ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NonUnitalStarAlgHom.fst_prod
@[simp]
theorem fst_prod (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : (fst R B C).comp (prod f g) = f 
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: (NonUnitalStarAlgHom.fst R B C).comp (f.prod g) = f
-- NonUnitalStarAlgHom.snd_prod
@[simp]
theorem snd_prod (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : (snd R B C).comp (prod f g) = 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: (NonUnitalStarAlgHom.snd R B C).comp (f.prod g) = g
-- NonUnitalStarAlgHom.prod_fst_snd
@[simp]
theorem prod_fst_snd : prod (fst R A B) (snd R A B) = 1 
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]
Goal: (NonUnitalStarAlgHom.fst R A B).prod (NonUnitalStarAlgHom.snd R A B) = 1
-- NonUnitalStarAlgHom.prodEquiv_apply
@[simps]
def prodEquiv : (A →⋆ₙₐ[R] B) × (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) × (A →⋆ₙₐ[R] C))
Goal: NonUnitalStarAlgHom.prodEquiv f = f.1.prod f.2
-- NonUnitalStarAlgHom.prodEquiv_symm_apply
@[simps]
def prodEquiv : (A →⋆ₙₐ[R] B) × (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 × C)
Goal: NonUnitalStarAlgHom.prodEquiv.symm f = ((NonUnitalStarAlgHom.fst R B C).comp f, (NonUnitalStarAlgHom.snd R B C).comp f)
-- NonUnitalStarAlgHom.coe_inl
@[simp]
theorem coe_inl : (inl R A B : A → A × B) = fun x => (x, 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: ⇑(NonUnitalStarAlgHom.inl R A B) = fun x => (x, 0)
-- NonUnitalStarAlgHom.inl_apply
theorem inl_apply (x : A) : inl R A B x = (x, 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] (x : A)
Goal: (NonUnitalStarAlgHom.inl R A B) x = (x, 0)
-- NonUnitalStarAlgHom.coe_inr
@[simp]
theorem coe_inr : (inr R A B : B → A × B) = Prod.mk 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: ⇑(NonUnitalStarAlgHom.inr R A B) = Prod.mk 0
-- NonUnitalStarAlgHom.inr_apply
theorem inr_apply (x : B) : inr R A B x = (0, x) 
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] (x : B)
Goal: (NonUnitalStarAlgHom.inr R A B) x = (0, x)
-- StarAlgHom.fst_apply
@[simps!]
def fst : A × B →⋆ₐ[R] A 
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 × B)
Goal: (StarAlgHom.fst R A B) self = self.1
-- StarAlgHom.snd_apply
@[simps!]
def snd : A × B →⋆ₐ[R] B 
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 × B)
Goal: (StarAlgHom.snd R A B) self = self.2
-- StarAlgHom.prod_apply
@[simps!]
def prod (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) : A →⋆ₐ[R] B × C 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) (x : A)
Goal: (f.prod g) x = (f x, g x)
-- StarAlgHom.coe_prod
theorem coe_prod (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[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} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C)
Goal: ⇑(f.prod g) = Pi.prod ⇑f ⇑g
-- StarAlgHom.fst_prod
@[simp]
theorem fst_prod (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) : (fst R B C).comp (prod f g) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C)
Goal: (StarAlgHom.fst R B C).comp (f.prod g) = f
-- StarAlgHom.snd_prod
@[simp]
theorem snd_prod (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) : (snd R B C).comp (prod f g) = g 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C)
Goal: (StarAlgHom.snd R B C).comp (f.prod g) = g
-- StarAlgHom.prod_fst_snd
@[simp]
theorem prod_fst_snd : prod (fst R A B) (snd R A B) = 1 
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]
Goal: (StarAlgHom.fst R A B).prod (StarAlgHom.snd R A B) = 1
-- StarAlgHom.prodEquiv_apply
