ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- multiple_mem_span_of_mem_localization_span
theorem multiple_mem_span_of_mem_localization_span
    {N : Type*} [AddCommMonoid N] [Module R N] [Module R' N]
    [IsScalarTower R R' N] [IsLocalization M R'] (s : Set N) (x : N)
    (hx : x ∈ Submodule.span R' s) : ∃ (t : M), t • x ∈ Submodule.span R s 
Arguments: {R : Type u} [CommRing R] (M : Submonoid R) (R' : Type u) [CommRing R'] [Algebra R R'] {N : Type u_1} [AddCommMonoid N] [Module R N] [Module R' N] [IsScalarTower R R' N] [IsLocalization M R'] (s : Set N) (x : N) (hx : x ∈ Submodule.span R' s)
Goal: ∃ t, t • x ∈ Submodule.span R s
-- multiple_mem_adjoin_of_mem_localization_adjoin
theorem multiple_mem_adjoin_of_mem_localization_adjoin [Algebra R' S] [Algebra R S]
    [IsScalarTower R R' S] [IsLocalization M R'] (s : Set S) (x : S)
    (hx : x ∈ Algebra.adjoin R' s) : ∃ t : M, t • x ∈ Algebra.adjoin R s 
Arguments: {R S : Type u} [CommRing R] [CommRing S] (M : Submonoid R) (R' : Type u) [CommRing R'] [Algebra R R'] [Algebra R' S] [Algebra R S] [IsScalarTower R R' S] [IsLocalization M R'] (s : Set S) (x : S) (hx : x ∈ Algebra.adjoin R' s)
Goal: ∃ t, t • x ∈ Algebra.adjoin R s
-- finite_ofLocalizationSpan
theorem finite_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.Finite 
Goal: RingHom.OfLocalizationSpan @RingHom.Finite
-- localization_finiteType
theorem localization_finiteType : RingHom.LocalizationPreserves @RingHom.FiniteType 
Goal: RingHom.LocalizationPreserves @RingHom.FiniteType
-- localization_away_map_finiteType
theorem localization_away_map_finiteType (r : R) [IsLocalization.Away r R']
    [IsLocalization.Away (f r) S'] (hf : f.FiniteType) :
    (IsLocalization.Away.map R' S' f r).FiniteType 
Arguments: {R S : Type u} [CommRing R] [CommRing S] (R' S' : Type u) [CommRing R'] [CommRing S'] (f : R →+* S) [Algebra R R'] [Algebra S S'] (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : f.FiniteType)
Goal: (IsLocalization.Away.map R' S' f r).FiniteType
-- IsLocalization.exists_smul_mem_of_mem_adjoin
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A 
Arguments: {R S : Type u} [CommRing R] [CommRing S] {S' : Type u} [CommRing S'] [Algebra S S'] [Algebra R S] [Algebra R S'] [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S') (A : Subalgebra R S) (hA₁ : ↑(IsLocalization.finsetIntegerMultiple M s) ⊆ ↑A) (hA₂ : M ≤ A.toSubmonoid) (hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s)
Goal: ∃ m, m • x ∈ A
-- IsLocalization.lift_mem_adjoin_finsetIntegerMultiple
theorem IsLocalization.lift_mem_adjoin_finsetIntegerMultiple [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] [IsLocalization (M.map (algebraMap R S)) S'] (x : S) (s : Finset S')
    (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈
      Algebra.adjoin R
        (IsLocalization.finsetIntegerMultiple (M.map (algebraMap R S)) s : Set S) 
Arguments: {R S : Type u} [CommRing R] [CommRing S] (M : Submonoid R) {S' : Type u} [CommRing S'] [Algebra S S'] [Algebra R S] [Algebra R S'] [IsScalarTower R S S'] [IsLocalization (Submonoid.map (algebraMap R S) M) S'] (x : S) (s : Finset S') (hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s)
Goal: ∃ m, m • x ∈ Algebra.adjoin R ↑(IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)
-- finiteType_ofLocalizationSpan
theorem finiteType_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.FiniteType 
Goal: RingHom.OfLocalizationSpan @RingHom.FiniteType
-- CoalgHom.counit_comp
structure CoalgHom (R A B : Type*) [CommSemiring R]
    [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B]
