ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DerivedCategory.HomologySequence.epi_homologyMap_mor₁_iff
lemma epi_homologyMap_mor₁_iff :
    Epi ((homologyFunctor C n₀).map T.mor₁) ↔ (homologyFunctor C n₀).map T.mor₂ = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] (T : CategoryTheory.Pretriangulated.Triangle (DerivedCategory C)) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ : ℤ)
Goal: CategoryTheory.Epi ((DerivedCategory.homologyFunctor C n₀).map T.mor₁) ↔
  (DerivedCategory.homologyFunctor C n₀).map T.mor₂ = 0
-- DerivedCategory.HomologySequence.mono_homologyMap_mor₁_iff
lemma mono_homologyMap_mor₁_iff :
    Mono ((homologyFunctor C n₁).map T.mor₁) ↔ δ T n₀ n₁ h = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] (T : CategoryTheory.Pretriangulated.Triangle (DerivedCategory C)) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: CategoryTheory.Mono ((DerivedCategory.homologyFunctor C n₁).map T.mor₁) ↔
  DerivedCategory.HomologySequence.δ T n₀ n₁ h = 0
-- DerivedCategory.HomologySequence.epi_homologyMap_mor₂_iff
lemma epi_homologyMap_mor₂_iff :
    Epi ((homologyFunctor C n₀).map T.mor₂) ↔ δ T n₀ n₁ h = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] (T : CategoryTheory.Pretriangulated.Triangle (DerivedCategory C)) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: CategoryTheory.Epi ((DerivedCategory.homologyFunctor C n₀).map T.mor₂) ↔
  DerivedCategory.HomologySequence.δ T n₀ n₁ h = 0
-- DerivedCategory.HomologySequence.mono_homologyMap_mor₂_iff
lemma mono_homologyMap_mor₂_iff :
    Mono ((homologyFunctor C n₀).map T.mor₂) ↔ (homologyFunctor C n₀).map T.mor₁ = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] (T : CategoryTheory.Pretriangulated.Triangle (DerivedCategory C)) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ : ℤ)
Goal: CategoryTheory.Mono ((DerivedCategory.homologyFunctor C n₀).map T.mor₂) ↔
  (DerivedCategory.homologyFunctor C n₀).map T.mor₁ = 0
-- CategoryTheory.ShortComplex.ShortExact.singleTriangle_obj₃
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangle.obj₃ = (DerivedCategory.singleFunctor C 0).obj S.X₃
-- CategoryTheory.ShortComplex.ShortExact.singleTriangle_mor₁
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangle.mor₁ = (DerivedCategory.singleFunctor C 0).map S.f
-- CategoryTheory.ShortComplex.ShortExact.singleTriangle_obj₂
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangle.obj₂ = (DerivedCategory.singleFunctor C 0).obj S.X₂
-- CategoryTheory.ShortComplex.ShortExact.singleTriangle_mor₂
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangle.mor₂ = (DerivedCategory.singleFunctor C 0).map S.g
-- CategoryTheory.ShortComplex.ShortExact.singleTriangle_mor₃
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangle.mor₃ = hS.singleδ
-- CategoryTheory.ShortComplex.ShortExact.singleTriangle_obj₁
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangle.obj₁ = (DerivedCategory.singleFunctor C 0).obj S.X₁
-- CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_hom_hom₁
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangleIso.hom.hom₁ = ((DerivedCategory.singleFunctorsPostcompQIso C).hom.hom 0).app S.X₁
-- CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_hom_hom₃
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangleIso.hom.hom₃ = ((DerivedCategory.singleFunctorsPostcompQIso C).hom.hom 0).app S.X₃
-- CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_inv_hom₃
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangleIso.inv.hom₃ = ((DerivedCategory.singleFunctorsPostcompQIso C).inv.hom 0).app S.X₃
-- CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_hom_hom₂
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangleIso.hom.hom₂ = ((DerivedCategory.singleFunctorsPostcompQIso C).hom.hom 0).app S.X₂
-- CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_inv_hom₂
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangleIso.inv.hom₂ = ((DerivedCategory.singleFunctorsPostcompQIso C).inv.hom 0).app S.X₂
-- CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_inv_hom₁
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangleIso.inv.hom₁ = ((DerivedCategory.singleFunctorsPostcompQIso C).inv.hom 0).app S.X₁
-- CategoryTheory.ShortComplex.ShortExact.singleTriangle_distinguished
lemma singleTriangle_distinguished :
    hS.singleTriangle ∈ distTriang (DerivedCategory C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact)
Goal: hS.singleTriangle ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
-- AddGrp.hasLimit_iff_small_sections
@[to_additive "A functor `F : J ⥤ AddGrp.{u}` has a limit iff
`(F ⋙ forget AddGrp).sections` is `u`-small."]
