ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LieModule.coe_maxTrivEquiv_apply
@[norm_cast, simp]
theorem coe_maxTrivEquiv_apply (e : M ≃ₗ⁅R,L⁆ N) (m : maxTrivSubmodule R L M) :
    (maxTrivEquiv e m : N) = e ↑m 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [AddCommGroup N] [Module R N] [LieRingModule L N] [LieModule R L N] (e : M ≃ₗ⁅R,L⁆ N) (m : ↥(LieModule.maxTrivSubmodule R L M))
Goal: ↑((LieModule.maxTrivEquiv e) m) = e ↑m
-- LieModule.maxTrivEquiv_of_refl_eq_refl
@[simp]
theorem maxTrivEquiv_of_refl_eq_refl :
    maxTrivEquiv (LieModuleEquiv.refl : M ≃ₗ⁅R,L⁆ M) = LieModuleEquiv.refl 
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.maxTrivEquiv LieModuleEquiv.refl = LieModuleEquiv.refl
-- LieModule.maxTrivEquiv_of_equiv_symm_eq_symm
@[simp]
theorem maxTrivEquiv_of_equiv_symm_eq_symm (e : M ≃ₗ⁅R,L⁆ N) :
    (maxTrivEquiv e).symm = maxTrivEquiv e.symm 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [AddCommGroup N] [Module R N] [LieRingModule L N] [LieModule R L N] (e : M ≃ₗ⁅R,L⁆ N)
Goal: (LieModule.maxTrivEquiv e).symm = LieModule.maxTrivEquiv e.symm
-- LieModule.coe_maxTrivLinearMapEquivLieModuleHom
@[simp]
theorem coe_maxTrivLinearMapEquivLieModuleHom (f : maxTrivSubmodule R L (M →ₗ[R] N)) :
    (maxTrivLinearMapEquivLieModuleHom f : M → N) = f 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [AddCommGroup N] [Module R N] [LieRingModule L N] [LieModule R L N] (f : ↥(LieModule.maxTrivSubmodule R L (M →ₗ[R] N)))
Goal: ⇑(LieModule.maxTrivLinearMapEquivLieModuleHom f) = ⇑↑f
-- LieModule.coe_maxTrivLinearMapEquivLieModuleHom_symm
@[simp]
theorem coe_maxTrivLinearMapEquivLieModuleHom_symm (f : M →ₗ⁅R,L⁆ N) :
    (maxTrivLinearMapEquivLieModuleHom.symm f : M → N) = f 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [AddCommGroup N] [Module R N] [LieRingModule L N] [LieModule R L N] (f : M →ₗ⁅R,L⁆ N)
Goal: ⇑↑(LieModule.maxTrivLinearMapEquivLieModuleHom.symm f) = ⇑f
-- LieModule.coe_linearMap_maxTrivLinearMapEquivLieModuleHom
@[simp]
theorem coe_linearMap_maxTrivLinearMapEquivLieModuleHom (f : maxTrivSubmodule R L (M →ₗ[R] N)) :
    (maxTrivLinearMapEquivLieModuleHom f : M →ₗ[R] N) = (f : M →ₗ[R] N) 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [AddCommGroup N] [Module R N] [LieRingModule L N] [LieModule R L N] (f : ↥(LieModule.maxTrivSubmodule R L (M →ₗ[R] N)))
Goal: ↑(LieModule.maxTrivLinearMapEquivLieModuleHom f) = ↑f
-- LieModule.coe_linearMap_maxTrivLinearMapEquivLieModuleHom_symm
@[simp]
theorem coe_linearMap_maxTrivLinearMapEquivLieModuleHom_symm (f : M →ₗ⁅R,L⁆ N) :
    (maxTrivLinearMapEquivLieModuleHom.symm f : M →ₗ[R] N) = (f : M →ₗ[R] N) 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [AddCommGroup N] [Module R N] [LieRingModule L N] [LieModule R L N] (f : M →ₗ⁅R,L⁆ N)
Goal: ↑(LieModule.maxTrivLinearMapEquivLieModuleHom.symm f) = ↑f
-- LieAlgebra.ad_ker_eq_self_module_ker
@[simp]
theorem ad_ker_eq_self_module_ker : (ad R L).ker = LieModule.ker R L L 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: (LieAlgebra.ad R L).ker = LieModule.ker R L L
-- LieAlgebra.self_module_ker_eq_center
