ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HopfAlgebraCat.forget₂_bialgebra_obj
@[simp]
theorem forget₂_bialgebra_obj (X : HopfAlgebraCat R) :
    (forget₂ (HopfAlgebraCat R) (BialgebraCat R)).obj X = BialgebraCat.of R X 
Arguments: {R : Type u} [CommRing R] (X : HopfAlgebraCat R)
Goal: (CategoryTheory.forget₂ (HopfAlgebraCat R) (BialgebraCat R)).obj X = BialgebraCat.of R ↑X.toBundled
-- HopfAlgebraCat.forget₂_bialgebra_map
@[simp]
theorem forget₂_bialgebra_map (X Y : HopfAlgebraCat R) (f : X ⟶ Y) :
    (forget₂ (HopfAlgebraCat R) (BialgebraCat R)).map f = BialgebraCat.ofHom f.toBialgHom 
Arguments: {R : Type u} [CommRing R] (X Y : HopfAlgebraCat R) (f : X ⟶ Y)
Goal: (CategoryTheory.forget₂ (HopfAlgebraCat R) (BialgebraCat R)).map f = BialgebraCat.ofHom f.toBialgHom
-- BialgEquiv.toHopfAlgebraCatIso_hom_toBialgHom
@[simps]
def toHopfAlgebraCatIso (e : X ≃ₐc[R] Y) : HopfAlgebraCat.of R X ≅ HopfAlgebraCat.of R Y 
Arguments: {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Ring Y] [HopfAlgebra R X] [HopfAlgebra R Y] (e : X ≃ₐc[R] Y)
Goal: e.toHopfAlgebraCatIso.hom.toBialgHom = ↑e
-- BialgEquiv.toHopfAlgebraCatIso_inv_toBialgHom
@[simps]
def toHopfAlgebraCatIso (e : X ≃ₐc[R] Y) : HopfAlgebraCat.of R X ≅ HopfAlgebraCat.of R Y 
Arguments: {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Ring Y] [HopfAlgebra R X] [HopfAlgebra R Y] (e : X ≃ₐc[R] Y)
Goal: e.toHopfAlgebraCatIso.inv.toBialgHom = ↑e.symm
-- BialgEquiv.toHopfAlgebraCatIso_refl
@[simp] theorem toHopfAlgebraCatIso_refl :
    toHopfAlgebraCatIso (BialgEquiv.refl R X) = .refl _ 
Arguments: {R : Type u} [CommRing R] {X : Type v} [Ring X] [HopfAlgebra R X]
Goal: (BialgEquiv.refl R X).toHopfAlgebraCatIso = CategoryTheory.Iso.refl (HopfAlgebraCat.of R X)
-- BialgEquiv.toHopfAlgebraCatIso_symm
@[simp] theorem toHopfAlgebraCatIso_symm (e : X ≃ₐc[R] Y) :
    toHopfAlgebraCatIso e.symm = (toHopfAlgebraCatIso e).symm 
Arguments: {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Ring Y] [HopfAlgebra R X] [HopfAlgebra R Y] (e : X ≃ₐc[R] Y)
Goal: e.symm.toHopfAlgebraCatIso = e.toHopfAlgebraCatIso.symm
-- BialgEquiv.toHopfAlgebraCatIso_trans
@[simp] theorem toHopfAlgebraCatIso_trans (e : X ≃ₐc[R] Y) (f : Y ≃ₐc[R] Z) :
    toHopfAlgebraCatIso (e.trans f) = toHopfAlgebraCatIso e ≪≫ toHopfAlgebraCatIso f 
Arguments: {R : Type u} [CommRing R] {X Y Z : Type v} [Ring X] [Ring Y] [Ring Z] [HopfAlgebra R X] [HopfAlgebra R Y] [HopfAlgebra R Z] (e : X ≃ₐc[R] Y) (f : Y ≃ₐc[R] Z)
Goal: (e.trans f).toHopfAlgebraCatIso = e.toHopfAlgebraCatIso ≪≫ f.toHopfAlgebraCatIso
-- CategoryTheory.Iso.toHopfAlgEquiv_toBialgHom
@[simp] theorem toHopfAlgEquiv_toBialgHom (i : X ≅ Y) :
    (i.toHopfAlgEquiv : X →ₐc[R] Y) = i.hom.1 
Arguments: {R : Type u} [CommRing R] {X Y : HopfAlgebraCat R} (i : X ≅ Y)
Goal: ↑i.toHopfAlgEquiv = i.hom.toBialgHom
-- CategoryTheory.Iso.toHopfAlgEquiv_refl
@[simp] theorem toHopfAlgEquiv_refl : toHopfAlgEquiv (.refl X) = .refl _ _ 
Arguments: {R : Type u} [CommRing R] {X : HopfAlgebraCat R}
Goal: (CategoryTheory.Iso.refl X).toHopfAlgEquiv = BialgEquiv.refl R ↑X.toBundled
-- CategoryTheory.Iso.toHopfAlgEquiv_symm
