ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Subalgebra.coe_star
@[simp]
theorem coe_star (S : Subalgebra R A) : ((star S : Subalgebra R A) : Set A) = star (S : Set A) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S : Subalgebra R A)
Goal: ↑(star S) = star ↑S
-- Subalgebra.star_mono
theorem star_mono : Monotone (star : Subalgebra R A → Subalgebra R A) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A]
Goal: Monotone star
-- Subalgebra.star_adjoin_comm
theorem star_adjoin_comm (s : Set A) : star (Algebra.adjoin R s) = Algebra.adjoin R (star s) 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (s : Set A)
Goal: star (Algebra.adjoin R s) = Algebra.adjoin R (star s)
-- Subalgebra.starClosure_carrier
@[simps!]
def starClosure (S : Subalgebra R A) : StarSubalgebra R A 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S : Subalgebra R A)
Goal: ↑S.starClosure = ⋂ t, ⋂ (_ : Set.range ⇑(algebraMap R A) ⊆ ↑t ∧ ↑S ⊆ ↑t ∧ star ↑S ⊆ ↑t), ↑t
-- Subalgebra.starClosure_toSubalgebra
theorem starClosure_toSubalgebra (S : Subalgebra R A) : S.starClosure.toSubalgebra = S ⊔ star S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S : Subalgebra R A)
Goal: S.starClosure.toSubalgebra = S ⊔ star S
-- Subalgebra.starClosure_le
theorem starClosure_le {S₁ : Subalgebra R A} {S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂.toSubalgebra) :
    S₁.starClosure ≤ S₂ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S₁ : Subalgebra R A} {S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂.toSubalgebra)
Goal: S₁.starClosure ≤ S₂
-- Subalgebra.starClosure_le_iff
theorem starClosure_le_iff {S₁ : Subalgebra R A} {S₂ : StarSubalgebra R A} :
    S₁.starClosure ≤ S₂ ↔ S₁ ≤ S₂.toSubalgebra 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S₁ : Subalgebra R A} {S₂ : StarSubalgebra R A}
Goal: S₁.starClosure ≤ S₂ ↔ S₁ ≤ S₂.toSubalgebra
-- StarAlgebra.adjoin_carrier
@[simps!]
def adjoin (s : Set A) : StarSubalgebra R A 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (s : Set A)
Goal: ↑(StarAlgebra.adjoin R s) = ⋂ t, ⋂ (_ : Set.range ⇑(algebraMap R A) ⊆ ↑t ∧ s ⊆ ↑t ∧ star s ⊆ ↑t), ↑t
-- StarAlgebra.adjoin_eq_starClosure_adjoin
theorem adjoin_eq_starClosure_adjoin (s : Set A) : adjoin R s = (Algebra.adjoin R s).starClosure 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (s : Set A)
Goal: StarAlgebra.adjoin R s = (Algebra.adjoin R s).starClosure
-- StarAlgebra.adjoin_toSubalgebra
theorem adjoin_toSubalgebra (s : Set A) :
    (adjoin R s).toSubalgebra = Algebra.adjoin R (s ∪ star s) 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (s : Set A)
Goal: (StarAlgebra.adjoin R s).toSubalgebra = Algebra.adjoin R (s ∪ star s)
-- StarAlgebra.subset_adjoin
@[aesop safe 20 apply (rule_sets := [SetLike])]
theorem subset_adjoin (s : Set A) : s ⊆ adjoin R s 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (s : Set A)
Goal: s ⊆ ↑(StarAlgebra.adjoin R s)
-- StarAlgebra.star_subset_adjoin
theorem star_subset_adjoin (s : Set A) : star s ⊆ adjoin R s 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (s : Set A)
Goal: star s ⊆ ↑(StarAlgebra.adjoin R s)
-- StarAlgebra.self_mem_adjoin_singleton
theorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (x : A)
Goal: x ∈ StarAlgebra.adjoin R {x}
-- StarAlgebra.star_self_mem_adjoin_singleton