@[simp]
theorem self_module_ker_eq_center : LieModule.ker R L L = center R L 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: LieModule.ker R L L = LieAlgebra.center R L
-- LieAlgebra.abelian_of_le_center
theorem abelian_of_le_center (I : LieIdeal R L) (h : I ≤ center R L) : IsLieAbelian I 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L) (h : I ≤ LieAlgebra.center R L)
Goal: IsLieAbelian ↥↑I
-- LieAlgebra.isLieAbelian_iff_center_eq_top
theorem isLieAbelian_iff_center_eq_top : IsLieAbelian L ↔ center R L = ⊤ 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: IsLieAbelian L ↔ LieAlgebra.center R L = ⊤
-- LieModule.commute_toEnd_of_mem_center_left
lemma commute_toEnd_of_mem_center_left :
    Commute (toEnd R L M x) (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] {x : L} (hx : x ∈ LieAlgebra.center R L) (y : L)
Goal: Commute ((LieModule.toEnd R L M) x) ((LieModule.toEnd R L M) y)
-- LieModule.commute_toEnd_of_mem_center_right
lemma commute_toEnd_of_mem_center_right :
    Commute (toEnd R L M y) (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] {x : L} (hx : x ∈ LieAlgebra.center R L) (y : L)
Goal: Commute ((LieModule.toEnd R L M) y) ((LieModule.toEnd R L M) x)
-- LieSubmodule.trivial_lie_oper_zero
@[simp]
theorem LieSubmodule.trivial_lie_oper_zero [LieModule.IsTrivial L M] : ⁅I, 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] (N : LieSubmodule R L M) (I : LieIdeal R L) [LieModule.IsTrivial L M]
Goal: ⁅I, N⁆ = ⊥
-- LieSubmodule.lie_abelian_iff_lie_self_eq_bot
theorem LieSubmodule.lie_abelian_iff_lie_self_eq_bot : IsLieAbelian I ↔ ⁅I, I⁆ = ⊥ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L)
Goal: IsLieAbelian ↥↑I ↔ ⁅I, I⁆ = ⊥
-- lie_eq_self_of_isAtom_of_ne_bot
lemma lie_eq_self_of_isAtom_of_ne_bot (hN : IsAtom N) (h : ⁅I, N⁆ ≠ ⊥) : ⁅I, N⁆ = 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] {N : LieSubmodule R L M} {I : LieIdeal R L} (hN : IsAtom N) (h : ⁅I, N⁆ ≠ ⊥)
Goal: ⁅I, N⁆ = N
-- lie_eq_self_of_isAtom_of_nonabelian
lemma lie_eq_self_of_isAtom_of_nonabelian {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
    (I : LieIdeal R L) (hI : IsAtom I) (h : ¬IsLieAbelian I) :
    ⁅I, I⁆ = I 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L) (hI : IsAtom I) (h : ¬IsLieAbelian ↥↑I)
Goal: ⁅I, I⁆ = I
-- LieAlgebra.matrix_trace_commutator_zero
@[simp]
theorem matrix_trace_commutator_zero [Fintype n] (X Y : Matrix n n R) : Matrix.trace ⁅X, Y⁆ = 0 
Arguments: (n : Type u_1) (R : Type u₂) [DecidableEq n] [CommRing R] [Fintype n] (X Y : Matrix n n R)
Goal: ⁅X, Y⁆.trace = 0
-- LieAlgebra.SpecialLinear.sl_bracket
theorem sl_bracket [Fintype n] (A B : sl n R) : ⁅A, B⁆.val = A.val * B.val - B.val * A.val 
Arguments: (n : Type u_1) (R : Type u₂) [DecidableEq n] [CommRing R] [Fintype n] (A B : ↥(LieAlgebra.SpecialLinear.sl n R))
Goal: ↑⁅A, B⁆ = ↑A * ↑B - ↑B * ↑A
-- LieAlgebra.SpecialLinear.eb_val
@[simp]
theorem eb_val (h : j ≠ i) : (Eb R i j h).val = Matrix.stdBasisMatrix i j 1 
Arguments: {n : Type u_1} (R : Type u₂) [DecidableEq n] [CommRing R] [Fintype n] (i j : n) (h : j ≠ i)
