_linearMap : ↑(0 : LieDerivation R L M) = (0 : L →ₗ[R] M) 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
Goal: ↑0 = 0
-- LieDerivation.zero_apply
theorem zero_apply (a : L) : (0 : LieDerivation R L M) a = 0 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (a : L)
Goal: 0 a = 0
-- LieDerivation.coe_add
@[simp]
theorem coe_add (D1 D2 : LieDerivation R L M) : ⇑(D1 + D2) = D1 + D2 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D1 D2 : LieDerivation R L M)
Goal: ⇑(D1 + D2) = ⇑D1 + ⇑D2
-- LieDerivation.coe_add_linearMap
@[simp]
theorem coe_add_linearMap (D1 D2 : LieDerivation R L M) : ↑(D1 + D2) = (D1 + D2 : L →ₗ[R] M) 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D1 D2 : LieDerivation R L M)
Goal: ↑(D1 + D2) = ↑D1 + ↑D2
-- LieDerivation.add_apply
theorem add_apply : (D1 + D2) a = D1 a + D2 a 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 D2 : LieDerivation R L M} (a : L)
Goal: (D1 + D2) a = D1 a + D2 a
-- LieDerivation.map_neg
protected theorem map_neg : D (-a) = -D a 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) (a : L)
Goal: D (-a) = -D a
-- LieDerivation.map_sub
protected theorem map_sub : D (a - b) = D a - D b 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) (a b : L)
Goal: D (a - b) = D a - D b
-- LieDerivation.coe_neg
@[simp]
theorem coe_neg (D : LieDerivation R L M) : ⇑(-D) = -D 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M)
Goal: ⇑(-D) = -⇑D
-- LieDerivation.coe_neg_linearMap
@[simp]
theorem coe_neg_linearMap (D : LieDerivation R L M) : ↑(-D) = (-D : L →ₗ[R] M) 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M)
Goal: ↑(-D) = -↑D
-- LieDerivation.neg_apply
theorem neg_apply : (-D) a = -D a 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) (a : L)
Goal: (-D) a = -D a
-- LieDerivation.coe_sub
@[simp]
theorem coe_sub (D1 D2 : LieDerivation R L M) : ⇑(D1 - D2) = D1 - D2 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D1 D2 : LieDerivation R L M)
Goal: ⇑(D1 - D2) = ⇑D1 - ⇑D2
-- LieDerivation.coe_sub_linearMap
@[simp]
theorem coe_sub_linearMap (D1 D2 : LieDerivation R L M) : ↑(D1 - D2) = (D1 - D2 : L →ₗ[R] M) 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D1 D2 : LieDerivation R L M)
Goal: ↑(D1 - D2) = ↑D1 - ↑D2
-- LieDerivation.sub_apply
theorem sub_apply {D1 D2 : LieDerivation R L M} : (D1 - D2) a = D1 a - D2 a 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (a : L) {D1 D2 : LieDerivation R L M}
Goal: (D1 - D2) a = D1 a - D2 a
-- LieDerivation.SMulBracketCommClass.smul_bracket_comm
class SMulBracketCommClass (S L α : Type*) [SMul S α] [LieRing L] [AddCommGroup α]
    [LieRingModule L α] : Prop where
    smul_bracket_comm : ∀ (s : S) (l : L) (a : α), s • ⁅l, a⁆ = ⁅l, s • a⁆
Arguments: {S : Type u_4} {L : Type u_5} {α : Type u_6} [SMul S α] [LieRing L] [AddCommGroup α] [LieRingModule L α] [self : LieDerivation.SMulBracketCommClass S L α] (s : S) (l : L) (a : α)
Goal: s • ⁅l, a⁆ = ⁅l, s • a⁆
-- LieDerivation.coe_smul
@[simp]
theorem coe_smul (r : S) (D : LieDerivation R L M) : ⇑(r • D) = r • ⇑D 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {S : Type u_4} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [LieDerivation.SMulBracketCommClass S L M] (r : S) (D : LieDerivation R L M)
Goal: ⇑(r • D) = r • ⇑D
-- LieDerivation.coe_smul_linearMap
@[simp]
theorem coe_smul_linearMap (r : S) (D : LieDerivation R L M) : ↑(r • D) = r • (D : L →ₗ[R] M) 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {S : Type u_4} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [LieDerivation.SMulBracketCommClass S L M] (r : S) (D : LieDerivation R L M)
Goal: ↑(r • D) = r • ↑D
-- LieDerivation.smul_apply
theorem smul_apply (r : S) (D : LieDerivation R L M) : (r • D) a = r • D a 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (a : L) {S : Type u_4} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [LieDerivation.SMulBracketCommClass S L M] (r : S) (D : LieDerivation R L M)
Goal: (r • D) a = r • D a
-- LieDerivation.commutator_coe_linear_map
@[simp]
lemma commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R L), (D2 : Module.End R L)⁆ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {D1 D2 : LieDerivation R L L}
Goal: ↑⁅D1, D2⁆ = ⁅↑D1, ↑D2⁆
-- LieDerivation.commutator_apply
lemma commutator_apply (a : L) : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {D1 D2 : LieDerivation R L L} (a : L)
Goal: ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a)
-- LieDerivation.lie_apply
@[simp] lemma lie_apply (D₁ D₂ : LieDerivation R L L) (x : L) :
    ⁅D₁, D₂⁆ x = D₁ (D₂ x) - D₂ (D₁ x) 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (D₁ D₂ : LieDerivation R L L) (x : L)
Goal: ⁅D₁, D₂⁆ x = D₁ (D₂ x) - D₂ (D₁ x)
-- LieDerivation.toLinearMapLieHom_injective
lemma toLinearMapLieHom_injective : Function.Injective (toLinearMapLieHom R L) 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: Function.Injective ⇑(LieDerivation.toLinearMapLieHom R L)
-- LieDerivation.inner_apply_apply
@[simps!]