@[simp] theorem toHopfAlgEquiv_symm (e : X ≅ Y) :
    toHopfAlgEquiv e.symm = (toHopfAlgEquiv e).symm 
Arguments: {R : Type u} [CommRing R] {X Y : HopfAlgebraCat R} (e : X ≅ Y)
Goal: e.symm.toHopfAlgEquiv = e.toHopfAlgEquiv.symm
-- CategoryTheory.Iso.toHopfAlgEquiv_trans
@[simp] theorem toHopfAlgEquiv_trans (e : X ≅ Y) (f : Y ≅ Z) :
    toHopfAlgEquiv (e ≪≫ f) = e.toHopfAlgEquiv.trans f.toHopfAlgEquiv 
Arguments: {R : Type u} [CommRing R] {X Y Z : HopfAlgebraCat R} (e : X ≅ Y) (f : Y ≅ Z)
Goal: (e ≪≫ f).toHopfAlgEquiv = e.toHopfAlgEquiv.trans f.toHopfAlgEquiv
-- NonUnitalStarAlgHom.map_star'
structure NonUnitalStarAlgHom (R A B : Type*) [Monoid R] [NonUnitalNonAssocSemiring A]
  [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B]
  [Star B] extends A →ₙₐ[R] B where
    map_star' : ∀ a : A, toFun (star a) = star (toFun a)
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (self : A →⋆ₙₐ[R] B) (a : A)
Goal: self.toFun (star a) = star (self.toFun a)
-- NonUnitalStarAlgHom.coe_coe
@[simp]
protected theorem coe_coe {F : Type*} [FunLike F A B] [NonUnitalAlgHomClass F R A B]
    [NonUnitalStarAlgHomClass F R A B] (f : F) :
    ⇑(f : A →⋆ₙₐ[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] {F : Type u_6} [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F)
Goal: ⇑↑f = ⇑f
-- NonUnitalStarAlgHom.coe_toNonUnitalAlgHom
@[simp]
theorem coe_toNonUnitalAlgHom {f : A →⋆ₙₐ[R] B} : (f.toNonUnitalAlgHom : A → B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] {f : A →⋆ₙₐ[R] B}
Goal: ⇑f.toNonUnitalAlgHom = ⇑f
-- NonUnitalStarAlgHom.ext
@[ext]
theorem ext {f g : A →⋆ₙₐ[R] B} (h : ∀ x, f x = g x) : f = g 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] {f g : A →⋆ₙₐ[R] B} (h : ∀ (x : A), f x = g x)
Goal: f = g
-- NonUnitalStarAlgHom.coe_copy
@[simp]
theorem coe_copy (f : A →⋆ₙₐ[R] B) (f' : A → B) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B) (f' : A → B) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- NonUnitalStarAlgHom.copy_eq
theorem copy_eq (f : A →⋆ₙₐ[R] B) (f' : A → B) (h : f' = f) : f.copy f' h = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B) (f' : A → B) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- NonUnitalStarAlgHom.coe_mk
@[simp]
theorem coe_mk (f : A → B) (h₁ h₂ h₃ h₄ h₅) :
    ((⟨⟨⟨⟨f, h₁⟩, h₂, h₃⟩, h₄⟩, h₅⟩ : A →⋆ₙₐ[R] B) : A → B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A → B) (h₁ : ∀ (m : R) (x : A), f (m • x) = (MonoidHom.id R) m • f x) (h₂ : { toFun := f, map_smul' := h₁ }.toFun 0 = 0) (h₃ :
  ∀ (x y : A),
    { toFun := f, map_smul' := h₁ }.toFun (x + y) =
      { toFun := f, map_smul' := h₁ }.toFun x + { toFun := f, map_smul' := h₁ }.toFun y) (h₄ :
  ∀ (x y : A),
    { toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun (x * y) =
      { toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun x *
        { toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun y) (h₅ :
  ∀ (a : A),
    { toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun (star a) =
      star ({ toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun a))
Goal: ⇑{ toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄, map_star' := h₅ } = f
