ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NonUnitalStarSubalgebra.map_mono
theorem map_mono {S₁ S₂ : NonUnitalStarSubalgebra R A} {f : F} :
    S₁ ≤ S₂ → (map f S₁ : NonUnitalStarSubalgebra R B) ≤ map f S₂ 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] {S₁ S₂ : NonUnitalStarSubalgebra R A} {f : F}
Goal: S₁ ≤ S₂ → NonUnitalStarSubalgebra.map f S₁ ≤ NonUnitalStarSubalgebra.map f S₂
-- NonUnitalStarSubalgebra.map_injective
theorem map_injective {f : F} (hf : Function.Injective f) :
    Function.Injective (map f : NonUnitalStarSubalgebra R A → NonUnitalStarSubalgebra R B) 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] {f : F} (hf : Function.Injective ⇑f)
Goal: Function.Injective (NonUnitalStarSubalgebra.map f)
-- NonUnitalStarSubalgebra.map_id
@[simp]
theorem map_id (S : NonUnitalStarSubalgebra R A) : map (NonUnitalStarAlgHom.id R A) S = S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A)
Goal: NonUnitalStarSubalgebra.map (NonUnitalStarAlgHom.id R A) S = S
-- NonUnitalStarSubalgebra.map_map
theorem map_map (S : NonUnitalStarSubalgebra R A) (g : B →⋆ₙₐ[R] C) (f : A →⋆ₙₐ[R] B) :
    (S.map f).map g = S.map (g.comp f) 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type w'} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [NonUnitalNonAssocSemiring C] [Module R C] [Star C] (S : NonUnitalStarSubalgebra R A) (g : B →⋆ₙₐ[R] C) (f : A →⋆ₙₐ[R] B)
Goal: NonUnitalStarSubalgebra.map g (NonUnitalStarSubalgebra.map f S) = NonUnitalStarSubalgebra.map (g.comp f) S
-- NonUnitalStarSubalgebra.mem_map
@[simp]
theorem mem_map {S : NonUnitalStarSubalgebra R A} {f : F} {y : B} :
    y ∈ map f S ↔ ∃ x ∈ S, f x = y 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] {S : NonUnitalStarSubalgebra R A} {f : F} {y : B}
Goal: y ∈ NonUnitalStarSubalgebra.map f S ↔ ∃ x ∈ S, f x = y
-- NonUnitalStarSubalgebra.map_toNonUnitalSubalgebra
theorem map_toNonUnitalSubalgebra {S : NonUnitalStarSubalgebra R A} {f : F} :
    (map f S : NonUnitalStarSubalgebra R B).toNonUnitalSubalgebra =
      NonUnitalSubalgebra.map f S.toNonUnitalSubalgebra 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] {S : NonUnitalStarSubalgebra R A} {f : F}
Goal: (NonUnitalStarSubalgebra.map f S).toNonUnitalSubalgebra = NonUnitalSubalgebra.map f S.toNonUnitalSubalgebra
-- NonUnitalStarSubalgebra.coe_map
@[simp]
theorem coe_map (S : NonUnitalStarSubalgebra R A) (f : F) : map f S = f '' S 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (S : NonUnitalStarSubalgebra R A) (f : F)
Goal: ↑(NonUnitalStarSubalgebra.map f S) = ⇑f '' ↑S
-- NonUnitalStarSubalgebra.map_le
theorem map_le {S : NonUnitalStarSubalgebra R A} {f : F} {U : NonUnitalStarSubalgebra R B} :
    map f S ≤ U ↔ S ≤ comap f U 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] {S : NonUnitalStarSubalgebra R A} {f : F} {U : NonUnitalStarSubalgebra R B}
Goal: NonUnitalStarSubalgebra.map f S ≤ U ↔ S ≤ NonUnitalStarSubalgebra.comap f U
-- NonUnitalStarSubalgebra.gc_map_comap
theorem gc_map_comap (f : F) : GaloisConnection (map f) (comap f) 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F)
Goal: GaloisConnection (NonUnitalStarSubalgebra.map f) (NonUnitalStarSubalgebra.comap f)
-- NonUnitalStarSubalgebra.mem_comap
@[simp]
theorem mem_comap (S : NonUnitalStarSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (S : NonUnitalStarSubalgebra R B) (f : F) (x : A)
Goal: x ∈ NonUnitalStarSubalgebra.comap f S ↔ f x ∈ S
-- NonUnitalStarSubalgebra.coe_comap
@[simp, norm_cast]
theorem coe_comap (S : NonUnitalStarSubalgebra R B) (f : F) : comap f S = f ⁻¹' (S : Set B) 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (S : NonUnitalStarSubalgebra R B) (f : F)
Goal: ↑(NonUnitalStarSubalgebra.comap f S) = ⇑f ⁻¹' ↑S
-- NonUnitalSubalgebra.mem_toNonUnitalStarSubalgebra
@[simp]