lemma hasLimit_iff_small_sections :
    HasLimit F ↔ Small.{u} (F ⋙ forget Grp).sections 
Arguments: {J : Type v} [CategoryTheory.Category.{w, v} J] (F : J ⥤ AddGrp)
Goal: CategoryTheory.Limits.HasLimit F ↔ Small.{u, max u v} ↑(F ⋙ CategoryTheory.forget AddGrp).sections
-- Grp.hasLimit_iff_small_sections
@[to_additive "A functor `F : J ⥤ AddGrp.{u}` has a limit iff
`(F ⋙ forget AddGrp).sections` is `u`-small."]
lemma hasLimit_iff_small_sections :
    HasLimit F ↔ Small.{u} (F ⋙ forget Grp).sections 
Arguments: {J : Type v} [CategoryTheory.Category.{w, v} J] (F : J ⥤ Grp)
Goal: CategoryTheory.Limits.HasLimit F ↔ Small.{u, max u v} ↑(F ⋙ CategoryTheory.forget Grp).sections
-- CommGrp.hasLimit_iff_small_sections
@[to_additive "A functor `F : J ⥤ AddCommGrp.{u}` has a limit iff
`(F ⋙ forget AddCommGrp).sections` is `u`-small."]
lemma hasLimit_iff_small_sections :
    HasLimit F ↔ Small.{u} (F ⋙ forget CommGrp).sections 
Arguments: {J : Type v} [CategoryTheory.Category.{w, v} J] (F : J ⥤ CommGrp)
Goal: CategoryTheory.Limits.HasLimit F ↔ Small.{u, max u v} ↑(F ⋙ CategoryTheory.forget CommGrp).sections
-- AddCommGrp.hasLimit_iff_small_sections
@[to_additive "A functor `F : J ⥤ AddCommGrp.{u}` has a limit iff
`(F ⋙ forget AddCommGrp).sections` is `u`-small."]
lemma hasLimit_iff_small_sections :
    HasLimit F ↔ Small.{u} (F ⋙ forget CommGrp).sections 
Arguments: {J : Type v} [CategoryTheory.Category.{w, v} J] (F : J ⥤ AddCommGrp)
Goal: CategoryTheory.Limits.HasLimit F ↔ Small.{u, max u v} ↑(F ⋙ CategoryTheory.forget AddCommGrp).sections
-- AddCommGrp.kernelIsoKer_hom_comp_subtype
@[simp]
theorem kernelIsoKer_hom_comp_subtype {G H : AddCommGrp.{u}} (f : G ⟶ H) :
    (kernelIsoKer f).hom ≫ AddSubgroup.subtype f.ker = kernel.ι f 
Arguments: {G H : AddCommGrp} (f : G ⟶ H)
Goal: (AddCommGrp.kernelIsoKer f).hom ≫ (AddMonoidHom.ker f).subtype = CategoryTheory.Limits.kernel.ι f
-- AddCommGrp.kernelIsoKer_inv_comp_ι
@[simp]
theorem kernelIsoKer_inv_comp_ι {G H : AddCommGrp.{u}} (f : G ⟶ H) :
    (kernelIsoKer f).inv ≫ kernel.ι f = AddSubgroup.subtype f.ker 
Arguments: {G H : AddCommGrp} (f : G ⟶ H)
Goal: (AddCommGrp.kernelIsoKer f).inv ≫ CategoryTheory.Limits.kernel.ι f = (AddMonoidHom.ker f).subtype
-- AddCommGrp.kernelIsoKerOver_inv_left_apply
