ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsSl2Triple.HasPrimitiveVectorWith.ne_zero
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: m ≠ 0
-- IsSl2Triple.HasPrimitiveVectorWith.lie_e
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: ⁅e, m⁆ = 0
-- IsSl2Triple.HasPrimitiveVectorWith.mk'
lemma HasPrimitiveVectorWith.mk' [NoZeroSMulDivisors ℤ M] (t : IsSl2Triple h e f) (m : M) (μ ρ : R)
    (hm : m ≠ 0) (hm' : ⁅h, m⁆ = μ • m) (he : ⁅e, m⁆ = ρ • m) :
    HasPrimitiveVectorWith t m μ  
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] {h e f : L} [NoZeroSMulDivisors ℤ M] (t : IsSl2Triple h e f) (m : M) (μ ρ : R) (hm : m ≠ 0) (hm' : ⁅h, m⁆ = μ • m) (he : ⁅e, m⁆ = ρ • m)
Goal: t.HasPrimitiveVectorWith m μ
-- IsSl2Triple.HasPrimitiveVectorWith.lie_h_pow_toEnd_f
lemma lie_h_pow_toEnd_f (n : ℕ) :
    ⁅h, ψ n⁆ = (μ - 2 * n) • ψ n 
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] {h e f : L} {m : M} {μ : R} {t : IsSl2Triple h e f} (P : t.HasPrimitiveVectorWith m μ) (n : ℕ)
Goal: ⁅h, ((LieModule.toEnd R L M) f ^ n) m⁆ = (μ - 2 * ↑n) • ((LieModule.toEnd R L M) f ^ n) m
-- IsSl2Triple.HasPrimitiveVectorWith.lie_f_pow_toEnd_f
@[nolint unusedArguments]
lemma lie_f_pow_toEnd_f (P : HasPrimitiveVectorWith t m μ) (n : ℕ) :
    ⁅f, ψ n⁆ = ψ (n + 1) 
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] {h e f : L} {m : M} {μ : R} {t : IsSl2Triple h e f} (P : t.HasPrimitiveVectorWith m μ) (n : ℕ)
Goal: ⁅f, ((LieModule.toEnd R L M) f ^ n) m⁆ = ((LieModule.toEnd R L M) f ^ (n + 1)) m
-- IsSl2Triple.HasPrimitiveVectorWith.lie_e_pow_succ_toEnd_f
lemma lie_e_pow_succ_toEnd_f (n : ℕ) :
    ⁅e, ψ (n + 1)⁆ = ((n + 1) * (μ - n)) • ψ n 
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] {h e f : L} {m : M} {μ : R} {t : IsSl2Triple h e f} (P : t.HasPrimitiveVectorWith m μ) (n : ℕ)
Goal: ⁅e, ((LieModule.toEnd R L M) f ^ (n + 1)) m⁆ = ((↑n + 1) * (μ - ↑n)) • ((LieModule.toEnd R L M) f ^ n) m
-- IsSl2Triple.HasPrimitiveVectorWith.exists_nat
lemma exists_nat [IsNoetherian R M] [NoZeroSMulDivisors R M] [IsDomain R] [CharZero R] :
    ∃ n : ℕ, μ = n 
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] {h e f : L} {m : M} {μ : R} {t : IsSl2Triple h e f} (P : t.HasPrimitiveVectorWith m μ) [IsNoetherian R M] [NoZeroSMulDivisors R M] [IsDomain R] [CharZero R]
Goal: ∃ n, μ = ↑n
-- IsSl2Triple.HasPrimitiveVectorWith.pow_toEnd_f_ne_zero_of_eq_nat
lemma pow_toEnd_f_ne_zero_of_eq_nat
    [CharZero R] [NoZeroSMulDivisors R M]
    {n : ℕ} (hn : μ = n) {i} (hi : i ≤ n) : (ψ i) ≠ 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] {h e f : L} {m : M} {μ : R} {t : IsSl2Triple h e f} (P : t.HasPrimitiveVectorWith m μ) [CharZero R] [NoZeroSMulDivisors R M] {n : ℕ} (hn : μ = ↑n) {i : ℕ} (hi : i ≤ n)
Goal: ((LieModule.toEnd R L M) f ^ i) m ≠ 0
-- IsSl2Triple.HasPrimitiveVectorWith.pow_toEnd_f_eq_zero_of_eq_nat
lemma pow_toEnd_f_eq_zero_of_eq_nat
    [IsNoetherian R M] [NoZeroSMulDivisors R M] [IsDomain R] [CharZero R]