def inner : M →ₗ[R] LieDerivation R L M 
Arguments: (R : Type u_1) (L : Type u_2) (M : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (m : M)
Goal: ∀ (m_1 : L), ((LieDerivation.inner R L M) m) m_1 = ⁅m_1, m⁆
-- LieDerivation.lie_lieDerivation_apply
@[simp] lemma lie_lieDerivation_apply (x y : L) (D : LieDerivation R L M) :
    ⁅x, D⁆ y = ⁅y, D x⁆ 
Arguments: (R : Type u_1) (L : Type u_2) (M : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x y : L) (D : LieDerivation R L M)
Goal: ⁅x, D⁆ y = ⁅y, D x⁆
-- LieDerivation.lie_coe_lieDerivation_apply
@[simp] lemma lie_coe_lieDerivation_apply (x : L) (D : LieDerivation R L M) :
    ⁅x, (D : L →ₗ[R] M)⁆ = ⁅x, D⁆ 
Arguments: (R : Type u_1) (L : Type u_2) (M : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L) (D : LieDerivation R L M)
Goal: ⁅x, ↑D⁆ = ↑⁅x, D⁆
-- LieDerivation.leibniz_lie
protected lemma leibniz_lie (x : L) (D₁ D₂ : LieDerivation R L L) :
    ⁅x, ⁅D₁, D₂⁆⁆ = ⁅⁅x, D₁⁆, D₂⁆ + ⁅D₁, ⁅x, D₂⁆⁆ 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (x : L) (D₁ D₂ : LieDerivation R L L)
Goal: ⁅x, ⁅D₁, D₂⁆⁆ = ⁅⁅x, D₁⁆, D₂⁆ + ⁅D₁, ⁅x, D₂⁆⁆
-- FreeLieAlgebra.Rel.addLeft
theorem Rel.addLeft (a : lib R X) {b c : lib R X} (h : Rel R X b c) : Rel R X (a + b) (a + c) 
Arguments: {R : Type u} {X : Type v} [CommRing R] (a : lib R X) {b c : lib R X} (h : FreeLieAlgebra.Rel R X b c)
Goal: FreeLieAlgebra.Rel R X (a + b) (a + c)
-- FreeLieAlgebra.Rel.neg
theorem Rel.neg {a b : lib R X} (h : Rel R X a b) : Rel R X (-a) (-b) 
Arguments: {R : Type u} {X : Type v} [CommRing R] {a b : lib R X} (h : FreeLieAlgebra.Rel R X a b)
Goal: FreeLieAlgebra.Rel R X (-a) (-b)
-- FreeLieAlgebra.Rel.subLeft
theorem Rel.subLeft (a : lib R X) {b c : lib R X} (h : Rel R X b c) : Rel R X (a - b) (a - c) 
Arguments: {R : Type u} {X : Type v} [CommRing R] (a : lib R X) {b c : lib R X} (h : FreeLieAlgebra.Rel R X b c)
Goal: FreeLieAlgebra.Rel R X (a - b) (a - c)
-- FreeLieAlgebra.Rel.subRight
theorem Rel.subRight {a b : lib R X} (c : lib R X) (h : Rel R X a b) : Rel R X (a - c) (b - c) 
Arguments: {R : Type u} {X : Type v} [CommRing R] {a b : lib R X} (c : lib R X) (h : FreeLieAlgebra.Rel R X a b)
Goal: FreeLieAlgebra.Rel R X (a - c) (b - c)
-- FreeLieAlgebra.Rel.smulOfTower
theorem Rel.smulOfTower {S : Type*} [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (t : S)
    (a b : lib R X) (h : Rel R X a b) : Rel R X (t • a) (t • b) 
Arguments: {R : Type u} {X : Type v} [CommRing R] {S : Type u_1} [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (t : S) (a b : lib R X) (h : FreeLieAlgebra.Rel R X a b)
Goal: FreeLieAlgebra.Rel R X (t • a) (t • b)
-- FreeLieAlgebra.liftAux_map_smul
theorem liftAux_map_smul (f : X → L) (t : R) (a : lib R X) :
    liftAux R f (t • a) = t • liftAux R f a 
Arguments: (R : Type u) {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (f : X → L) (t : R) (a : lib R X)
Goal: (FreeLieAlgebra.liftAux R f) (t • a) = t • (FreeLieAlgebra.liftAux R f) a
-- FreeLieAlgebra.liftAux_map_add
theorem liftAux_map_add (f : X → L) (a b : lib R X) :
    liftAux R f (a + b) = liftAux R f a + liftAux R f b 
Arguments: (R : Type u) {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (f : X → L) (a b : lib R X)
Goal: (FreeLieAlgebra.liftAux R f) (a + b) = (FreeLieAlgebra.liftAux R f) a + (FreeLieAlgebra.liftAux R f) b