Goal: ↑(LieAlgebra.SpecialLinear.Eb R i j h) = Matrix.stdBasisMatrix i j 1
-- LieAlgebra.SpecialLinear.sl_non_abelian
theorem sl_non_abelian [Fintype n] [Nontrivial R] (h : 1 < Fintype.card n) :
    ¬IsLieAbelian (sl n R) 
Arguments: (n : Type u_1) (R : Type u₂) [DecidableEq n] [CommRing R] [Fintype n] [Nontrivial R] (h : 1 < Fintype.card n)
Goal: ¬IsLieAbelian ↥(LieAlgebra.SpecialLinear.sl n R)
-- LieAlgebra.Orthogonal.mem_so
@[simp]
theorem mem_so [Fintype n] (A : Matrix n n R) : A ∈ so n R ↔ Aᵀ = -A 
Arguments: (n : Type u_1) (R : Type u₂) [DecidableEq n] [CommRing R] [Fintype n] (A : Matrix n n R)
Goal: A ∈ LieAlgebra.Orthogonal.so n R ↔ A.transpose = -A
-- LieAlgebra.Orthogonal.pso_inv
theorem pso_inv {i : R} (hi : i * i = -1) : Pso p q R i * Pso p q R (-i) = 1 
Arguments: (p : Type u_2) (q : Type u_3) (R : Type u₂) [DecidableEq p] [DecidableEq q] [CommRing R] [Fintype p] [Fintype q] {i : R} (hi : i * i = -1)
Goal: LieAlgebra.Orthogonal.Pso p q R i * LieAlgebra.Orthogonal.Pso p q R (-i) = 1
-- LieAlgebra.Orthogonal.indefiniteDiagonal_transform
theorem indefiniteDiagonal_transform {i : R} (hi : i * i = -1) :
    (Pso p q R i)ᵀ * indefiniteDiagonal p q R * Pso p q R i = 1 
Arguments: (p : Type u_2) (q : Type u_3) (R : Type u₂) [DecidableEq p] [DecidableEq q] [CommRing R] [Fintype p] [Fintype q] {i : R} (hi : i * i = -1)
Goal: (LieAlgebra.Orthogonal.Pso p q R i).transpose * LieAlgebra.Orthogonal.indefiniteDiagonal p q R *
    LieAlgebra.Orthogonal.Pso p q R i =
  1
-- LieAlgebra.Orthogonal.soIndefiniteEquiv_apply
theorem soIndefiniteEquiv_apply {i : R} (hi : i * i = -1) (A : so' p q R) :
    (soIndefiniteEquiv p q R hi A : Matrix (p ⊕ q) (p ⊕ q) R) =
      (Pso p q R i)⁻¹ * (A : Matrix (p ⊕ q) (p ⊕ q) R) * Pso p q R i 
Arguments: (p : Type u_2) (q : Type u_3) (R : Type u₂) [DecidableEq p] [DecidableEq q] [CommRing R] [Fintype p] [Fintype q] {i : R} (hi : i * i = -1) (A : ↥(LieAlgebra.Orthogonal.so' p q R))
Goal: ↑((LieAlgebra.Orthogonal.soIndefiniteEquiv p q R hi) A) =
  (LieAlgebra.Orthogonal.Pso p q R i)⁻¹ * ↑A * LieAlgebra.Orthogonal.Pso p q R i
-- LieAlgebra.Orthogonal.s_as_blocks
theorem s_as_blocks : S l R = Matrix.fromBlocks 1 0 0 (-1) 
Arguments: (l : Type u_4) (R : Type u₂) [DecidableEq l] [CommRing R]
Goal: LieAlgebra.Orthogonal.S l R = Matrix.fromBlocks 1 0 0 (-1)
-- LieAlgebra.Orthogonal.jd_transform
theorem jd_transform [Fintype l] : (PD l R)ᵀ * JD l R * PD l R = (2 : R) • S l R 
Arguments: (l : Type u_4) (R : Type u₂) [DecidableEq l] [CommRing R] [Fintype l]
Goal: (LieAlgebra.Orthogonal.PD l R).transpose * LieAlgebra.Orthogonal.JD l R * LieAlgebra.Orthogonal.PD l R =
  2 • LieAlgebra.Orthogonal.S l R
-- LieAlgebra.Orthogonal.pd_inv
theorem pd_inv [Fintype l] [Invertible (2 : R)] : PD l R * ⅟ (2 : R) • (PD l R)ᵀ = 1 
Arguments: (l : Type u_4) (R : Type u₂) [DecidableEq l] [CommRing R] [Fintype l] [Invertible 2]
Goal: LieAlgebra.Orthogonal.PD l R * ⅟2 • (LieAlgebra.Orthogonal.PD l R).transpose = 1
-- LieAlgebra.Orthogonal.pb_inv
theorem pb_inv [Invertible (2 : R)] : PB l R * Matrix.fromBlocks 1 0 0 (⅟ (PD l R)) = 1 