    {n : ℕ} (hn : μ = n) : (ψ (n + 1)) = 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] {h e f : L} {m : M} {μ : R} {t : IsSl2Triple h e f} (P : t.HasPrimitiveVectorWith m μ) [IsNoetherian R M] [NoZeroSMulDivisors R M] [IsDomain R] [CharZero R] {n : ℕ} (hn : μ = ↑n)
Goal: ((LieModule.toEnd R L M) f ^ (n + 1)) m = 0
-- UniversalEnvelopingAlgebra.ι_apply
@[simps!] -- Porting note (#11445): added
def ι : L →ₗ⁅R⁆ UniversalEnvelopingAlgebra R L 
Arguments: (R : Type u₁) {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: ∀ (a : L), (UniversalEnvelopingAlgebra.ι R) a = (UniversalEnvelopingAlgebra.mkAlgHom R L) (ιₜ a)
-- UniversalEnvelopingAlgebra.lift_symm_apply
@[simp]
theorem lift_symm_apply (F : UniversalEnvelopingAlgebra R L →ₐ[R] A) :
    (lift R).symm F = (F : UniversalEnvelopingAlgebra R L →ₗ⁅R⁆ A).comp (ι R) 
Arguments: (R : Type u₁) {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] {A : Type u₃} [Ring A] [Algebra R A] (F : UniversalEnvelopingAlgebra R L →ₐ[R] A)
Goal: (UniversalEnvelopingAlgebra.lift R).symm F = F.toLieHom.comp (UniversalEnvelopingAlgebra.ι R)
-- UniversalEnvelopingAlgebra.ι_comp_lift
@[simp]
theorem ι_comp_lift : lift R f ∘ ι R = f 
Arguments: (R : Type u₁) {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] {A : Type u₃} [Ring A] [Algebra R A] (f : L →ₗ⁅R⁆ A)
Goal: ⇑((UniversalEnvelopingAlgebra.lift R) f) ∘ ⇑(UniversalEnvelopingAlgebra.ι R) = ⇑f
-- UniversalEnvelopingAlgebra.lift_ι_apply
theorem lift_ι_apply (x : L) : lift R f (ι R x) = f x 
Arguments: (R : Type u₁) {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] {A : Type u₃} [Ring A] [Algebra R A] (f : L →ₗ⁅R⁆ A) (x : L)
Goal: ((UniversalEnvelopingAlgebra.lift R) f) ((UniversalEnvelopingAlgebra.ι R) x) = f x
-- UniversalEnvelopingAlgebra.lift_ι_apply'
@[simp]
theorem lift_ι_apply' (x : L) :
    lift R f ((UniversalEnvelopingAlgebra.mkAlgHom R L) (ιₜ x)) = f x 
Arguments: (R : Type u₁) {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] {A : Type u₃} [Ring A] [Algebra R A] (f : L →ₗ⁅R⁆ A) (x : L)
Goal: ((UniversalEnvelopingAlgebra.lift R) f) ((UniversalEnvelopingAlgebra.mkAlgHom R L) (ιₜ x)) = f x
-- UniversalEnvelopingAlgebra.lift_unique
theorem lift_unique (g : UniversalEnvelopingAlgebra R L →ₐ[R] A) : g ∘ ι R = f ↔ g = lift R f 
Arguments: (R : Type u₁) {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] {A : Type u₃} [Ring A] [Algebra R A] (f : L →ₗ⁅R⁆ A) (g : UniversalEnvelopingAlgebra R L →ₐ[R] A)
Goal: ⇑g ∘ ⇑(UniversalEnvelopingAlgebra.ι R) = ⇑f ↔ g = (UniversalEnvelopingAlgebra.lift R) f
-- UniversalEnvelopingAlgebra.hom_ext
@[ext]
theorem hom_ext {g₁ g₂ : UniversalEnvelopingAlgebra R L →ₐ[R] A}
    (h :
      (g₁ : UniversalEnvelopingAlgebra R L →ₗ⁅R⁆ A).comp (ι R) =
        (g₂ : UniversalEnvelopingAlgebra R L →ₗ⁅R⁆ A).comp (ι R)) :
    g₁ = g₂ 
Arguments: (R : Type u₁) {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] {A : Type u₃} [Ring A] [Algebra R A] {g₁ g₂ : UniversalEnvelopingAlgebra R L →ₐ[R] A} (h : g₁.toLieHom.comp (UniversalEnvelopingAlgebra.ι R) = g₂.toLieHom.comp (UniversalEnvelopingAlgebra.ι R))
Goal: g₁ = g₂
-- CategoryTheory.Bicategory.rightZigzag_idempotent_of_left_triangle