theorem mem_toNonUnitalStarSubalgebra {s : NonUnitalSubalgebra R A} {h_star} {x} :
    x ∈ s.toNonUnitalStarSubalgebra h_star ↔ x ∈ s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [Star A] {s : NonUnitalSubalgebra R A} {h_star : ∀ x ∈ s, star x ∈ s} {x : A}
Goal: x ∈ s.toNonUnitalStarSubalgebra h_star ↔ x ∈ s
-- NonUnitalSubalgebra.coe_toNonUnitalStarSubalgebra
@[simp]
theorem coe_toNonUnitalStarSubalgebra (s : NonUnitalSubalgebra R A) (h_star) :
    (s.toNonUnitalStarSubalgebra h_star : Set A) = s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [Star A] (s : NonUnitalSubalgebra R A) (h_star : ∀ x ∈ s, star x ∈ s)
Goal: ↑(s.toNonUnitalStarSubalgebra h_star) = ↑s
-- NonUnitalSubalgebra.toNonUnitalStarSubalgebra_toNonUnitalSubalgebra
@[simp]
theorem toNonUnitalStarSubalgebra_toNonUnitalSubalgebra (s : NonUnitalSubalgebra R A) (h_star) :
    (s.toNonUnitalStarSubalgebra h_star).toNonUnitalSubalgebra = s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [Star A] (s : NonUnitalSubalgebra R A) (h_star : ∀ x ∈ s, star x ∈ s)
Goal: (s.toNonUnitalStarSubalgebra h_star).toNonUnitalSubalgebra = s
-- NonUnitalStarSubalgebra.toNonUnitalSubalgebra_toNonUnitalStarSubalgebra
@[simp]
theorem _root_.NonUnitalStarSubalgebra.toNonUnitalSubalgebra_toNonUnitalStarSubalgebra
    (S : NonUnitalStarSubalgebra R A) :
    (S.toNonUnitalSubalgebra.toNonUnitalStarSubalgebra fun _ => star_mem (s := S)) = S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [Star A] (S : NonUnitalStarSubalgebra R A)
Goal: S.toNonUnitalStarSubalgebra ⋯ = S
-- NonUnitalStarAlgHom.mem_range
@[simp]
theorem mem_range (φ : F) {y : B} :
    y ∈ (NonUnitalStarAlgHom.range φ : NonUnitalStarSubalgebra R B) ↔ ∃ x : A, φ x = y 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (φ : F) {y : B}
Goal: y ∈ NonUnitalStarAlgHom.range φ ↔ ∃ x, φ x = y
-- NonUnitalStarAlgHom.mem_range_self
theorem mem_range_self (φ : F) (x : A) :
    φ x ∈ (NonUnitalStarAlgHom.range φ : NonUnitalStarSubalgebra R B) 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (φ : F) (x : A)
Goal: φ x ∈ NonUnitalStarAlgHom.range φ
-- NonUnitalStarAlgHom.coe_range
@[simp]
theorem coe_range (φ : F) :
    ((NonUnitalStarAlgHom.range φ : NonUnitalStarSubalgebra R B) : Set B) = Set.range (φ : A → B) 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (φ : F)
Goal: ↑(NonUnitalStarAlgHom.range φ) = Set.range ⇑φ
-- NonUnitalStarAlgHom.range_comp
theorem range_comp (f : A →⋆ₙₐ[R] B) (g : B →⋆ₙₐ[R] C) :
    NonUnitalStarAlgHom.range (g.comp f) = (NonUnitalStarAlgHom.range f).map g 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type w'} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [NonUnitalNonAssocSemiring C] [Module R C] [Star C] (f : A →⋆ₙₐ[R] B) (g : B →⋆ₙₐ[R] C)
Goal: NonUnitalStarAlgHom.range (g.comp f) = NonUnitalStarSubalgebra.map g (NonUnitalStarAlgHom.range f)
-- NonUnitalStarAlgHom.range_comp_le_range
theorem range_comp_le_range (f : A →⋆ₙₐ[R] B) (g : B →⋆ₙₐ[R] C) :
    NonUnitalStarAlgHom.range (g.comp f) ≤ NonUnitalStarAlgHom.range g 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type w'} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [NonUnitalNonAssocSemiring C] [Module R C] [Star C] (f : A →⋆ₙₐ[R] B) (g : B →⋆ₙₐ[R] C)
Goal: NonUnitalStarAlgHom.range (g.comp f) ≤ NonUnitalStarAlgHom.range g
-- NonUnitalStarAlgHom.subtype_comp_codRestrict
@[simp]
theorem subtype_comp_codRestrict (f : F) (S : NonUnitalStarSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :
    (NonUnitalStarSubalgebraClass.subtype S).comp (NonUnitalStarAlgHom.codRestrict f S hf) = f 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F) (S : NonUnitalStarSubalgebra R B) (hf : ∀ (x : A), f x ∈ S)
Goal: (NonUnitalStarSubalgebraClass.subtype S).comp (NonUnitalStarAlgHom.codRestrict f S hf) = ↑f