Arguments: (l : Type u_4) (R : Type u₂) [DecidableEq l] [CommRing R] [Fintype l] [Invertible 2]
Goal: LieAlgebra.Orthogonal.PB l R * Matrix.fromBlocks 1 0 0 ⅟(LieAlgebra.Orthogonal.PD l R) = 1
-- LieAlgebra.Orthogonal.jb_transform
theorem jb_transform : (PB l R)ᵀ * JB l R * PB l R = (2 : R) • Matrix.fromBlocks 1 0 0 (S l R) 
Arguments: (l : Type u_4) (R : Type u₂) [DecidableEq l] [CommRing R] [Fintype l]
Goal: (LieAlgebra.Orthogonal.PB l R).transpose * LieAlgebra.Orthogonal.JB l R * LieAlgebra.Orthogonal.PB l R =
  2 • Matrix.fromBlocks 1 0 0 (LieAlgebra.Orthogonal.S l R)
-- LieAlgebra.Orthogonal.indefiniteDiagonal_assoc
theorem indefiniteDiagonal_assoc :
    indefiniteDiagonal (Unit ⊕ l) l R =
      Matrix.reindexLieEquiv (Equiv.sumAssoc Unit l l).symm
        (Matrix.fromBlocks 1 0 0 (indefiniteDiagonal l l R)) 
Arguments: (l : Type u_4) (R : Type u₂) [DecidableEq l] [CommRing R] [Fintype l]
Goal: LieAlgebra.Orthogonal.indefiniteDiagonal (Unit ⊕ l) l R =
  (Matrix.reindexLieEquiv (Equiv.sumAssoc Unit l l).symm)
    (Matrix.fromBlocks 1 0 0 (LieAlgebra.Orthogonal.indefiniteDiagonal l l R))
-- LieSubalgebra.IsCartanSubalgebra.self_normalizing
class IsCartanSubalgebra : Prop where
  nilpotent : LieAlgebra.IsNilpotent R H
  self_normalizing : H.normalizer = H
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {H : LieSubalgebra R L} [self : H.IsCartanSubalgebra]
Goal: H.normalizer = H
-- LieSubalgebra.IsCartanSubalgebra.nilpotent
class IsCartanSubalgebra : Prop where
  nilpotent : LieAlgebra.IsNilpotent R H
  self_normalizing : H.normalizer = H
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {H : LieSubalgebra R L} [self : H.IsCartanSubalgebra]
Goal: LieAlgebra.IsNilpotent R ↥H
-- LieSubalgebra.normalizer_eq_self_of_isCartanSubalgebra
@[simp]
theorem normalizer_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
    H.toLieSubmodule.normalizer = H.toLieSubmodule 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra]
Goal: H.toLieSubmodule.normalizer = H.toLieSubmodule
-- LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra
@[simp]
theorem ucs_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ) :
    H.toLieSubmodule.ucs k = H.toLieSubmodule 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ)
Goal: LieSubmodule.ucs k H.toLieSubmodule = H.toLieSubmodule
-- LieSubalgebra.isCartanSubalgebra_iff_isUcsLimit
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L)
Goal: H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit
-- LieSubalgebra.ne_bot_of_isCartanSubalgebra
lemma ne_bot_of_isCartanSubalgebra [Nontrivial L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
    H ≠ ⊥ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] [Nontrivial L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra]
Goal: H ≠ ⊥
-- LieIdeal.normalizer_eq_top
@[simp]
theorem LieIdeal.normalizer_eq_top {R : Type u} {L : Type v} [CommRing R] [LieRing L]
    [LieAlgebra R L] (I : LieIdeal R L) : (I : LieSubalgebra R L).normalizer = ⊤ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L)
Goal: (lieIdealSubalgebra R L I).normalizer = ⊤
-- LieDerivation.ad_apply_apply
@[simps!]