theorem rightZigzag_idempotent_of_left_triangle
    (η : 𝟙 a ⟶ f ≫ g) (ε : g ≫ f ⟶ 𝟙 b) (h : leftZigzag η ε = (λ_ _).hom ≫ (ρ_ _).inv) :
    rightZigzag η ε ⊗≫ rightZigzag η ε = rightZigzag η ε 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (η : CategoryTheory.CategoryStruct.id a ⟶ CategoryTheory.CategoryStruct.comp f g) (ε : CategoryTheory.CategoryStruct.comp g f ⟶ CategoryTheory.CategoryStruct.id b) (h :
  CategoryTheory.Bicategory.leftZigzag η ε =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor f).hom
      (CategoryTheory.Bicategory.rightUnitor f).inv)
Goal: CategoryTheory.bicategoricalComp (CategoryTheory.Bicategory.rightZigzag η ε)
    (CategoryTheory.Bicategory.rightZigzag η ε) =
  CategoryTheory.Bicategory.rightZigzag η ε
-- CategoryTheory.Bicategory.Adjunction.left_triangle
structure Adjunction (f : a ⟶ b) (g : b ⟶ a) where
    unit : 𝟙 a ⟶ f ≫ g
    counit : g ≫ f ⟶ 𝟙 b
    left_triangle : leftZigzag unit counit = (λ_ _).hom ≫ (ρ_ _).inv := by aesop_cat
    right_triangle : rightZigzag unit counit = (ρ_ _).hom ≫ (λ_ _).inv := by aesop_cat
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (self : CategoryTheory.Bicategory.Adjunction f g)
Goal: CategoryTheory.Bicategory.leftZigzag self.unit self.counit =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor f).hom
    (CategoryTheory.Bicategory.rightUnitor f).inv
-- CategoryTheory.Bicategory.Adjunction.right_triangle
structure Adjunction (f : a ⟶ b) (g : b ⟶ a) where
    unit : 𝟙 a ⟶ f ≫ g
    counit : g ≫ f ⟶ 𝟙 b
    left_triangle : leftZigzag unit counit = (λ_ _).hom ≫ (ρ_ _).inv := by aesop_cat
    right_triangle : rightZigzag unit counit = (ρ_ _).hom ≫ (λ_ _).inv := by aesop_cat
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (self : CategoryTheory.Bicategory.Adjunction f g)
Goal: CategoryTheory.Bicategory.rightZigzag self.unit self.counit =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor g).hom
    (CategoryTheory.Bicategory.leftUnitor g).inv
-- CategoryTheory.Bicategory.Adjunction.comp_left_triangle_aux
theorem comp_left_triangle_aux (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂) :
    leftZigzag (compUnit adj₁ adj₂) (compCounit adj₁ adj₂) = (λ_ _).hom ≫ (ρ_ _).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f₁ : a ⟶ b} {g₁ : b ⟶ a} {f₂ : b ⟶ c} {g₂ : c ⟶ b} (adj₁ : CategoryTheory.Bicategory.Adjunction f₁ g₁) (adj₂ : CategoryTheory.Bicategory.Adjunction f₂ g₂)
Goal: CategoryTheory.Bicategory.leftZigzag (adj₁.compUnit adj₂) (adj₁.compCounit adj₂) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.leftUnitor (CategoryTheory.CategoryStruct.comp f₁ f₂)).hom
    (CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.comp f₁ f₂)).inv
-- CategoryTheory.Bicategory.Adjunction.comp_right_triangle_aux
theorem comp_right_triangle_aux (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂) :
    rightZigzag (compUnit adj₁ adj₂) (compCounit adj₁ adj₂) = (ρ_ _).hom ≫ (λ_ _).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f₁ : a ⟶ b} {g₁ : b ⟶ a} {f₂ : b ⟶ c} {g₂ : c ⟶ b} (adj₁ : CategoryTheory.Bicategory.Adjunction f₁ g₁) (adj₂ : CategoryTheory.Bicategory.Adjunction f₂ g₂)