-- NonUnitalStarAlgHom.coe_codRestrict
@[simp]
theorem coe_codRestrict (f : F) (S : NonUnitalStarSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :
    ↑(NonUnitalStarAlgHom.codRestrict f S hf x) = f x 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F) (S : NonUnitalStarSubalgebra R B) (hf : ∀ (x : A), f x ∈ S) (x : A)
Goal: ↑((NonUnitalStarAlgHom.codRestrict f S hf) x) = f x
-- NonUnitalStarAlgHom.injective_codRestrict
theorem injective_codRestrict (f : F) (S : NonUnitalStarSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :
    Function.Injective (NonUnitalStarAlgHom.codRestrict f S hf) ↔ Function.Injective f 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F) (S : NonUnitalStarSubalgebra R B) (hf : ∀ (x : A), f x ∈ S)
Goal: Function.Injective ⇑(NonUnitalStarAlgHom.codRestrict f S hf) ↔ Function.Injective ⇑f
-- NonUnitalStarAlgHom.mem_equalizer
@[simp]
theorem mem_equalizer (φ ψ : F) (x : A) :
    x ∈ NonUnitalStarAlgHom.equalizer φ ψ ↔ φ x = ψ x 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Star A] [NonUnitalNonAssocSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (φ ψ : F) (x : A)
Goal: x ∈ NonUnitalStarAlgHom.equalizer φ ψ ↔ φ x = ψ x
-- StarAlgEquiv.ofLeftInverse'_apply
@[simp]
theorem ofLeftInverse'_apply {g : B → A} {f : F} (h : Function.LeftInverse g f) (x : A) :
    ofLeftInverse' h x = f x 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [Star A] [NonUnitalSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] {g : B → A} {f : F} (h : Function.LeftInverse g ⇑f) (x : A)
Goal: ↑((StarAlgEquiv.ofLeftInverse' h) x) = f x
-- StarAlgEquiv.ofLeftInverse'_symm_apply
@[simp]
theorem ofLeftInverse'_symm_apply {g : B → A} {f : F} (h : Function.LeftInverse g f)
    (x : NonUnitalStarAlgHom.range f) : (ofLeftInverse' h).symm x = g x 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [Star A] [NonUnitalSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] {g : B → A} {f : F} (h : Function.LeftInverse g ⇑f) (x : ↥(NonUnitalStarAlgHom.range f))
Goal: (StarAlgEquiv.ofLeftInverse' h).symm x = g ↑x
-- StarAlgEquiv.ofInjective'_apply
@[simp]
theorem ofInjective'_apply (f : F) (hf : Function.Injective f) (x : A) :
    ofInjective' f hf x = f x 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [Star A] [NonUnitalSemiring B] [Module R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F) (hf : Function.Injective ⇑f) (x : A)
Goal: ↑((StarAlgEquiv.ofInjective' f hf) x) = f x
-- NonUnitalSubalgebra.mem_star_iff
@[simp]
theorem mem_star_iff (S : NonUnitalSubalgebra R A) (x : A) : x ∈ star S ↔ star x ∈ S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [StarModule R A] (S : NonUnitalSubalgebra R A) (x : A)
Goal: x ∈ star S ↔ star x ∈ S
-- NonUnitalSubalgebra.star_mem_star_iff
theorem star_mem_star_iff (S : NonUnitalSubalgebra R A) (x : A) : star x ∈ star S ↔ x ∈ S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [StarModule R A] (S : NonUnitalSubalgebra R A) (x : A)
Goal: star x ∈ star S ↔ x ∈ S
-- NonUnitalSubalgebra.coe_star
@[simp]
theorem coe_star (S : NonUnitalSubalgebra R A) : star S = star (S : Set A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [StarModule R A] (S : NonUnitalSubalgebra R A)
Goal: ↑(star S) = star ↑S
-- NonUnitalSubalgebra.star_mono
theorem star_mono : Monotone (star : NonUnitalSubalgebra R A → NonUnitalSubalgebra R A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A]
Goal: Monotone star
-- NonUnitalSubalgebra.star_adjoin_comm
theorem star_adjoin_comm (s : Set A) :
    star (NonUnitalAlgebra.adjoin R s) = NonUnitalAlgebra.adjoin R (star s) 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (s : Set A)
Goal: star (NonUnitalAlgebra.adjoin R s) = NonUnitalAlgebra.adjoin R (star s)
-- NonUnitalSubalgebra.starClosure_carrier
@[simps!]