@[simps]
def prodEquiv : (A →⋆ₐ[R] B) × (A →⋆ₐ[R] C) ≃ (A →⋆ₐ[R] B × C) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : (A →⋆ₐ[R] B) × (A →⋆ₐ[R] C))
Goal: StarAlgHom.prodEquiv f = f.1.prod f.2
-- StarAlgHom.prodEquiv_symm_apply
@[simps]
def prodEquiv : (A →⋆ₐ[R] B) × (A →⋆ₐ[R] C) ≃ (A →⋆ₐ[R] B × C) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B × C)
Goal: StarAlgHom.prodEquiv.symm f = ((StarAlgHom.fst R B C).comp f, (StarAlgHom.snd R B C).comp f)
-- StarAlgEquiv.map_smul'
structure StarAlgEquiv (R A B : Type*) [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B]
  [Star A] [Star B] extends A ≃+* B where
    map_star' : ∀ a : A, toFun (star a) = star (toFun a)
    map_smul' : ∀ (r : R) (a : A), toFun (r • a) = r • toFun a
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (self : A ≃⋆ₐ[R] B) (r : R) (a : A)
Goal: self.toFun (r • a) = r • self.toFun a
-- StarAlgEquiv.map_star'
structure StarAlgEquiv (R A B : Type*) [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B]
  [Star A] [Star B] extends A ≃+* B where
    map_star' : ∀ a : A, toFun (star a) = star (toFun a)
    map_smul' : ∀ (r : R) (a : A), toFun (r • a) = r • toFun a
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (self : A ≃⋆ₐ[R] B) (a : A)
Goal: self.toFun (star a) = star (self.toFun a)
-- StarAlgEquivClass.map_star
class StarAlgEquivClass (F : Type*) (R A B : outParam Type*)
  [Add A] [Mul A] [SMul R A] [Star A] [Add B] [Mul B] [SMul R B]
  [Star B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] : Prop where
    map_star : ∀ (f : F) (a : A), f (star a) = star (f a)
Arguments: {F : Type u_1} {R : outParam (Type u_2)} {A : outParam (Type u_3)} {B : outParam (Type u_4)} [Add A] [Mul A] [SMul R A] [Star A] [Add B] [Mul B] [SMul R B] [Star B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] [self : StarAlgEquivClass F R A B] (f : F) (a : A)
Goal: f (star a) = star (f a)
-- StarAlgEquiv.toRingEquiv_eq_coe
@[simp]
theorem toRingEquiv_eq_coe (e : A ≃⋆ₐ[R] B) : e.toRingEquiv = e 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B)
Goal: e.toRingEquiv = ↑e
-- StarAlgEquiv.ext
@[ext]
theorem ext {f g : A ≃⋆ₐ[R] B} (h : ∀ a, f a = g a) : f = g 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] {f g : A ≃⋆ₐ[R] B} (h : ∀ (a : A), f a = g a)
Goal: f = g
-- StarAlgEquiv.ext_iff
theorem ext_iff {f g : A ≃⋆ₐ[R] B} : f = g ↔ ∀ a, f a = g a 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] {f g : A ≃⋆ₐ[R] B}
Goal: f = g ↔ ∀ (a : A), f a = g a
-- StarAlgEquiv.coe_refl
@[simp]
theorem coe_refl : ⇑(refl : A ≃⋆ₐ[R] A) = id 
Arguments: {R : Type u_2} {A : Type u_3} [Add A] [Mul A] [SMul R A] [Star A]
Goal: ⇑StarAlgEquiv.refl = id
-- StarAlgEquiv.invFun_eq_symm
@[simp]
theorem invFun_eq_symm {e : A ≃⋆ₐ[R] B} : EquivLike.inv e = e.symm 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] {e : A ≃⋆ₐ[R] B}
Goal: EquivLike.inv e = ⇑e.symm
-- StarAlgEquiv.symm_symm
@[simp]
theorem symm_symm (e : A ≃⋆ₐ[R] B) : e.symm.symm = e 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B)
Goal: e.symm.symm = e
-- StarAlgEquiv.symm_bijective
theorem symm_bijective : Function.Bijective (symm : (A ≃⋆ₐ[R] B) → B ≃⋆ₐ[R] A) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B]
Goal: Function.Bijective StarAlgEquiv.symm
-- StarAlgEquiv.coe_mk
@[simp]
theorem coe_mk (e h₁ h₂) : ⇑(⟨e, h₁, h₂⟩ : A ≃⋆ₐ[R] B) = e 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃+* B) (h₁ : ∀ (a : A), e.toFun (star a) = star (e.toFun a)) (h₂ : ∀ (r : R) (a : A), e.toFun (r • a) = r • e.toFun a)