    [CoalgebraStruct R A] [CoalgebraStruct R B] extends A →ₗ[R] B where
  counit_comp : counit ∘ₗ toLinearMap = counit
  map_comp_comul : TensorProduct.map toLinearMap toLinearMap ∘ₗ comul = comul ∘ₗ toLinearMap
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (self : A →ₗc[R] B)
Goal: Coalgebra.counit ∘ₗ self.toLinearMap = Coalgebra.counit
-- CoalgHom.map_comp_comul
structure CoalgHom (R A B : Type*) [CommSemiring R]
    [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B]
    [CoalgebraStruct R A] [CoalgebraStruct R B] extends A →ₗ[R] B where
  counit_comp : counit ∘ₗ toLinearMap = counit
  map_comp_comul : TensorProduct.map toLinearMap toLinearMap ∘ₗ comul = comul ∘ₗ toLinearMap
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (self : A →ₗc[R] B)
Goal: TensorProduct.map self.toLinearMap self.toLinearMap ∘ₗ Coalgebra.comul = Coalgebra.comul ∘ₗ self.toLinearMap
-- CoalgHomClass.map_comp_comul
class CoalgHomClass (F : Type*) (R A B : outParam Type*)
    [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B]
    [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B]
    extends SemilinearMapClass F (RingHom.id R) A B : Prop where
  counit_comp : ∀ f : F, counit ∘ₗ (f : A →ₗ[R] B) = counit
  map_comp_comul : ∀ f : F, TensorProduct.map (f : A →ₗ[R] B)
    (f : A →ₗ[R] B) ∘ₗ comul = comul ∘ₗ (f : A →ₗ[R] B)
Arguments: {F : Type u_1} {R : outParam (Type u_2)} {A : outParam (Type u_3)} {B : outParam (Type u_4)} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [self : CoalgHomClass F R A B] (f : F)
Goal: TensorProduct.map ↑f ↑f ∘ₗ Coalgebra.comul = Coalgebra.comul ∘ₗ ↑f
-- CoalgHomClass.counit_comp
class CoalgHomClass (F : Type*) (R A B : outParam Type*)
    [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B]
    [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B]
    extends SemilinearMapClass F (RingHom.id R) A B : Prop where
  counit_comp : ∀ f : F, counit ∘ₗ (f : A →ₗ[R] B) = counit
  map_comp_comul : ∀ f : F, TensorProduct.map (f : A →ₗ[R] B)
    (f : A →ₗ[R] B) ∘ₗ comul = comul ∘ₗ (f : A →ₗ[R] B)
Arguments: {F : Type u_1} {R : outParam (Type u_2)} {A : outParam (Type u_3)} {B : outParam (Type u_4)} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [self : CoalgHomClass F R A B] (f : F)
Goal: Coalgebra.counit ∘ₗ ↑f = Coalgebra.counit
-- CoalgHomClass.counit_comp_apply
@[simp]
theorem counit_comp_apply (f : F) (x : A) : counit (f x) = counit (R := R) x 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [CoalgHomClass F R A B] (f : F) (x : A)
Goal: Coalgebra.counit (f x) = Coalgebra.counit x
-- CoalgHomClass.map_comp_comul_apply
@[simp]
theorem map_comp_comul_apply (f : F) (x : A) :
    TensorProduct.map f f (comul x) = comul (R := R) (f x) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [CoalgHomClass F R A B] (f : F) (x : A)
Goal: (TensorProduct.map ↑f ↑f) (Coalgebra.comul x) = Coalgebra.comul (f x)
-- CoalgHom.coe_coe
@[simp]
protected theorem coe_coe {F : Type*} [FunLike F A B] [CoalgHomClass F R A B] (f : F) :
    ⇑(f : A →ₗc[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {F : Type u_6} [FunLike F A B] [CoalgHomClass F R A B] (f : F)
Goal: ⇑↑f = ⇑f
-- CoalgHom.coe_mk
@[simp]
theorem coe_mk {f : A →ₗ[R] B} (h h₁) : ((⟨f, h, h₁⟩ : A →ₗc[R] B) : A → B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₗ[R] B} (h : Coalgebra.counit ∘ₗ f = Coalgebra.counit) (h₁ : TensorProduct.map f f ∘ₗ Coalgebra.comul = Coalgebra.comul ∘ₗ f)
Goal: ⇑{ toLinearMap := f, counit_comp := h, map_comp_comul := h₁ } = ⇑f
-- CoalgHom.coe_mks
@[norm_cast]
theorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄) : ⇑(⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : A →ₗc[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A → B} (h₁ : ∀ (x y : A), f (x + y) = f x + f y) (h₂ :
  ∀ (m : R) (x : A),
    { toFun := f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := f, map_add' := h₁ }.toFun x) (h₃ : Coalgebra.counit ∘ₗ { toFun := f, map_add' := h₁, map_smul' := h₂ } = Coalgebra.counit) (h₄ :
  TensorProduct.map { toFun := f, map_add' := h₁, map_smul' := h₂ } { toFun := f, map_add' := h₁, map_smul' := h₂ } ∘ₗ
      Coalgebra.comul =
    Coalgebra.comul ∘ₗ { toFun := f, map_add' := h₁, map_smul' := h₂ })
Goal: ⇑{ toFun := f, map_add' := h₁, map_smul' := h₂, counit_comp := h₃, map_comp_comul := h₄ } = f
-- CoalgHom.coe_linearMap_mk
@[simp, norm_cast]
theorem coe_linearMap_mk {f : A →ₗ[R] B} (h h₁) : ((⟨f, h, h₁⟩ : A →ₗc[R] B) : A →ₗ[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₗ[R] B} (h : Coalgebra.counit ∘ₗ f = Coalgebra.counit) (h₁ : TensorProduct.map f f ∘ₗ Coalgebra.comul = Coalgebra.comul ∘ₗ f)
Goal: ↑{ toLinearMap := f, counit_comp := h, map_comp_comul := h₁ } = f
-- CoalgHom.toLinearMap_eq_coe
@[simp]
theorem toLinearMap_eq_coe (f : A →ₗc[R] B) : f.toLinearMap = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B)
Goal: f.toLinearMap = ↑f
-- CoalgHom.coe_toLinearMap
@[simp, norm_cast]
theorem coe_toLinearMap (f : A →ₗc[R] B) : ⇑(f : A →ₗ[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B)
Goal: ⇑↑f = ⇑f
-- CoalgHom.coe_toAddMonoidHom
@[norm_cast]
theorem coe_toAddMonoidHom (f : A →ₗc[R] B) : ⇑(f : A →+ B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B)
Goal: ⇑↑f = ⇑f
-- CoalgHom.coe_fn_injective
theorem coe_fn_injective : @Function.Injective (A →ₗc[R] B) (A → B) (↑) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Function.Injective DFunLike.coe
-- CoalgHom.coe_fn_inj
theorem coe_fn_inj {φ₁ φ₂ : A →ₗc[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₗc[R] B}
Goal: ⇑φ₁ = ⇑φ₂ ↔ φ₁ = φ₂
-- CoalgHom.coe_linearMap_injective
theorem coe_linearMap_injective : Function.Injective ((↑) : (A →ₗc[R] B) → A →ₗ[R] B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Function.Injective fun x => ↑x
-- CoalgHom.coe_addMonoidHom_injective
theorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₗc[R] B) → A →+ B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Function.Injective AddMonoidHomClass.toAddMonoidHom
-- CoalgHom.congr_fun
protected theorem congr_fun {φ₁ φ₂ : A →ₗc[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₗc[R] B} (H : φ₁ = φ₂) (x : A)
Goal: φ₁ x = φ₂ x
-- CoalgHom.congr_arg
protected theorem congr_arg (φ : A →ₗc[R] B) {x y : A} (h : x = y) : φ x = φ y 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₗc[R] B) {x y : A} (h : x = y)
Goal: φ x = φ y
-- CoalgHom.ext
@[ext]
theorem ext {φ₁ φ₂ : A →ₗc[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₗc[R] B} (H : ∀ (x : A), φ₁ x = φ₂ x)
Goal: φ₁ = φ₂
-- CoalgHom.ext_iff
theorem ext_iff {φ₁ φ₂ : A →ₗc[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₗc[R] B}
Goal: φ₁ = φ₂ ↔ ∀ (x : A), φ₁ x = φ₂ x
-- CoalgHom.ext_of_ring
@[ext high]
theorem ext_of_ring {f g : R →ₗc[R] A} (h : f 1 = g 1) : f = g 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] {f g : R →ₗc[R] A} (h : f 1 = g 1)
Goal: f = g
-- CoalgHom.mk_coe
@[simp]
theorem mk_coe {f : A →ₗc[R] B} (h₁ h₂ h₃ h₄) : (⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : A →ₗc[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₗc[R] B} (h₁ : ∀ (x y : A), f (x + y) = f x + f y) (h₂ :
  ∀ (m : R) (x : A),