def ad : L →ₗ⁅R⁆ LieDerivation R L L 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (a : L)
Goal: ∀ (a_1 : L), ((LieDerivation.ad R L) a) a_1 = ⁅a, a_1⁆
-- LieDerivation.coe_ad_apply_eq_ad_apply
@[simp] lemma coe_ad_apply_eq_ad_apply (x : L) : ad R L x = LieAlgebra.ad R L x 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (x : L)
Goal: ↑((LieDerivation.ad R L) x) = (LieAlgebra.ad R L) x
-- LieDerivation.ad_apply_lieDerivation
lemma ad_apply_lieDerivation (x : L) (D : LieDerivation R L L) : ad R L (D x) = - ⁅x, D⁆ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (x : L) (D : LieDerivation R L L)
Goal: (LieDerivation.ad R L) (D x) = -⁅x, D⁆
-- LieDerivation.lie_ad
lemma lie_ad (x : L) (D : LieDerivation R L L) : ⁅ad R L x, D⁆ = ⁅x, D⁆ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (x : L) (D : LieDerivation R L L)
Goal: ⁅(LieDerivation.ad R L) x, D⁆ = ⁅x, D⁆
-- LieDerivation.ad_ker_eq_center
lemma ad_ker_eq_center : (ad R L).ker = LieAlgebra.center R L 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: (LieDerivation.ad R L).ker = LieAlgebra.center R L
-- LieDerivation.injective_ad_of_center_eq_bot
lemma injective_ad_of_center_eq_bot (h : LieAlgebra.center R L = ⊥) :
    Function.Injective (ad R L) 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (h : LieAlgebra.center R L = ⊥)
Goal: Function.Injective ⇑(LieDerivation.ad R L)
-- LieDerivation.lie_der_ad_eq_ad_der
lemma lie_der_ad_eq_ad_der (D : LieDerivation R L L) (x : L) : ⁅D, ad R L x⁆ = ad R L (D x) 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (D : LieDerivation R L L) (x : L)
Goal: ⁅D, (LieDerivation.ad R L) x⁆ = (LieDerivation.ad R L) (D x)
-- LieDerivation.ad_isIdealMorphism
lemma ad_isIdealMorphism : (ad R L).IsIdealMorphism 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: (LieDerivation.ad R L).IsIdealMorphism
-- LieDerivation.mem_ad_idealRange_iff
lemma mem_ad_idealRange_iff {D : LieDerivation R L L} :
    D ∈ (ad R L).idealRange ↔ ∃ x : L, ad R L x = D 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {D : LieDerivation R L L}
Goal: D ∈ (LieDerivation.ad R L).idealRange ↔ ∃ x, (LieDerivation.ad R L) x = D
-- LieDerivation.maxTrivSubmodule_eq_bot_of_center_eq_bot
lemma maxTrivSubmodule_eq_bot_of_center_eq_bot (h : LieAlgebra.center R L = ⊥) :
    LieModule.maxTrivSubmodule R L (LieDerivation R L L) = ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (h : LieAlgebra.center R L = ⊥)
Goal: LieModule.maxTrivSubmodule R L (LieDerivation R L L) = ⊥
-- LinearMap.BilinForm.isSkewAdjoint_bracket
theorem LinearMap.BilinForm.isSkewAdjoint_bracket {f g : Module.End R M}
    (hf : f ∈ B.skewAdjointSubmodule) (hg : g ∈ B.skewAdjointSubmodule) :
    ⁅f, g⁆ ∈ B.skewAdjointSubmodule 
Arguments: {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] (B : LinearMap.BilinForm R M) {f g : Module.End R M} (hf : f ∈ LinearMap.skewAdjointSubmodule B) (hg : g ∈ LinearMap.skewAdjointSubmodule B)
Goal: ⁅f, g⁆ ∈ LinearMap.skewAdjointSubmodule B
-- skewAdjointLieSubalgebraEquiv_apply
@[simp]
theorem skewAdjointLieSubalgebraEquiv_apply