@[simps! hom_left_apply_coe inv_left_apply]
def kernelIsoKerOver {G H : AddCommGrp.{u}} (f : G ⟶ H) :
    Over.mk (kernel.ι f) ≅ @Over.mk _ _ G (AddCommGrp.of f.ker) (AddSubgroup.subtype f.ker) 
Arguments: {G H : AddCommGrp} (f : G ⟶ H)
Goal: ∀ (a : ↑(CategoryTheory.Limits.KernelFork.ofι (AddMonoidHom.ker f).subtype ⋯).pt),
  (AddCommGrp.kernelIsoKerOver f).inv.left a =
    (CategoryTheory.Limits.limit.lift (CategoryTheory.Limits.parallelPair f 0)
        (CategoryTheory.Limits.KernelFork.ofι (AddMonoidHom.ker f).subtype ⋯))
      a
-- AddCommGrp.kernelIsoKerOver_hom_left_apply_coe
@[simps! hom_left_apply_coe inv_left_apply]
def kernelIsoKerOver {G H : AddCommGrp.{u}} (f : G ⟶ H) :
    Over.mk (kernel.ι f) ≅ @Over.mk _ _ G (AddCommGrp.of f.ker) (AddSubgroup.subtype f.ker) 
Arguments: {G H : AddCommGrp} (f : G ⟶ H) (g : ↑(CategoryTheory.Limits.kernel f))
Goal: ↑((AddCommGrp.kernelIsoKerOver f).hom.left g) = (CategoryTheory.Limits.kernel.ι f) g
-- ComplexShape.next_eq
@[ext]
structure ComplexShape (ι : Type*) where
    Rel : ι → ι → Prop
    next_eq : ∀ {i j j'}, Rel i j → Rel i j' → j = j'
    prev_eq : ∀ {i i' j}, Rel i j → Rel i' j → i = i'
Arguments: {ι : Type u_1} (self : ComplexShape ι) {i j j' : ι}
Goal: self.Rel i j → self.Rel i j' → j = j'
-- ComplexShape.prev_eq
@[ext]
structure ComplexShape (ι : Type*) where
    Rel : ι → ι → Prop
    next_eq : ∀ {i j j'}, Rel i j → Rel i j' → j = j'
    prev_eq : ∀ {i i' j}, Rel i j → Rel i' j → i = i'
Arguments: {ι : Type u_1} (self : ComplexShape ι) {i i' j : ι}
Goal: self.Rel i j → self.Rel i' j → i = i'
-- ComplexShape.refl_Rel
@[simps]
def refl (ι : Type*) : ComplexShape ι 
Arguments: (ι : Type u_2) (i j : ι)
Goal: (ComplexShape.refl ι).Rel i j = (i = j)
-- ComplexShape.symm_Rel
@[simps]
def symm (c : ComplexShape ι) : ComplexShape ι 
Arguments: {ι : Type u_1} (c : ComplexShape ι) (i j : ι)
Goal: c.symm.Rel i j = c.Rel j i
-- ComplexShape.symm_symm
@[simp]
theorem symm_symm (c : ComplexShape ι) : c.symm.symm = c 
Arguments: {ι : Type u_1} (c : ComplexShape ι)
Goal: c.symm.symm = c
-- ComplexShape.symm_bijective
theorem symm_bijective :
    Function.Bijective (ComplexShape.symm : ComplexShape ι → ComplexShape ι) 
Arguments: {ι : Type u_1}
Goal: Function.Bijective ComplexShape.symm