-- NonUnitalStarAlgHom.coe_mk'
@[simp]
theorem coe_mk' (f : A →ₙₐ[R] B) (h) :
    ((⟨f, h⟩ : A →⋆ₙₐ[R] B) : A → B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →ₙₐ[R] B) (h : ∀ (a : A), f.toFun (star a) = star (f.toFun a))
Goal: ⇑{ toNonUnitalAlgHom := f, map_star' := h } = ⇑f
-- NonUnitalStarAlgHom.mk_coe
@[simp]
theorem mk_coe (f : A →⋆ₙₐ[R] B) (h₁ h₂ h₃ h₄ h₅) :
    (⟨⟨⟨⟨f, h₁⟩, h₂, h₃⟩, h₄⟩, h₅⟩ : A →⋆ₙₐ[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B) (h₁ : ∀ (m : R) (x : A), f (m • x) = (MonoidHom.id R) m • f x) (h₂ : { toFun := ⇑f, map_smul' := h₁ }.toFun 0 = 0) (h₃ :
  ∀ (x y : A),
    { toFun := ⇑f, map_smul' := h₁ }.toFun (x + y) =
      { toFun := ⇑f, map_smul' := h₁ }.toFun x + { toFun := ⇑f, map_smul' := h₁ }.toFun y) (h₄ :
  ∀ (x y : A),
    { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun (x * y) =
      { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun x *
        { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun y) (h₅ :
  ∀ (a : A),
    { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun (star a) =
      star ({ toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun a))
Goal: { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄, map_star' := h₅ } = f
-- NonUnitalStarAlgHom.coe_id
@[simp]
theorem coe_id : ⇑(NonUnitalStarAlgHom.id R A) = id 
Arguments: (R : Type u_1) (A : Type u_2) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A]
Goal: ⇑(NonUnitalStarAlgHom.id R A) = id
-- NonUnitalStarAlgHom.coe_comp
@[simp]
theorem coe_comp (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B) : ⇑(comp f g) = f ∘ g 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- NonUnitalStarAlgHom.comp_apply
@[simp]
theorem comp_apply (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B) (a : A) : comp f g a = f (g a) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B) (a : A)
Goal: (f.comp g) a = f (g a)
-- NonUnitalStarAlgHom.comp_assoc
@[simp]
theorem comp_assoc (f : C →⋆ₙₐ[R] D) (g : B →⋆ₙₐ[R] C) (h : A →⋆ₙₐ[R] B) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] [NonUnitalNonAssocSemiring D] [DistribMulAction R D] [Star D] (f : C →⋆ₙₐ[R] D) (g : B →⋆ₙₐ[R] C) (h : A →⋆ₙₐ[R] B)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- NonUnitalStarAlgHom.id_comp
@[simp]
theorem id_comp (f : A →⋆ₙₐ[R] B) : (NonUnitalStarAlgHom.id _ _).comp f = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B)
Goal: (NonUnitalStarAlgHom.id R B).comp f = f
-- NonUnitalStarAlgHom.comp_id
@[simp]
theorem comp_id (f : A →⋆ₙₐ[R] B) : f.comp (NonUnitalStarAlgHom.id _ _) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B)
Goal: f.comp (NonUnitalStarAlgHom.id R A) = f
-- NonUnitalStarAlgHom.coe_one
@[simp]
theorem coe_one : ((1 : A →⋆ₙₐ[R] A) : A → A) = id 
Arguments: {R : Type u_1} {A : Type u_2} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A]
Goal: ⇑1 = id
-- NonUnitalStarAlgHom.one_apply
theorem one_apply (a : A) : (1 : A →⋆ₙₐ[R] A) a = a 
Arguments: {R : Type u_1} {A : Type u_2} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] (a : A)
Goal: 1 a = a
-- NonUnitalStarAlgHom.coe_zero
@[simp]