Goal: ⇑{ toRingEquiv := e, map_star' := h₁, map_smul' := h₂ } = ⇑e
-- StarAlgEquiv.mk_coe
@[simp]
theorem mk_coe (e : A ≃⋆ₐ[R] B) (e' h₁ h₂ h₃ h₄ h₅ h₆) :
    (⟨⟨⟨e, e', h₁, h₂⟩, h₃, h₄⟩, h₅, h₆⟩ : A ≃⋆ₐ[R] B) = e 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) (e' : B → A) (h₁ : Function.LeftInverse e' ⇑e) (h₂ : Function.RightInverse e' ⇑e) (h₃ :
  ∀ (x y : A),
    { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x * y) =
      { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x *
        { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ :
  ∀ (x y : A),
    { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x + y) =
      { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x +
        { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₅ :
  ∀ (a : A),
    { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (star a) =
      star ({ toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a)) (h₆ :
  ∀ (r : R) (a : A),
    { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (r • a) =
      r • { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a)
Goal: { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, map_star' := h₅,
    map_smul' := h₆ } =
  e
-- StarAlgEquiv.symm_mk
@[simp]
theorem symm_mk (f f') (h₁ h₂ h₃ h₄ h₅ h₆) :
    (⟨⟨⟨f, f', h₁, h₂⟩, h₃, h₄⟩, h₅, h₆⟩ : A ≃⋆ₐ[R] B).symm =
      { symm_mk.aux f f' h₁ h₂ h₃ h₄ h₅ h₆ with
        toFun := f'
        invFun := f } 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (f : A → B) (f' : B → A) (h₁ : Function.LeftInverse f' f) (h₂ : Function.RightInverse f' f) (h₃ :
  ∀ (x y : A),
    { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun (x * y) =
      { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun x *
        { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ :
  ∀ (x y : A),
    { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun (x + y) =
      { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun x +
        { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun y) (h₅ :
  ∀ (a : A),
    { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (star a) =
      star ({ toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a)) (h₆ :
  ∀ (r : R) (a : A),
    { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (r • a) =
      r • { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a)
Goal: { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, map_star' := h₅,
      map_smul' := h₆ }.symm =
  let __src := StarAlgEquiv.symm_mk.aux f f' h₁ h₂ h₃ h₄ h₅ h₆;
  { toFun := f', invFun := f, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯, map_star' := ⋯,
    map_smul' := ⋯ }
-- StarAlgEquiv.refl_symm
@[simp]
theorem refl_symm : (StarAlgEquiv.refl : A ≃⋆ₐ[R] A).symm = StarAlgEquiv.refl 
Arguments: {R : Type u_2} {A : Type u_3} [Add A] [Mul A] [SMul R A] [Star A]
Goal: StarAlgEquiv.refl.symm = StarAlgEquiv.refl
-- StarAlgEquiv.to_ringEquiv_symm
theorem to_ringEquiv_symm (f : A ≃⋆ₐ[R] B) : (f : A ≃+* B).symm = f.symm 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (f : A ≃⋆ₐ[R] B)
Goal: (↑f).symm = ↑f.symm
-- StarAlgEquiv.symm_to_ringEquiv
@[simp]
theorem symm_to_ringEquiv (e : A ≃⋆ₐ[R] B) : (e.symm : B ≃+* A) = (e : A ≃+* B).symm 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B)
Goal: ↑e.symm = (↑e).symm
-- StarAlgEquiv.apply_symm_apply
@[simp]
theorem apply_symm_apply (e : A ≃⋆ₐ[R] B) : ∀ x, e (e.symm x) = x 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) (x : B)
Goal: e (e.symm x) = x
-- StarAlgEquiv.symm_apply_apply
@[simp]