Goal: CategoryTheory.Bicategory.rightZigzag (adj₁.compUnit adj₂) (adj₁.compCounit adj₂) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.comp g₂ g₁)).hom
    (CategoryTheory.Bicategory.leftUnitor (CategoryTheory.CategoryStruct.comp g₂ g₁)).inv
-- CategoryTheory.Bicategory.Adjunction.comp_unit
@[simps]
def comp (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂) : f₁ ≫ f₂ ⊣ g₂ ≫ g₁ 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f₁ : a ⟶ b} {g₁ : b ⟶ a} {f₂ : b ⟶ c} {g₂ : c ⟶ b} (adj₁ : CategoryTheory.Bicategory.Adjunction f₁ g₁) (adj₂ : CategoryTheory.Bicategory.Adjunction f₂ g₂)
Goal: (adj₁.comp adj₂).unit = adj₁.compUnit adj₂
-- CategoryTheory.Bicategory.Adjunction.comp_counit
@[simps]
def comp (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂) : f₁ ≫ f₂ ⊣ g₂ ≫ g₁ 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f₁ : a ⟶ b} {g₁ : b ⟶ a} {f₂ : b ⟶ c} {g₂ : c ⟶ b} (adj₁ : CategoryTheory.Bicategory.Adjunction f₁ g₁) (adj₂ : CategoryTheory.Bicategory.Adjunction f₂ g₂)
Goal: (adj₁.comp adj₂).counit = adj₁.compCounit adj₂
-- CategoryTheory.Bicategory.leftZigzagIso_hom
@[simp]
theorem leftZigzagIso_hom : (leftZigzagIso η ε).hom = leftZigzag η.hom ε.hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (η : CategoryTheory.CategoryStruct.id a ≅ CategoryTheory.CategoryStruct.comp f g) (ε : CategoryTheory.CategoryStruct.comp g f ≅ CategoryTheory.CategoryStruct.id b)
Goal: (CategoryTheory.Bicategory.leftZigzagIso η ε).hom = CategoryTheory.Bicategory.leftZigzag η.hom ε.hom
-- CategoryTheory.Bicategory.rightZigzagIso_hom
@[simp]
theorem rightZigzagIso_hom : (rightZigzagIso η ε).hom = rightZigzag η.hom ε.hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (η : CategoryTheory.CategoryStruct.id a ≅ CategoryTheory.CategoryStruct.comp f g) (ε : CategoryTheory.CategoryStruct.comp g f ≅ CategoryTheory.CategoryStruct.id b)
Goal: (CategoryTheory.Bicategory.rightZigzagIso η ε).hom = CategoryTheory.Bicategory.rightZigzag η.hom ε.hom
-- CategoryTheory.Bicategory.leftZigzagIso_inv
@[simp]
theorem leftZigzagIso_inv : (leftZigzagIso η ε).inv = rightZigzag ε.inv η.inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (η : CategoryTheory.CategoryStruct.id a ≅ CategoryTheory.CategoryStruct.comp f g) (ε : CategoryTheory.CategoryStruct.comp g f ≅ CategoryTheory.CategoryStruct.id b)
Goal: (CategoryTheory.Bicategory.leftZigzagIso η ε).inv = CategoryTheory.Bicategory.rightZigzag ε.inv η.inv
-- CategoryTheory.Bicategory.rightZigzagIso_inv
@[simp]
theorem rightZigzagIso_inv : (rightZigzagIso η ε).inv = leftZigzag ε.inv η.inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (η : CategoryTheory.CategoryStruct.id a ≅ CategoryTheory.CategoryStruct.comp f g) (ε : CategoryTheory.CategoryStruct.comp g f ≅ CategoryTheory.CategoryStruct.id b)
Goal: (CategoryTheory.Bicategory.rightZigzagIso η ε).inv = CategoryTheory.Bicategory.leftZigzag ε.inv η.inv
-- CategoryTheory.Bicategory.leftZigzagIso_symm
@[simp]