theorem coe_zero : ((0 : A →⋆ₙₐ[R] B) : A → B) = 0 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B]
Goal: ⇑0 = 0
-- NonUnitalStarAlgHom.zero_apply
theorem zero_apply (a : A) : (0 : A →⋆ₙₐ[R] B) a = 0 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] (a : A)
Goal: 0 a = 0
-- NonUnitalStarAlgHom.restrictScalars_apply
@[simp]
lemma restrictScalars_apply (f : A →⋆ₙₐ[S] B) (x : A) : f.restrictScalars R x = f x 
Arguments: (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [Monoid S] [Star A] [Star B] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [MulAction R S] [DistribMulAction S A] [DistribMulAction S B] [DistribMulAction R A] [DistribMulAction R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →⋆ₙₐ[S] B) (x : A)
Goal: (NonUnitalStarAlgHom.restrictScalars R f) x = f x
-- NonUnitalStarAlgHom.coe_restrictScalars
lemma coe_restrictScalars (f : A →⋆ₙₐ[S] B) : (f.restrictScalars R : A →ₙ+* B) = f 
Arguments: (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [Monoid S] [Star A] [Star B] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [MulAction R S] [DistribMulAction S A] [DistribMulAction S B] [DistribMulAction R A] [DistribMulAction R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →⋆ₙₐ[S] B)
Goal: ↑(NonUnitalStarAlgHom.restrictScalars R f) = ↑f
-- NonUnitalStarAlgHom.coe_restrictScalars'
lemma coe_restrictScalars' (f : A →⋆ₙₐ[S] B) : (f.restrictScalars R : A → B) = f 
Arguments: (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [Monoid S] [Star A] [Star B] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [MulAction R S] [DistribMulAction S A] [DistribMulAction S B] [DistribMulAction R A] [DistribMulAction R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →⋆ₙₐ[S] B)
Goal: ⇑(NonUnitalStarAlgHom.restrictScalars R f) = ⇑f
-- NonUnitalStarAlgHom.restrictScalars_injective
theorem restrictScalars_injective :
    Function.Injective (restrictScalars R : (A →⋆ₙₐ[S] B) → A →⋆ₙₐ[R] B) 
Arguments: (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [Monoid S] [Star A] [Star B] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [MulAction R S] [DistribMulAction S A] [DistribMulAction S B] [DistribMulAction R A] [DistribMulAction R B] [IsScalarTower R S A] [IsScalarTower R S B]
Goal: Function.Injective (NonUnitalStarAlgHom.restrictScalars R)
-- StarAlgHom.map_star'
structure StarAlgHom (R A B : Type*) [CommSemiring R] [Semiring A] [Algebra R A] [Star A]
  [Semiring B] [Algebra R B] [Star B] extends AlgHom R A B where
    map_star' : ∀ x : A, toFun (star x) = star (toFun x)
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (self : A →⋆ₐ[R] B) (x : A)
Goal: (↑↑self.toRingHom).toFun (star x) = star ((↑↑self.toRingHom).toFun x)
-- StarAlgHom.coe_coe
@[simp]
protected theorem coe_coe {F : Type*} [FunLike F A B] [AlgHomClass F R A B]
    [StarAlgHomClass F R A B] (f : F) :
    ⇑(f : A →⋆ₐ[R] B) = f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] {F : Type u_7} [FunLike F A B] [AlgHomClass F R A B] [StarAlgHomClass F R A B] (f : F)
Goal: ⇑↑f = ⇑f
-- StarAlgHom.coe_toAlgHom
@[simp]
theorem coe_toAlgHom {f : A →⋆ₐ[R] B} : (f.toAlgHom : A → B) = f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] {f : A →⋆ₐ[R] B}