theorem symm_apply_apply (e : A ≃⋆ₐ[R] B) : ∀ x, e.symm (e x) = x 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) (x : A)
Goal: e.symm (e x) = x
-- StarAlgEquiv.symm_trans_apply
@[simp]
theorem symm_trans_apply (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) (x : C) :
    (e₁.trans e₂).symm x = e₁.symm (e₂.symm x) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] [Add C] [Mul C] [SMul R C] [Star C] (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) (x : C)
Goal: (e₁.trans e₂).symm x = e₁.symm (e₂.symm x)
-- StarAlgEquiv.coe_trans
@[simp]
theorem coe_trans (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) : ⇑(e₁.trans e₂) = e₂ ∘ e₁ 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] [Add C] [Mul C] [SMul R C] [Star C] (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C)
Goal: ⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁
-- StarAlgEquiv.trans_apply
@[simp]
theorem trans_apply (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) (x : A) : (e₁.trans e₂) x = e₂ (e₁ x) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] [Add C] [Mul C] [SMul R C] [Star C] (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) (x : A)
Goal: (e₁.trans e₂) x = e₂ (e₁ x)
-- StarAlgEquiv.leftInverse_symm
theorem leftInverse_symm (e : A ≃⋆ₐ[R] B) : Function.LeftInverse e.symm e 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B)
Goal: Function.LeftInverse ⇑e.symm ⇑e
-- StarAlgEquiv.rightInverse_symm
theorem rightInverse_symm (e : A ≃⋆ₐ[R] B) : Function.RightInverse e.symm e 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B)
Goal: Function.RightInverse ⇑e.symm ⇑e
-- StarAlgEquiv.ofStarAlgHom_symm_apply
@[simps]
def ofStarAlgHom (f : F) (g : G) (h₁ : ∀ x, g (f x) = x) (h₂ : ∀ x, f (g x) = x) : A ≃⋆ₐ[R] B 
Arguments: {F : Type u_1} {G : Type u_2} {R : Type u_3} {A : Type u_4} {B : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] [FunLike G B A] (f : F) (g : G) (h₁ : ∀ (x : A), g (f x) = x) (h₂ : ∀ (x : B), f (g x) = x) (a : B)
Goal: (StarAlgEquiv.ofStarAlgHom f g h₁ h₂).symm a = g a
-- StarAlgEquiv.ofStarAlgHom_apply
@[simps]
def ofStarAlgHom (f : F) (g : G) (h₁ : ∀ x, g (f x) = x) (h₂ : ∀ x, f (g x) = x) : A ≃⋆ₐ[R] B 
Arguments: {F : Type u_1} {G : Type u_2} {R : Type u_3} {A : Type u_4} {B : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] [FunLike G B A] (f : F) (g : G) (h₁ : ∀ (x : A), g (f x) = x) (h₂ : ∀ (x : B), f (g x) = x) (a : A)
Goal: (StarAlgEquiv.ofStarAlgHom f g h₁ h₂) a = f a
-- StarAlgEquiv.coe_ofBijective
@[simp]
theorem coe_ofBijective {f : F} (hf : Function.Bijective f) :
    (StarAlgEquiv.ofBijective f hf : A → B) = f 
Arguments: {F : Type u_1} {R : Type u_3} {A : Type u_4} {B : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] {f : F} (hf : Function.Bijective ⇑f)
Goal: ⇑(StarAlgEquiv.ofBijective f hf) = ⇑f
-- StarAlgEquiv.ofBijective_apply
theorem ofBijective_apply {f : F} (hf : Function.Bijective f) (a : A) :
    (StarAlgEquiv.ofBijective f hf) a = f a 
Arguments: {F : Type u_1} {R : Type u_3} {A : Type u_4} {B : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] {f : F} (hf : Function.Bijective ⇑f) (a : A)
Goal: (StarAlgEquiv.ofBijective f hf) a = f a
-- CategoryTheory.ExactPairing.coevaluation_evaluation'
class ExactPairing (X Y : C) where
    coevaluation' : 𝟙_ C ⟶ X ⊗ Y
    evaluation' : Y ⊗ X ⟶ 𝟙_ C
  coevaluation_evaluation' :
    Y ◁ coevaluation' ≫ (α_ _ _ _).inv ≫ evaluation' ▷ Y = (ρ_ Y).hom ≫ (λ_ Y).inv := by
    aesop_cat
  evaluation_coevaluation' :