    (f : skewAdjointLieSubalgebra (B.compl₁₂ (Qₗ := N) (Qₗ' := N) ↑e ↑e)) :
    ↑(skewAdjointLieSubalgebraEquiv B e f) = e.lieConj f 
Arguments: {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] (B : LinearMap.BilinForm R M) {N : Type w} [AddCommGroup N] [Module R N] (e : N ≃ₗ[R] M) (f : ↥(skewAdjointLieSubalgebra (LinearMap.compl₁₂ B ↑e ↑e)))
Goal: ↑((skewAdjointLieSubalgebraEquiv B e) f) = e.lieConj ↑f
-- skewAdjointLieSubalgebraEquiv_symm_apply
@[simp]
theorem skewAdjointLieSubalgebraEquiv_symm_apply (f : skewAdjointLieSubalgebra B) :
    ↑((skewAdjointLieSubalgebraEquiv B e).symm f) = e.symm.lieConj f 
Arguments: {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] (B : LinearMap.BilinForm R M) {N : Type w} [AddCommGroup N] [Module R N] (e : N ≃ₗ[R] M) (f : ↥(skewAdjointLieSubalgebra B))
Goal: ↑((skewAdjointLieSubalgebraEquiv B e).symm f) = e.symm.lieConj ↑f
-- Matrix.lie_transpose
theorem Matrix.lie_transpose (A B : Matrix n n R) : ⁅A, B⁆ᵀ = ⁅Bᵀ, Aᵀ⁆ 
Arguments: {R : Type u} {n : Type w} [CommRing R] [DecidableEq n] [Fintype n] (A B : Matrix n n R)
Goal: ⁅A, B⁆.transpose = ⁅B.transpose, A.transpose⁆
-- Matrix.isSkewAdjoint_bracket
theorem Matrix.isSkewAdjoint_bracket {A B : Matrix n n R} (hA : A ∈ skewAdjointMatricesSubmodule J)
    (hB : B ∈ skewAdjointMatricesSubmodule J) : ⁅A, B⁆ ∈ skewAdjointMatricesSubmodule J 
Arguments: {R : Type u} {n : Type w} [CommRing R] [DecidableEq n] [Fintype n] (J : Matrix n n R) {A B : Matrix n n R} (hA : A ∈ skewAdjointMatricesSubmodule J) (hB : B ∈ skewAdjointMatricesSubmodule J)
Goal: ⁅A, B⁆ ∈ skewAdjointMatricesSubmodule J
-- mem_skewAdjointMatricesLieSubalgebra
@[simp]
theorem mem_skewAdjointMatricesLieSubalgebra (A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra J ↔ A ∈ skewAdjointMatricesSubmodule J 
Arguments: {R : Type u} {n : Type w} [CommRing R] [DecidableEq n] [Fintype n] (J A : Matrix n n R)
Goal: A ∈ skewAdjointMatricesLieSubalgebra J ↔ A ∈ skewAdjointMatricesSubmodule J
-- skewAdjointMatricesLieSubalgebraEquiv_apply
theorem skewAdjointMatricesLieSubalgebraEquiv_apply (P : Matrix n n R) (h : Invertible P)
    (A : skewAdjointMatricesLieSubalgebra J) :
    ↑(skewAdjointMatricesLieSubalgebraEquiv J P h A) = P⁻¹ * (A : Matrix n n R) * P 
Arguments: {R : Type u} {n : Type w} [CommRing R] [DecidableEq n] [Fintype n] (J P : Matrix n n R) (h : Invertible P) (A : ↥(skewAdjointMatricesLieSubalgebra J))
Goal: ↑((skewAdjointMatricesLieSubalgebraEquiv J P h) A) = P⁻¹ * ↑A * P
-- skewAdjointMatricesLieSubalgebraEquivTranspose_apply
@[simp]
theorem skewAdjointMatricesLieSubalgebraEquivTranspose_apply {m : Type w} [DecidableEq m]
    [Fintype m] (e : Matrix n n R ≃ₐ[R] Matrix m m R) (h : ∀ A, (e A)ᵀ = e Aᵀ)
    (A : skewAdjointMatricesLieSubalgebra J) :
    (skewAdjointMatricesLieSubalgebraEquivTranspose J e h A : Matrix m m R) = e A 
Arguments: {R : Type u} {n : Type w} [CommRing R] [DecidableEq n] [Fintype n] (J : Matrix n n R) {m : Type w} [DecidableEq m] [Fintype m] (e : Matrix n n R ≃ₐ[R] Matrix m m R) (h : ∀ (A : Matrix n n R), (e A).transpose = e A.transpose) (A : ↥(skewAdjointMatricesLieSubalgebra J))