def starClosure (S : NonUnitalSubalgebra R A) : NonUnitalStarSubalgebra R A 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (S : NonUnitalSubalgebra R A)
Goal: ↑S.starClosure = ⋂ s, ⋂ (_ : ↑(NonUnitalSubsemiring.closure (↑S ∪ star ↑S)) ⊆ ↑s), ↑s
-- NonUnitalSubalgebra.starClosure_le
theorem starClosure_le {S₁ : NonUnitalSubalgebra R A} {S₂ : NonUnitalStarSubalgebra R A}
    (h : S₁ ≤ S₂.toNonUnitalSubalgebra) : S₁.starClosure ≤ S₂ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S₁ : NonUnitalSubalgebra R A} {S₂ : NonUnitalStarSubalgebra R A} (h : S₁ ≤ S₂.toNonUnitalSubalgebra)
Goal: S₁.starClosure ≤ S₂
-- NonUnitalSubalgebra.starClosure_le_iff
theorem starClosure_le_iff {S₁ : NonUnitalSubalgebra R A} {S₂ : NonUnitalStarSubalgebra R A} :
    S₁.starClosure ≤ S₂ ↔ S₁ ≤ S₂.toNonUnitalSubalgebra 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S₁ : NonUnitalSubalgebra R A} {S₂ : NonUnitalStarSubalgebra R A}
Goal: S₁.starClosure ≤ S₂ ↔ S₁ ≤ S₂.toNonUnitalSubalgebra
-- NonUnitalSubalgebra.starClosure_toNonunitalSubalgebra
@[simp]
theorem starClosure_toNonunitalSubalgebra {S : NonUnitalSubalgebra R A} :
    S.starClosure.toNonUnitalSubalgebra = S ⊔ star S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S : NonUnitalSubalgebra R A}
Goal: S.starClosure.toNonUnitalSubalgebra = S ⊔ star S
-- NonUnitalSubalgebra.starClosure_mono
@[mono]
theorem starClosure_mono : Monotone (starClosure (R := R) (A := A)) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A]
Goal: Monotone NonUnitalSubalgebra.starClosure
-- NonUnitalStarAlgebra.adjoin_eq_starClosure_adjoin
theorem adjoin_eq_starClosure_adjoin (s : Set A) :
    adjoin R s = (NonUnitalAlgebra.adjoin R s).starClosure 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (s : Set A)
Goal: NonUnitalStarAlgebra.adjoin R s = (NonUnitalAlgebra.adjoin R s).starClosure
-- NonUnitalStarAlgebra.adjoin_toNonUnitalSubalgebra
theorem adjoin_toNonUnitalSubalgebra (s : Set A) :
    (adjoin R s).toNonUnitalSubalgebra = NonUnitalAlgebra.adjoin R (s ∪ star s) 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (s : Set A)
Goal: (NonUnitalStarAlgebra.adjoin R s).toNonUnitalSubalgebra = NonUnitalAlgebra.adjoin R (s ∪ star s)
-- NonUnitalStarAlgebra.subset_adjoin
@[aesop safe 20 apply (rule_sets := [SetLike])]
theorem subset_adjoin (s : Set A) : s ⊆ adjoin R s 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (s : Set A)
Goal: s ⊆ ↑(NonUnitalStarAlgebra.adjoin R s)
-- NonUnitalStarAlgebra.star_subset_adjoin
theorem star_subset_adjoin (s : Set A) : star s ⊆ adjoin R s 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (s : Set A)
Goal: star s ⊆ ↑(NonUnitalStarAlgebra.adjoin R s)
-- NonUnitalStarAlgebra.self_mem_adjoin_singleton
theorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (x : A)
Goal: x ∈ NonUnitalStarAlgebra.adjoin R {x}
-- NonUnitalStarAlgebra.star_self_mem_adjoin_singleton