theorem leftZigzagIso_symm : (leftZigzagIso η ε).symm = rightZigzagIso ε.symm η.symm 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (η : CategoryTheory.CategoryStruct.id a ≅ CategoryTheory.CategoryStruct.comp f g) (ε : CategoryTheory.CategoryStruct.comp g f ≅ CategoryTheory.CategoryStruct.id b)
Goal: (CategoryTheory.Bicategory.leftZigzagIso η ε).symm = CategoryTheory.Bicategory.rightZigzagIso ε.symm η.symm
-- CategoryTheory.Bicategory.rightZigzagIso_symm
@[simp]
theorem rightZigzagIso_symm : (rightZigzagIso η ε).symm = leftZigzagIso ε.symm η.symm 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (η : CategoryTheory.CategoryStruct.id a ≅ CategoryTheory.CategoryStruct.comp f g) (ε : CategoryTheory.CategoryStruct.comp g f ≅ CategoryTheory.CategoryStruct.id b)
Goal: (CategoryTheory.Bicategory.rightZigzagIso η ε).symm = CategoryTheory.Bicategory.leftZigzagIso ε.symm η.symm
-- CategoryTheory.Bicategory.right_triangle_of_left_triangle
theorem right_triangle_of_left_triangle (h : leftZigzag η.hom ε.hom = (λ_ f).hom ≫ (ρ_ f).inv) :
    rightZigzag η.hom ε.hom = (ρ_ g).hom ≫ (λ_ g).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (η : CategoryTheory.CategoryStruct.id a ≅ CategoryTheory.CategoryStruct.comp f g) (ε : CategoryTheory.CategoryStruct.comp g f ≅ CategoryTheory.CategoryStruct.id b) (h :
  CategoryTheory.Bicategory.leftZigzag η.hom ε.hom =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor f).hom
      (CategoryTheory.Bicategory.rightUnitor f).inv)
Goal: CategoryTheory.Bicategory.rightZigzag η.hom ε.hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor g).hom
    (CategoryTheory.Bicategory.leftUnitor g).inv
-- CategoryTheory.Bicategory.adjointifyCounit_left_triangle
theorem adjointifyCounit_left_triangle (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f ≅ 𝟙 b) :
    leftZigzagIso η (adjointifyCounit η ε) = λ_ f ≪≫ (ρ_ f).symm 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (η : CategoryTheory.CategoryStruct.id a ≅ CategoryTheory.CategoryStruct.comp f g) (ε : CategoryTheory.CategoryStruct.comp g f ≅ CategoryTheory.CategoryStruct.id b)
Goal: CategoryTheory.Bicategory.leftZigzagIso η (CategoryTheory.Bicategory.adjointifyCounit η ε) =
  CategoryTheory.Bicategory.leftUnitor f ≪≫ (CategoryTheory.Bicategory.rightUnitor f).symm
-- CategoryTheory.Bicategory.Equivalence.left_triangle
structure Equivalence (a b : B) where
    hom : a ⟶ b
    inv : b ⟶ a
    unit : 𝟙 a ≅ hom ≫ inv
    counit : inv ≫ hom ≅ 𝟙 b
    left_triangle : leftZigzagIso unit counit = λ_ hom ≪≫ (ρ_ hom).symm := by aesop_cat
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} (self : CategoryTheory.Bicategory.Equivalence a b)
Goal: CategoryTheory.Bicategory.leftZigzagIso self.unit self.counit =
  CategoryTheory.Bicategory.leftUnitor self.hom ≪≫ (CategoryTheory.Bicategory.rightUnitor self.hom).symm
-- CategoryTheory.Bicategory.Equivalence.left_triangle_hom
theorem left_triangle_hom (e : a ≌ b) :
    leftZigzag e.unit.hom e.counit.hom = (λ_ e.hom).hom ≫ (ρ_ e.hom).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} (e : CategoryTheory.Bicategory.Equivalence a b)