Goal: ⇑f.toAlgHom = ⇑f
-- StarAlgHom.ext
@[ext]
theorem ext {f g : A →⋆ₐ[R] B} (h : ∀ x, f x = g x) : f = g 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] {f g : A →⋆ₐ[R] B} (h : ∀ (x : A), f x = g x)
Goal: f = g
-- StarAlgHom.coe_copy
@[simp]
theorem coe_copy (f : A →⋆ₐ[R] B) (f' : A → B) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) (f' : A → B) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- StarAlgHom.copy_eq
theorem copy_eq (f : A →⋆ₐ[R] B) (f' : A → B) (h : f' = f) : f.copy f' h = f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) (f' : A → B) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- StarAlgHom.coe_mk
@[simp]
theorem coe_mk (f : A → B) (h₁ h₂ h₃ h₄ h₅ h₆) :
    ((⟨⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩, h₆⟩ : A →⋆ₐ[R] B) : A → B) = f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A → B) (h₁ : f 1 = 1) (h₂ :
  ∀ (x y : A),
    { toFun := f, map_one' := h₁ }.toFun (x * y) =
      { toFun := f, map_one' := h₁ }.toFun x * { toFun := f, map_one' := h₁ }.toFun y) (h₃ : (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun 0 = 0) (h₄ :
  ∀ (x y : A),
    (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun (x + y) =
      (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun x +
        (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun y) (h₅ :
  ∀ (r : R),
    (↑↑{ toFun := f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄ }).toFun ((algebraMap R A) r) =
      (algebraMap R B) r) (h₆ :
  ∀ (x : A),
    (↑↑{ toFun := f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅ }.toRingHom).toFun
        (star x) =
      star
        ((↑↑{ toFun := f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄,
                    commutes' := h₅ }.toRingHom).toFun
          x))
Goal: ⇑{ toFun := f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅, map_star' := h₆ } = f
-- StarAlgHom.coe_mk'
@[simp]
theorem coe_mk' (f : A →ₐ[R] B) (h) :
    ((⟨f, h⟩ : A →⋆ₐ[R] B) : A → B) = f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →ₐ[R] B) (h : ∀ (x : A), (↑↑f.toRingHom).toFun (star x) = star ((↑↑f.toRingHom).toFun x))
Goal: ⇑{ toAlgHom := f, map_star' := h } = ⇑f
-- StarAlgHom.mk_coe
@[simp]
theorem mk_coe (f : A →⋆ₐ[R] B) (h₁ h₂ h₃ h₄ h₅ h₆) :
    (⟨⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩, h₆⟩ : A →⋆ₐ[R] B) = f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) (h₁ : f 1 = 1) (h₂ :
  ∀ (x y : A),
    { toFun := ⇑f, map_one' := h₁ }.toFun (x * y) =
      { toFun := ⇑f, map_one' := h₁ }.toFun x * { toFun := ⇑f, map_one' := h₁ }.toFun y) (h₃ : (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun 0 = 0) (h₄ :
  ∀ (x y : A),
    (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun (x + y) =
      (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun x +
        (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun y) (h₅ :
  ∀ (r : R),
    (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄ }).toFun ((algebraMap R A) r) =
      (algebraMap R B) r) (h₆ :
  ∀ (x : A),
    (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄,