Goal: ↑((skewAdjointMatricesLieSubalgebraEquivTranspose J e h) A) = e ↑A
-- mem_skewAdjointMatricesLieSubalgebra_unit_smul
theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J 
Arguments: {R : Type u} {n : Type w} [CommRing R] [DecidableEq n] [Fintype n] (u : Rˣ) (J A : Matrix n n R)
Goal: A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J
-- IsSl2Triple.lie_e_f
structure IsSl2Triple (h e f : L) : Prop where
  h_ne_zero : h ≠ 0
  lie_e_f : ⁅e, f⁆ = h
  lie_h_e_nsmul : ⁅h, e⁆ = 2 • e
  lie_h_f_nsmul : ⁅h, f⁆ = - (2 • f)
Arguments: {L : Type u_2} [LieRing L] {h e f : L} (self : IsSl2Triple h e f)
Goal: ⁅e, f⁆ = h
-- IsSl2Triple.h_ne_zero
structure IsSl2Triple (h e f : L) : Prop where
  h_ne_zero : h ≠ 0
  lie_e_f : ⁅e, f⁆ = h
  lie_h_e_nsmul : ⁅h, e⁆ = 2 • e
  lie_h_f_nsmul : ⁅h, f⁆ = - (2 • f)
Arguments: {L : Type u_2} [LieRing L] {h e f : L} (self : IsSl2Triple h e f)
Goal: h ≠ 0
-- IsSl2Triple.lie_h_e_nsmul
structure IsSl2Triple (h e f : L) : Prop where
  h_ne_zero : h ≠ 0
  lie_e_f : ⁅e, f⁆ = h
  lie_h_e_nsmul : ⁅h, e⁆ = 2 • e
  lie_h_f_nsmul : ⁅h, f⁆ = - (2 • f)
Arguments: {L : Type u_2} [LieRing L] {h e f : L} (self : IsSl2Triple h e f)
Goal: ⁅h, e⁆ = 2 • e
-- IsSl2Triple.lie_h_f_nsmul
structure IsSl2Triple (h e f : L) : Prop where
  h_ne_zero : h ≠ 0
  lie_e_f : ⁅e, f⁆ = h
  lie_h_e_nsmul : ⁅h, e⁆ = 2 • e
  lie_h_f_nsmul : ⁅h, f⁆ = - (2 • f)
Arguments: {L : Type u_2} [LieRing L] {h e f : L} (self : IsSl2Triple h e f)
Goal: ⁅h, f⁆ = -(2 • f)
-- IsSl2Triple.symm
lemma symm (ht : IsSl2Triple h e f) : IsSl2Triple (-h) f e 
Arguments: {L : Type u_2} [LieRing L] {h e f : L} (ht : IsSl2Triple h e f)
Goal: IsSl2Triple (-h) f e
-- IsSl2Triple.symm_iff
@[simp] lemma symm_iff : IsSl2Triple (-h) f e ↔ IsSl2Triple h e f 
Arguments: {L : Type u_2} [LieRing L] {h e f : L}
Goal: IsSl2Triple (-h) f e ↔ IsSl2Triple h e f
-- IsSl2Triple.lie_h_e_smul
lemma lie_h_e_smul (t : IsSl2Triple h e f) : ⁅h, e⁆ = (2 : R) • e 
Arguments: (R : Type u_1) {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {h e f : L} (t : IsSl2Triple h e f)
Goal: ⁅h, e⁆ = 2 • e
-- IsSl2Triple.lie_lie_smul_f
lemma lie_lie_smul_f (t : IsSl2Triple h e f) : ⁅h, f⁆ = -((2 : R) • f) 
Arguments: (R : Type u_1) {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {h e f : L} (t : IsSl2Triple h e f)
Goal: ⁅h, f⁆ = -(2 • f)
-- IsSl2Triple.e_ne_zero
lemma e_ne_zero (t : IsSl2Triple h e f) : e ≠ 0 
Arguments: {L : Type u_2} [LieRing L] {h e f : L} (t : IsSl2Triple h e f)
Goal: e ≠ 0
-- IsSl2Triple.f_ne_zero
lemma f_ne_zero (t : IsSl2Triple h e f) : f ≠ 0 
Arguments: {L : Type u_2} [LieRing L] {h e f : L} (t : IsSl2Triple h e f)
Goal: f ≠ 0
-- IsSl2Triple.HasPrimitiveVectorWith.lie_h
structure HasPrimitiveVectorWith (t : IsSl2Triple h e f) (m : M) (μ : R) : Prop where
  ne_zero : m ≠ 0
  lie_h : ⁅h, m⁆ = μ • m
  lie_e : ⁅e, m⁆ = 0
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {h e f : L} {t : IsSl2Triple h e f} {m : M} {μ : R} (self : t.HasPrimitiveVectorWith m μ)
Goal: ⁅h, m⁆ = μ • m