-- ComplexShape.next_eq'
theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j 
Arguments: {ι : Type u_1} (c : ComplexShape ι) {i j : ι} (h : c.Rel i j)
Goal: c.next i = j
-- ComplexShape.prev_eq'
theorem prev_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.prev j = i 
Arguments: {ι : Type u_1} (c : ComplexShape ι) {i j : ι} (h : c.Rel i j)
Goal: c.prev j = i
-- ComplexShape.next_eq_self'
lemma next_eq_self' (c : ComplexShape ι) (j : ι) (hj : ∀ k, ¬ c.Rel j k) :
    c.next j = j 
Arguments: {ι : Type u_1} (c : ComplexShape ι) (j : ι) (hj : ∀ (k : ι), ¬c.Rel j k)
Goal: c.next j = j
-- ComplexShape.prev_eq_self'
lemma prev_eq_self' (c : ComplexShape ι) (j : ι) (hj : ∀ i, ¬ c.Rel i j) :
    c.prev j = j 
Arguments: {ι : Type u_1} (c : ComplexShape ι) (j : ι) (hj : ∀ (i : ι), ¬c.Rel i j)
Goal: c.prev j = j
-- ComplexShape.next_eq_self
lemma next_eq_self (c : ComplexShape ι) (j : ι) (hj : ¬ c.Rel j (c.next j)) :
    c.next j = j 
Arguments: {ι : Type u_1} (c : ComplexShape ι) (j : ι) (hj : ¬c.Rel j (c.next j))
Goal: c.next j = j
-- ComplexShape.prev_eq_self
lemma prev_eq_self (c : ComplexShape ι) (j : ι) (hj : ¬ c.Rel (c.prev j) j) :
    c.prev j = j 
Arguments: {ι : Type u_1} (c : ComplexShape ι) (j : ι) (hj : ¬c.Rel (c.prev j) j)
Goal: c.prev j = j
-- ComplexShape.up'_Rel
@[simps]
def up' {α : Type*} [AddRightCancelSemigroup α] (a : α) : ComplexShape α 
Arguments: {α : Type u_2} [AddRightCancelSemigroup α] (a i j : α)
Goal: (ComplexShape.up' a).Rel i j = (i + a = j)
-- ComplexShape.down'_Rel
@[simps]
def down' {α : Type*} [AddRightCancelSemigroup α] (a : α) : ComplexShape α 
Arguments: {α : Type u_2} [AddRightCancelSemigroup α] (a i j : α)
Goal: (ComplexShape.down' a).Rel i j = (j + a = i)
-- ComplexShape.down'_mk
theorem down'_mk {α : Type*} [AddRightCancelSemigroup α] (a : α) (i j : α) (h : j + a = i) :
    (down' a).Rel i j 
Arguments: {α : Type u_2} [AddRightCancelSemigroup α] (a i j : α) (h : j + a = i)
Goal: (ComplexShape.down' a).Rel i j
-- ComplexShape.up_Rel
@[simps!]
def up (α : Type*) [AddRightCancelSemigroup α] [One α] : ComplexShape α 
Arguments: (α : Type u_2) [AddRightCancelSemigroup α] [One α] (i j : α)
Goal: (ComplexShape.up α).Rel i j = (i + 1 = j)
-- ComplexShape.down_Rel
@[simps!]