theorem star_self_mem_adjoin_singleton (x : A) : star x ∈ adjoin R ({x} : Set A) 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (x : A)
Goal: star x ∈ NonUnitalStarAlgebra.adjoin R {x}
-- NonUnitalStarAlgebra.adjoin_induction'
@[elab_as_elim]
lemma adjoin_induction' {s : Set A} {p : ∀ x, x ∈ adjoin R s → Prop} {a : A}
    (ha : a ∈ adjoin R s) (mem : ∀ (x : A) (hx : x ∈ s), p x (subset_adjoin R s hx))
    (add : ∀ x hx y hy, p x hx → p y hy → p (x + y) (add_mem hx hy))
    (zero : p 0 (zero_mem _)) (mul : ∀ x hx y hy, p x hx → p y hy → p (x * y) (mul_mem hx hy))
    (smul : ∀ (r : R) x hx, p x hx → p (r • x) (SMulMemClass.smul_mem r hx))
    (star : ∀ x hx, p x hx → p (star x) (star_mem hx)) : p a ha 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {s : Set A} {p : (x : A) → x ∈ NonUnitalStarAlgebra.adjoin R s → Prop} {a : A} (ha : a ∈ NonUnitalStarAlgebra.adjoin R s) (mem : ∀ (x : A) (hx : x ∈ s), p x ⋯) (add :
  ∀ (x : A) (hx : x ∈ NonUnitalStarAlgebra.adjoin R s) (y : A) (hy : y ∈ NonUnitalStarAlgebra.adjoin R s),
    p x hx → p y hy → p (x + y) ⋯) (zero : p 0 ⋯) (mul :
  ∀ (x : A) (hx : x ∈ NonUnitalStarAlgebra.adjoin R s) (y : A) (hy : y ∈ NonUnitalStarAlgebra.adjoin R s),
    p x hx → p y hy → p (x * y) ⋯) (smul : ∀ (r : R) (x : A) (hx : x ∈ NonUnitalStarAlgebra.adjoin R s), p x hx → p (r • x) ⋯) (star : ∀ (x : A) (hx : x ∈ NonUnitalStarAlgebra.adjoin R s), p x hx → p (star x) ⋯)
Goal: p a ha
-- NonUnitalStarAlgebra.gc
protected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalStarSubalgebra R A) (↑) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A]
Goal: GaloisConnection (NonUnitalStarAlgebra.adjoin R) SetLike.coe
-- NonUnitalStarAlgebra.adjoin_le
theorem adjoin_le {S : NonUnitalStarSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S : NonUnitalStarSubalgebra R A} {s : Set A} (hs : s ⊆ ↑S)
Goal: NonUnitalStarAlgebra.adjoin R s ≤ S
-- NonUnitalStarAlgebra.adjoin_le_iff
theorem adjoin_le_iff {S : NonUnitalStarSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S : NonUnitalStarSubalgebra R A} {s : Set A}
Goal: NonUnitalStarAlgebra.adjoin R s ≤ S ↔ s ⊆ ↑S
-- NonUnitalStarAlgebra.adjoin_eq
lemma adjoin_eq (s : NonUnitalStarSubalgebra R A) : adjoin R (s : Set A) = s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (s : NonUnitalStarSubalgebra R A)
Goal: NonUnitalStarAlgebra.adjoin R ↑s = s
-- NonUnitalStarAlgebra.adjoin_eq_span
lemma adjoin_eq_span (s : Set A) :
    (adjoin R s).toSubmodule = Submodule.span R (Subsemigroup.closure (s ∪ star s)) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (s : Set A)
Goal: (NonUnitalStarAlgebra.adjoin R s).toSubmodule = Submodule.span R ↑(Subsemigroup.closure (s ∪ star s))
-- NonUnitalStarAlgebra.span_eq_toSubmodule
@[simp]
lemma span_eq_toSubmodule (s : NonUnitalStarSubalgebra R A) :
    Submodule.span R (s : Set A) = s.toSubmodule 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] (s : NonUnitalStarSubalgebra R A)
Goal: Submodule.span R ↑s = s.toSubmodule