    { toFun := ⇑f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := ⇑f, map_add' := h₁ }.toFun x) (h₃ : Coalgebra.counit ∘ₗ { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ } = Coalgebra.counit) (h₄ :
  TensorProduct.map { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ } { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ } ∘ₗ
      Coalgebra.comul =
    Coalgebra.comul ∘ₗ { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ })
Goal: { toFun := ⇑f, map_add' := h₁, map_smul' := h₂, counit_comp := h₃, map_comp_comul := h₄ } = f
-- CoalgHom.coe_copy
@[simp]
theorem coe_copy (f : A →ₗc[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} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) (f' : A → B) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- CoalgHom.copy_eq
theorem copy_eq (f : A →ₗc[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} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) (f' : A → B) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- CoalgHom.id_apply
@[simps!] protected def id : A →ₗc[R] A 
Arguments: (R : Type u_1) (A : Type u_2) [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] (a : A)
Goal: (CoalgHom.id R A) a = a
-- CoalgHom.coe_id
@[simp]
theorem coe_id : ⇑(CoalgHom.id R A) = id 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A]
Goal: ⇑(CoalgHom.id R A) = id
-- CoalgHom.id_toLinearMap
@[simp]
theorem id_toLinearMap : (CoalgHom.id R A : A →ₗ[R] A) = LinearMap.id 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A]
Goal: ↑(CoalgHom.id R A) = LinearMap.id
-- CoalgHom.comp_apply
@[simps!] def comp (φ₁ : B →ₗc[R] C) (φ₂ : A →ₗc[R] B) : A →ₗc[R] C 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [AddCommMonoid C] [Module R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₗc[R] C) (φ₂ : A →ₗc[R] B)
Goal: ∀ (a : A), (φ₁.comp φ₂) a = φ₁ (φ₂ a)
-- CoalgHom.coe_comp
@[simp]
theorem coe_comp (φ₁ : B →ₗc[R] C) (φ₂ : A →ₗc[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [AddCommMonoid C] [Module R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₗc[R] C) (φ₂ : A →ₗc[R] B)
Goal: ⇑(φ₁.comp φ₂) = ⇑φ₁ ∘ ⇑φ₂
-- CoalgHom.comp_toLinearMap
@[simp]
theorem comp_toLinearMap (φ₁ : B →ₗc[R] C) (φ₂ : A →ₗc[R] B) :
    φ₁.comp φ₂ = (φ₁ : B →ₗ[R] C) ∘ₗ (φ₂ : A →ₗ[R] B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [AddCommMonoid C] [Module R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₗc[R] C) (φ₂ : A →ₗc[R] B)
Goal: ↑(φ₁.comp φ₂) = ↑φ₁ ∘ₗ ↑φ₂
-- CoalgHom.comp_id
@[simp]
theorem comp_id : φ.comp (CoalgHom.id R A) = φ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₗc[R] B)
Goal: φ.comp (CoalgHom.id R A) = φ
-- CoalgHom.id_comp
@[simp]
theorem id_comp : (CoalgHom.id R B).comp φ = φ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₗc[R] B)
Goal: (CoalgHom.id R B).comp φ = φ
-- CoalgHom.comp_assoc
theorem comp_assoc (φ₁ : C →ₗc[R] D) (φ₂ : B →ₗc[R] C) (φ₃ : A →ₗc[R] B) :
    (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [AddCommMonoid C] [Module R C] [AddCommMonoid D] [Module R D] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] [CoalgebraStruct R D] (φ₁ : C →ₗc[R] D) (φ₂ : B →ₗc[R] C) (φ₃ : A →ₗc[R] B)
Goal: (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃)
-- CoalgHom.map_smul_of_tower
theorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')
    (x : A) : φ (r • x) = r • φ x 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₗc[R] B) {R' : Type u_6} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R') (x : A)
Goal: φ (r • x) = r • φ x
-- CoalgHom.End_toSemigroup_toMul_mul
@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]
instance End : Monoid (A →ₗc[R] A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] (φ₁ φ₂ : A →ₗc[R] A)