def down (α : Type*) [AddRightCancelSemigroup α] [One α] : ComplexShape α 
Arguments: (α : Type u_2) [AddRightCancelSemigroup α] [One α] (i j : α)
Goal: (ComplexShape.down α).Rel i j = (j + 1 = i)
-- ComplexShape.down_mk
theorem down_mk {α : Type*} [AddRightCancelSemigroup α] [One α] (i j : α) (h : j + 1 = i) :
    (down α).Rel i j 
Arguments: {α : Type u_2} [AddRightCancelSemigroup α] [One α] (i j : α) (h : j + 1 = i)
Goal: (ComplexShape.down α).Rel i j
-- NonUnitalStarSubalgebraClass.coeSubtype
@[simp]
theorem coeSubtype : (subtype s : s → A) = Subtype.val 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Star A] [Module R A] {S : Type w''} [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] [StarMemClass S A] (s : S)
Goal: ⇑(NonUnitalStarSubalgebraClass.subtype s) = Subtype.val
-- NonUnitalStarSubalgebra.star_mem'
structure NonUnitalStarSubalgebra (R : Type u) (A : Type v) [CommSemiring R]
    [NonUnitalNonAssocSemiring A] [Module R A] [Star A]
    extends NonUnitalSubalgebra R A : Type v where
    star_mem' : ∀ {a : A} (_ha : a ∈ carrier), star a ∈ carrier
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (self : NonUnitalStarSubalgebra R A) {a : A} (_ha : a ∈ self.carrier)
Goal: star a ∈ self.carrier
-- NonUnitalStarSubalgebra.mem_carrier
theorem mem_carrier {s : NonUnitalStarSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] {s : NonUnitalStarSubalgebra R A} {x : A}
Goal: x ∈ s.carrier ↔ x ∈ s
-- NonUnitalStarSubalgebra.ext
@[ext]
theorem ext {S T : NonUnitalStarSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] {S T : NonUnitalStarSubalgebra R A} (h : ∀ (x : A), x ∈ S ↔ x ∈ T)
Goal: S = T
-- NonUnitalStarSubalgebra.mem_toNonUnitalSubalgebra
@[simp]
theorem mem_toNonUnitalSubalgebra {S : NonUnitalStarSubalgebra R A} {x} :
    x ∈ S.toNonUnitalSubalgebra ↔ x ∈ S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] {S : NonUnitalStarSubalgebra R A} {x : A}
Goal: x ∈ S.toNonUnitalSubalgebra ↔ x ∈ S
-- NonUnitalStarSubalgebra.coe_toNonUnitalSubalgebra
@[simp]
theorem coe_toNonUnitalSubalgebra (S : NonUnitalStarSubalgebra R A) :
    (↑S.toNonUnitalSubalgebra : Set A) = S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A)
Goal: ↑S.toNonUnitalSubalgebra = ↑S
-- NonUnitalStarSubalgebra.toNonUnitalSubalgebra_injective
theorem toNonUnitalSubalgebra_injective :
    Function.Injective
      (toNonUnitalSubalgebra : NonUnitalStarSubalgebra R A → NonUnitalSubalgebra R A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A]
Goal: Function.Injective NonUnitalStarSubalgebra.toNonUnitalSubalgebra
-- NonUnitalStarSubalgebra.toNonUnitalSubalgebra_inj
theorem toNonUnitalSubalgebra_inj {S U : NonUnitalStarSubalgebra R A} :
    S.toNonUnitalSubalgebra = U.toNonUnitalSubalgebra ↔ S = U 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] {S U : NonUnitalStarSubalgebra R A}
Goal: S.toNonUnitalSubalgebra = U.toNonUnitalSubalgebra ↔ S = U
-- NonUnitalStarSubalgebra.toNonUnitalSubalgebra_le_iff
theorem toNonUnitalSubalgebra_le_iff {S₁ S₂ : NonUnitalStarSubalgebra R A} :
    S₁.toNonUnitalSubalgebra ≤ S₂.toNonUnitalSubalgebra ↔ S₁ ≤ S₂ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] {S₁ S₂ : NonUnitalStarSubalgebra R A}
Goal: S₁.toNonUnitalSubalgebra ≤ S₂.toNonUnitalSubalgebra ↔ S₁ ≤ S₂
-- NonUnitalStarSubalgebra.coe_copy
@[simp]
theorem coe_copy (S : NonUnitalStarSubalgebra R A) (s : Set A) (hs : s = ↑S) :
    (S.copy s hs : Set A) = s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A) (s : Set A) (hs : s = ↑S)
Goal: ↑(S.copy s hs) = s
-- NonUnitalStarSubalgebra.copy_eq
theorem copy_eq (S : NonUnitalStarSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A) (s : Set A) (hs : s = ↑S)
Goal: S.copy s hs = S
-- NonUnitalStarSubalgebra.mem_toNonUnitalSubring
@[simp]
theorem mem_toNonUnitalSubring {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A]