Goal: CategoryTheory.Bicategory.leftZigzag e.unit.hom e.counit.hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor e.hom).hom
    (CategoryTheory.Bicategory.rightUnitor e.hom).inv
-- CategoryTheory.Bicategory.Equivalence.right_triangle
theorem right_triangle (e : a ≌ b) :
    rightZigzagIso e.unit e.counit = ρ_ e.inv ≪≫ (λ_ e.inv).symm 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} (e : CategoryTheory.Bicategory.Equivalence a b)
Goal: CategoryTheory.Bicategory.rightZigzagIso e.unit e.counit =
  CategoryTheory.Bicategory.rightUnitor e.inv ≪≫ (CategoryTheory.Bicategory.leftUnitor e.inv).symm
-- CategoryTheory.Bicategory.Equivalence.right_triangle_hom
theorem right_triangle_hom (e : a ≌ b) :
    rightZigzag e.unit.hom e.counit.hom = (ρ_ e.inv).hom ≫ (λ_ e.inv).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} (e : CategoryTheory.Bicategory.Equivalence a b)
Goal: CategoryTheory.Bicategory.rightZigzag e.unit.hom e.counit.hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor e.inv).hom
    (CategoryTheory.Bicategory.leftUnitor e.inv).inv
-- CategoryTheory.Bicategory.IsLeftAdjoint.nonempty
class IsLeftAdjoint (left : a ⟶ b) : Prop where mk' ::
  nonempty : Nonempty (RightAdjoint left)
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {left : a ⟶ b} [self : CategoryTheory.Bicategory.IsLeftAdjoint left]
Goal: Nonempty (CategoryTheory.Bicategory.RightAdjoint left)
-- CategoryTheory.Bicategory.IsLeftAdjoint.mk
theorem IsLeftAdjoint.mk (adj : f ⊣ g) : IsLeftAdjoint f 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (adj : CategoryTheory.Bicategory.Adjunction f g)
Goal: CategoryTheory.Bicategory.IsLeftAdjoint f
-- CategoryTheory.Bicategory.IsRightAdjoint.nonempty
class IsRightAdjoint (right : b ⟶ a) : Prop where mk' ::
  nonempty : Nonempty (LeftAdjoint right)
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {right : b ⟶ a} [self : CategoryTheory.Bicategory.IsRightAdjoint right]
Goal: Nonempty (CategoryTheory.Bicategory.LeftAdjoint right)
-- CategoryTheory.Bicategory.IsRightAdjoint.mk
theorem IsRightAdjoint.mk (adj : f ⊣ g) : IsRightAdjoint g 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f : a ⟶ b} {g : b ⟶ a} (adj : CategoryTheory.Bicategory.Adjunction f g)
Goal: CategoryTheory.Bicategory.IsRightAdjoint g
-- LieModule.nontrivial_of_isIrreducible
lemma LieModule.nontrivial_of_isIrreducible [LieModule.IsIrreducible R L M] : Nontrivial M 
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] [LieModule.IsIrreducible R L M]
Goal: Nontrivial M
-- LieAlgebra.HasTrivialRadical.eq_bot_of_isSolvable
theorem HasTrivialRadical.eq_bot_of_isSolvable [HasTrivialRadical R L]
    (I : LieIdeal R L) [hI : IsSolvable R I] : I = ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.HasTrivialRadical R L] (I : LieIdeal R L) [hI : LieAlgebra.IsSolvable R ↥↑I]
Goal: I = ⊥
-- LieAlgebra.HasTrivialRadical.center_eq_bot
@[simp]
theorem HasTrivialRadical.center_eq_bot [HasTrivialRadical R L] : center R L = ⊥ 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.HasTrivialRadical R L]
Goal: LieAlgebra.center R L = ⊥