Goal: φ₁ * φ₂ = φ₁.comp φ₂
-- CoalgHom.End_toOne_one
@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]
instance End : Monoid (A →ₗc[R] A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A]
Goal: 1 = CoalgHom.id R A
-- CoalgHom.one_apply
@[simp]
theorem one_apply (x : A) : (1 : A →ₗc[R] A) x = x 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] (x : A)
Goal: 1 x = x
-- CoalgHom.mul_apply
@[simp]
theorem mul_apply (φ ψ : A →ₗc[R] A) (x : A) : (φ * ψ) x = φ (ψ x) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] (φ ψ : A →ₗc[R] A) (x : A)
Goal: (φ * ψ) x = φ (ψ x)
-- Coalgebra.counitCoalgHom_apply
@[simp]
theorem counitCoalgHom_apply (x : A) :
    counitCoalgHom R A x = counit x 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (x : A)
Goal: (Coalgebra.counitCoalgHom R A) x = Coalgebra.counit x
-- Coalgebra.counitCoalgHom_toLinearMap
@[simp]
theorem counitCoalgHom_toLinearMap :
    counitCoalgHom R A = counit (R := R) (A := A) 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A]
Goal: ↑(Coalgebra.counitCoalgHom R A) = Coalgebra.counit
-- Coalgebra.ext_to_ring
@[ext high]
theorem ext_to_ring (f g : A →ₗc[R] R) : f = g 
Arguments: {R : Type u} (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (f g : A →ₗc[R] R)
Goal: f = g
-- CoalgEquiv.toEquiv_injective
theorem toEquiv_injective : Function.Injective (toEquiv : (A ≃ₗc[R] B) → A ≃ B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Function.Injective CoalgEquiv.toEquiv
-- CoalgEquiv.toEquiv_inj
@[simp]
theorem toEquiv_inj {e₁ e₂ : A ≃ₗc[R] B} : e₁.toEquiv = e₂.toEquiv ↔ e₁ = e₂ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {e₁ e₂ : A ≃ₗc[R] B}
Goal: e₁.toEquiv = e₂.toEquiv ↔ e₁ = e₂
-- CoalgEquiv.toCoalgHom_injective
theorem toCoalgHom_injective : Function.Injective (toCoalgHom : (A ≃ₗc[R] B) → A →ₗc[R] B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Function.Injective CoalgEquiv.toCoalgHom
-- CoalgEquiv.toCoalgHom_inj
@[simp, norm_cast]
theorem toCoalgHom_inj {e₁ e₂ : A ≃ₗc[R] B} : (↑e₁ : A →ₗc[R] B) = e₂ ↔ e₁ = e₂ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {e₁ e₂ : A ≃ₗc[R] B}
Goal: ↑e₁ = ↑e₂ ↔ e₁ = e₂
-- CoalgEquiv.coe_mk
@[simp]
theorem coe_mk {f h h₀ h₁ h₂ h₃ h₄ h₅} :
    (⟨⟨⟨⟨f, h⟩, h₀⟩, h₁, h₂⟩, h₃, h₄, h₅⟩ : A ≃ₗc[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A → B} {h : ∀ (x y : A), f (x + y) = f x + f y} {h₀ :
  ∀ (m : R) (x : A),
    { toFun := f, map_add' := h }.toFun (m • x) = (RingHom.id R) m • { toFun := f, map_add' := h }.toFun x} {h₁ : Coalgebra.counit ∘ₗ { toFun := f, map_add' := h, map_smul' := h₀ } = Coalgebra.counit} {h₂ :
  TensorProduct.map { toFun := f, map_add' := h, map_smul' := h₀ } { toFun := f, map_add' := h, map_smul' := h₀ } ∘ₗ
      Coalgebra.comul =
    Coalgebra.comul ∘ₗ { toFun := f, map_add' := h, map_smul' := h₀ }} {h₃ : B → A} {h₄ :
  Function.LeftInverse h₃ { toFun := f, map_add' := h, map_smul' := h₀, counit_comp := h₁, map_comp_comul := h₂ }.toFun} {h₅ :
  Function.RightInverse h₃
    { toFun := f, map_add' := h, map_smul' := h₀, counit_comp := h₁, map_comp_comul := h₂ }.toFun}
Goal: ⇑{ toFun := f, map_add' := h, map_smul' := h₀, counit_comp := h₁, map_comp_comul := h₂, invFun := h₃, left_inv := h₄,
      right_inv := h₅ } =
  f
-- CoalgEquiv.coe_coe
@[simp, norm_cast]
theorem coe_coe : ⇑(e : A →ₗc[R] B) = e 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (e : A ≃ₗc[R] B)
Goal: ⇑↑e = ⇑e
-- CoalgEquiv.toLinearEquiv_eq_coe
@[simp]
theorem toLinearEquiv_eq_coe (f : A ≃ₗc[R] B) : f.toLinearEquiv = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A ≃ₗc[R] B)
Goal: f.toLinearEquiv = ↑f