    [Star A] {S : NonUnitalStarSubalgebra R A} {x} : x ∈ S.toNonUnitalSubring ↔ x ∈ S 
Arguments: {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A] [Star A] {S : NonUnitalStarSubalgebra R A} {x : A}
Goal: x ∈ S.toNonUnitalSubring ↔ x ∈ S
-- NonUnitalStarSubalgebra.coe_toNonUnitalSubring
@[simp]
theorem coe_toNonUnitalSubring {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A]
    [Star A] (S : NonUnitalStarSubalgebra R A) : (↑S.toNonUnitalSubring : Set A) = S 
Arguments: {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A)
Goal: ↑S.toNonUnitalSubring = ↑S
-- NonUnitalStarSubalgebra.toNonUnitalSubring_injective
theorem toNonUnitalSubring_injective {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A]
    [Module R A] [Star A] :
    Function.Injective (toNonUnitalSubring : NonUnitalStarSubalgebra R A → NonUnitalSubring A) 
Arguments: {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A] [Star A]
Goal: Function.Injective NonUnitalStarSubalgebra.toNonUnitalSubring
-- NonUnitalStarSubalgebra.toNonUnitalSubring_inj
theorem toNonUnitalSubring_inj {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A]
    [Star A] {S U : NonUnitalStarSubalgebra R A} :
    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U 
Arguments: {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A] [Star A] {S U : NonUnitalStarSubalgebra R A}
Goal: S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U
-- NonUnitalStarSubalgebra.coe_add
protected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A) (x y : ↥S)
Goal: ↑(x + y) = ↑x + ↑y
-- NonUnitalStarSubalgebra.coe_mul
protected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A) (x y : ↥S)
Goal: ↑(x * y) = ↑x * ↑y
-- NonUnitalStarSubalgebra.coe_zero
protected theorem coe_zero : ((0 : S) : A) = 0 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A)
Goal: ↑0 = 0
-- NonUnitalStarSubalgebra.coe_neg
protected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A]
    [Star A] {S : NonUnitalStarSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x 
Arguments: {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A] [Star A] {S : NonUnitalStarSubalgebra R A} (x : ↥S)
Goal: ↑(-x) = -↑x
-- NonUnitalStarSubalgebra.coe_sub
protected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A]
    [Star A] {S : NonUnitalStarSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y 
Arguments: {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A] [Star A] {S : NonUnitalStarSubalgebra R A} (x y : ↥S)
Goal: ↑(x - y) = ↑x - ↑y
-- NonUnitalStarSubalgebra.coe_smul
@[simp, norm_cast]
theorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :
    ↑(r • x) = r • (x : A) 
Arguments: {R' : Type u'} {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A) [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : ↥S)
Goal: ↑(r • x) = r • ↑x
-- NonUnitalStarSubalgebra.coe_eq_zero
protected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A) {x : ↥S}
Goal: ↑x = 0 ↔ x = 0
-- NonUnitalStarSubalgebra.toNonUnitalSubalgebra_subtype
@[simp]
theorem toNonUnitalSubalgebra_subtype :
    NonUnitalSubalgebraClass.subtype S = NonUnitalStarSubalgebraClass.subtype S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A)
Goal: NonUnitalSubalgebraClass.subtype S = NonUnitalAlgHomClass.toNonUnitalAlgHom (NonUnitalStarSubalgebraClass.subtype S)
-- NonUnitalStarSubalgebra.toSubring_subtype
@[simp]
theorem toSubring_subtype {R A : Type*} [CommRing R] [NonUnitalRing A] [Module R A] [Star A]
    (S : NonUnitalStarSubalgebra R A) :
    NonUnitalSubringClass.subtype S = NonUnitalStarSubalgebraClass.subtype S 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [NonUnitalRing A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A)
Goal: NonUnitalSubringClass.subtype S = ↑(NonUnitalStarSubalgebraClass.subtype S)