    coevaluation' ▷ X ≫ (α_ _ _ _).hom ≫ X ◁ evaluation' = (λ_ X).hom ≫ (ρ_ X).inv := by
    aesop_cat
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X Y : C} [self : CategoryTheory.ExactPairing X Y]
Goal: Y ◁ CategoryTheory.ExactPairing.coevaluation' ≫ (α_ Y X Y).inv ≫ CategoryTheory.ExactPairing.evaluation' ▷ Y =
  (ρ_ Y).hom ≫ (λ_ Y).inv
-- CategoryTheory.ExactPairing.evaluation_coevaluation'
class ExactPairing (X Y : C) where
    coevaluation' : 𝟙_ C ⟶ X ⊗ Y
    evaluation' : Y ⊗ X ⟶ 𝟙_ C
  coevaluation_evaluation' :
    Y ◁ coevaluation' ≫ (α_ _ _ _).inv ≫ evaluation' ▷ Y = (ρ_ Y).hom ≫ (λ_ Y).inv := by
    aesop_cat
  evaluation_coevaluation' :
    coevaluation' ▷ X ≫ (α_ _ _ _).hom ≫ X ◁ evaluation' = (λ_ X).hom ≫ (ρ_ X).inv := by
    aesop_cat
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X Y : C} [self : CategoryTheory.ExactPairing X Y]
Goal: CategoryTheory.ExactPairing.coevaluation' ▷ X ≫ (α_ X Y X).hom ≫ X ◁ CategoryTheory.ExactPairing.evaluation' =
  (λ_ X).hom ≫ (ρ_ X).inv
-- CategoryTheory.ExactPairing.coevaluation_evaluation
lemma coevaluation_evaluation :
    Y ◁ η_ _ _ ≫ (α_ _ _ _).inv ≫ ε_ X _ ▷ Y = (ρ_ Y).hom ≫ (λ_ Y).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y : C) [CategoryTheory.ExactPairing X Y]
Goal: Y ◁ η_ X Y ≫ (α_ Y X Y).inv ≫ ε_ X Y ▷ Y = (ρ_ Y).hom ≫ (λ_ Y).inv
-- CategoryTheory.ExactPairing.evaluation_coevaluation
lemma evaluation_coevaluation :
    η_ _ _ ▷ X ≫ (α_ _ _ _).hom ≫ X ◁ ε_ _ Y = (λ_ X).hom ≫ (ρ_ X).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y : C) [CategoryTheory.ExactPairing X Y]
Goal: η_ X Y ▷ X ≫ (α_ X Y X).hom ≫ X ◁ ε_ X Y = (λ_ X).hom ≫ (ρ_ X).inv
-- CategoryTheory.ExactPairing.coevaluation_evaluation''
lemma coevaluation_evaluation'' :
    Y ◁ η_ X Y ⊗≫ ε_ X Y ▷ Y = ⊗𝟙 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y : C) [CategoryTheory.ExactPairing X Y]
Goal: Y ◁ η_ X Y ⊗≫ ε_ X Y ▷ Y = ⊗𝟙
-- CategoryTheory.ExactPairing.evaluation_coevaluation''
lemma evaluation_coevaluation'' :
    η_ X Y ▷ X ⊗≫ X ◁ ε_ X Y = ⊗𝟙 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y : C) [CategoryTheory.ExactPairing X Y]
Goal: η_ X Y ▷ X ⊗≫ X ◁ ε_ X Y = ⊗𝟙
-- CategoryTheory.leftDual_rightDual
@[simp]
theorem leftDual_rightDual {X : C} [HasRightDual X] : ᘁXᘁ = X 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X : C} [CategoryTheory.HasRightDual X]
Goal: ᘁXᘁ = X
-- CategoryTheory.rightDual_leftDual
@[simp]
theorem rightDual_leftDual {X : C} [HasLeftDual X] : (ᘁX)ᘁ = X 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X : C} [CategoryTheory.HasLeftDual X]
Goal: (ᘁX)ᘁ = X
-- CategoryTheory.rightAdjointMate_id
@[simp]
theorem rightAdjointMate_id {X : C} [HasRightDual X] : (𝟙 X)ᘁ = 𝟙 (Xᘁ) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X : C} [CategoryTheory.HasRightDual X]
Goal: 𝟙 Xᘁ = 𝟙 Xᘁ
-- CategoryTheory.leftAdjointMate_id
@[simp]
theorem leftAdjointMate_id {X : C} [HasLeftDual X] : (ᘁ(𝟙 X)) = 𝟙 (ᘁX) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X : C} [CategoryTheory.HasLeftDual X]
Goal: (ᘁ𝟙 X) = 𝟙 ᘁX
-- CategoryTheory.rightAdjointMate_comp
theorem rightAdjointMate_comp {X Y Z : C} [HasRightDual X] [HasRightDual Y] {f : X ⟶ Y}
    {g : Xᘁ ⟶ Z} :
    fᘁ ≫ g =
      (ρ_ (Yᘁ)).inv ≫
        _ ◁ η_ X (Xᘁ) ≫ _ ◁ (f ⊗ g) ≫ (α_ (Yᘁ) Y Z).inv ≫ ε_ Y (Yᘁ) ▷ _ ≫ (λ_ Z).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X Y Z : C} [CategoryTheory.HasRightDual X] [CategoryTheory.HasRightDual Y] {f : X ⟶ Y} {g : Xᘁ ⟶ Z}
Goal: fᘁ ≫ g = (ρ_ Yᘁ).inv ≫ Yᘁ ◁ η_ X Xᘁ ≫ Yᘁ ◁ (f ⊗ g) ≫ (α_ Yᘁ Y Z).inv ≫ ε_ Y Yᘁ ▷ Z ≫ (λ_ Z).hom