                  commutes' := h₅ }.toRingHom).toFun
        (star x) =
      star
        ((↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄,
                    commutes' := h₅ }.toRingHom).toFun
          x))
Goal: { toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅, map_star' := h₆ } = f
-- StarAlgHom.coe_id
@[simp]
theorem coe_id : ⇑(StarAlgHom.id R A) = id 
Arguments: (R : Type u_2) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Star A]
Goal: ⇑(StarAlgHom.id R A) = id
-- StarAlgHom.ofId_apply
@[simps]
def ofId (R A : Type*) [CommSemiring R] [StarRing R] [Semiring A] [StarMul A]
    [Algebra R A] [StarModule R A] : R →⋆ₐ[R] A 
Arguments: (R : Type u_7) (A : Type u_8) [CommSemiring R] [StarRing R] [Semiring A] [StarMul A] [Algebra R A] [StarModule R A] (a : R)
Goal: (StarAlgHom.ofId R A) a = (algebraMap R A) a
-- StarAlgHom.coe_comp
@[simp]
theorem coe_comp (f : B →⋆ₐ[R] C) (g : A →⋆ₐ[R] B) : ⇑(comp f g) = f ∘ g 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : B →⋆ₐ[R] C) (g : A →⋆ₐ[R] B)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- StarAlgHom.comp_apply
@[simp]
theorem comp_apply (f : B →⋆ₐ[R] C) (g : A →⋆ₐ[R] B) (a : A) : comp f g a = f (g a) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : B →⋆ₐ[R] C) (g : A →⋆ₐ[R] B) (a : A)
Goal: (f.comp g) a = f (g a)
-- StarAlgHom.comp_assoc
@[simp]
theorem comp_assoc (f : C →⋆ₐ[R] D) (g : B →⋆ₐ[R] C) (h : A →⋆ₐ[R] B) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} {D : Type u_6} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] [Semiring D] [Algebra R D] [Star D] (f : C →⋆ₐ[R] D) (g : B →⋆ₐ[R] C) (h : A →⋆ₐ[R] B)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- StarAlgHom.id_comp
@[simp]
theorem id_comp (f : A →⋆ₐ[R] B) : (StarAlgHom.id _ _).comp f = f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B)
Goal: (StarAlgHom.id R B).comp f = f
-- StarAlgHom.comp_id
@[simp]
theorem comp_id (f : A →⋆ₐ[R] B) : f.comp (StarAlgHom.id _ _) = f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B)
Goal: f.comp (StarAlgHom.id R A) = f
-- StarAlgHom.coe_toNonUnitalStarAlgHom
@[simp]
theorem coe_toNonUnitalStarAlgHom (f : A →⋆ₐ[R] B) : (f.toNonUnitalStarAlgHom : A → B) = f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B)
Goal: ⇑f.toNonUnitalStarAlgHom = ⇑f
-- NonUnitalStarAlgHom.fst_apply
@[simps!]
def fst : A × B →⋆ₙₐ[R] A 
Arguments: (R : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (self : A × B)
Goal: (NonUnitalStarAlgHom.fst R A B) self = self.1
-- NonUnitalStarAlgHom.snd_apply
@[simps!]
def snd : A × B →⋆ₙₐ[R] B 
Arguments: (R : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (self : A × B)
Goal: (NonUnitalStarAlgHom.snd R A B) self = self.2
-- NonUnitalStarAlgHom.prod_apply
@[simps!]
def prod (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : A →⋆ₙₐ[R] B × C 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) (i : A)
Goal: (f.prod g) i = (f i, g i)
-- NonUnitalStarAlgHom.coe_prod
theorem coe_prod (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : ⇑(f.prod g) = Pi.prod f g 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C)
Goal: ⇑(f.prod g) = Pi.prod ⇑f ⇑g