-- LieAlgebra.hasTrivialRadical_of_no_solvable_ideals
theorem hasTrivialRadical_of_no_solvable_ideals (h : ∀ I : LieIdeal R L, IsSolvable R I → I = ⊥) :
    HasTrivialRadical R L 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (h : ∀ (I : LieIdeal R L), LieAlgebra.IsSolvable R ↥↑I → I = ⊥)
Goal: LieAlgebra.HasTrivialRadical R L
-- LieAlgebra.hasTrivialRadical_iff_no_solvable_ideals
theorem hasTrivialRadical_iff_no_solvable_ideals :
    HasTrivialRadical R L ↔ ∀ I : LieIdeal R L, IsSolvable R I → I = ⊥ 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: LieAlgebra.HasTrivialRadical R L ↔ ∀ (I : LieIdeal R L), LieAlgebra.IsSolvable R ↥↑I → I = ⊥
-- LieAlgebra.hasTrivialRadical_iff_no_abelian_ideals
theorem hasTrivialRadical_iff_no_abelian_ideals :
    HasTrivialRadical R L ↔ ∀ I : LieIdeal R L, IsLieAbelian I → I = ⊥ 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: LieAlgebra.HasTrivialRadical R L ↔ ∀ (I : LieIdeal R L), IsLieAbelian ↥↑I → I = ⊥
-- LieAlgebra.IsSimple.eq_top_of_isAtom
lemma eq_top_of_isAtom (I : LieIdeal R L) (hI : IsAtom I) : I = ⊤ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsSimple R L] (I : LieIdeal R L) (hI : IsAtom I)
Goal: I = ⊤
-- LieAlgebra.IsSimple.isAtom_top
lemma isAtom_top : IsAtom (⊤ : LieIdeal R L) 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsSimple R L]
Goal: IsAtom ⊤
-- LieAlgebra.IsSimple.isAtom_iff_eq_top
@[simp] lemma isAtom_iff_eq_top (I : LieIdeal R L) : IsAtom I ↔ I = ⊤ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsSimple R L] (I : LieIdeal R L)
Goal: IsAtom I ↔ I = ⊤
-- LieAlgebra.IsSemisimple.isSimple_of_isAtom
lemma isSimple_of_isAtom (I : LieIdeal R L) (hI : IsAtom I) : IsSimple R I 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsSemisimple R L] (I : LieIdeal R L) (hI : IsAtom I)
Goal: LieAlgebra.IsSimple R ↥↑I
-- _private.Mathlib.Algebra.Lie.Semisimple.Basic.0.LieAlgebra.IsSemisimple.finitelyAtomistic
private
lemma finitelyAtomistic : ∀ s : Finset (LieIdeal R L), ↑s ⊆ {I : LieIdeal R L | IsAtom I} →
    ∀ I : LieIdeal R L, I ≤ s.sup id → ∃ t ⊆ s, I = t.sup id 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsSemisimple R L] (s : Finset (LieIdeal R L))
Goal: ↑s ⊆ {I | IsAtom I} → ∀ I ≤ s.sup id, ∃ t ⊆ s, I = t.sup id
-- LieAlgebra.IsSemisimple.booleanGenerators
lemma booleanGenerators : BooleanGenerators {I : LieIdeal R L | IsAtom I} 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsSemisimple R L]
Goal: IsCompactlyGenerated.BooleanGenerators {I | IsAtom I}
-- LieAlgebra.subsingleton_of_hasTrivialRadical_lie_abelian
theorem subsingleton_of_hasTrivialRadical_lie_abelian [HasTrivialRadical R L] [h : IsLieAbelian L] :
    Subsingleton L 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.HasTrivialRadical R L] [h : IsLieAbelian L]
Goal: Subsingleton L
-- LieAlgebra.abelian_radical_of_hasTrivialRadical
theorem abelian_radical_of_hasTrivialRadical [HasTrivialRadical R L] :
    IsLieAbelian (radical R L) 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.HasTrivialRadical R L]
Goal: IsLieAbelian ↥↑(LieAlgebra.radical R L)