theorem star_self_mem_adjoin_singleton (x : A) : star x ∈ adjoin R ({x} : Set A) 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (x : A)
Goal: star x ∈ StarAlgebra.adjoin R {x}
-- StarAlgebra.gc
protected theorem gc : GaloisConnection (adjoin R : Set A → StarSubalgebra R A) (↑) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A]
Goal: GaloisConnection (StarAlgebra.adjoin R) SetLike.coe
-- StarAlgebra.adjoin_le
theorem adjoin_le {S : StarSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S : StarSubalgebra R A} {s : Set A} (hs : s ⊆ ↑S)
Goal: StarAlgebra.adjoin R s ≤ S
-- StarAlgebra.adjoin_le_iff
theorem adjoin_le_iff {S : StarSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S : StarSubalgebra R A} {s : Set A}
Goal: StarAlgebra.adjoin R s ≤ S ↔ s ⊆ ↑S
-- StarAlgebra.adjoin_eq
lemma adjoin_eq (S : StarSubalgebra R A) : adjoin R (S : Set A) = S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S : StarSubalgebra R A)
Goal: StarAlgebra.adjoin R ↑S = S
-- StarAlgebra.adjoin_eq_span
lemma adjoin_eq_span (s : Set A) :
    Subalgebra.toSubmodule (adjoin R s).toSubalgebra = span R (Submonoid.closure (s ∪ star s)) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (s : Set A)
Goal: Subalgebra.toSubmodule (StarAlgebra.adjoin R s).toSubalgebra = Submodule.span R ↑(Submonoid.closure (s ∪ star s))
-- Subalgebra.starClosure_eq_adjoin
theorem _root_.Subalgebra.starClosure_eq_adjoin (S : Subalgebra R A) :
    S.starClosure = adjoin R (S : Set A) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S : Subalgebra R A)
Goal: S.starClosure = StarAlgebra.adjoin R ↑S
-- StarAlgebra.adjoin_induction
@[elab_as_elim]
theorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)
    (mem : ∀ x : A, x ∈ s → p x) (algebraMap : ∀ r : R, p (algebraMap R A r))
    (add : ∀ x y : A, p x → p y → p (x + y)) (mul : ∀ x y : A, p x → p y → p (x * y))
    (star : ∀ x : A, p x → p (star x)) : p a 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {s : Set A} {p : A → Prop} {a : A} (h : a ∈ StarAlgebra.adjoin R s) (mem : ∀ x ∈ s, p x) (algebraMap : ∀ (r : R), p ((algebraMap R A) r)) (add : ∀ (x y : A), p x → p y → p (x + y)) (mul : ∀ (x y : A), p x → p y → p (x * y)) (star : ∀ (x : A), p x → p (star x))
Goal: p a
-- StarAlgebra.adjoin_induction₂
@[elab_as_elim]
theorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)
    (hb : b ∈ adjoin R s) (Hs : ∀ x : A, x ∈ s → ∀ y : A, y ∈ s → p x y)
    (Halg : ∀ r₁ r₂ : R, p (algebraMap R A r₁) (algebraMap R A r₂))
    (Halg_left : ∀ (r : R) (x : A), x ∈ s → p (algebraMap R A r) x)
    (Halg_right : ∀ (r : R) (x : A), x ∈ s → p x (algebraMap R A r))
    (Hadd_left : ∀ x₁ x₂ y : A, p x₁ y → p x₂ y → p (x₁ + x₂) y)
    (Hadd_right : ∀ x y₁ y₂ : A, p x y₁ → p x y₂ → p x (y₁ + y₂))
    (Hmul_left : ∀ x₁ x₂ y : A, p x₁ y → p x₂ y → p (x₁ * x₂) y)
    (Hmul_right : ∀ x y₁ y₂ : A, p x y₁ → p x y₂ → p x (y₁ * y₂))
    (Hstar : ∀ x y : A, p x y → p (star x) (star y)) (Hstar_left : ∀ x y : A, p x y → p (star x) y)
    (Hstar_right : ∀ x y : A, p x y → p x (star y)) : p a b 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ StarAlgebra.adjoin R s) (hb : b ∈ StarAlgebra.adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (Halg : ∀ (r₁ r₂ : R), p ((algebraMap R A) r₁) ((algebraMap R A) r₂)) (Halg_left : ∀ (r : R), ∀ x ∈ s, p ((algebraMap R A) r) x) (Halg_right : ∀ (r : R), ∀ x ∈ s, p x ((algebraMap R A) r)) (Hadd_left : ∀ (x₁ x₂ y : A), p x₁ y → p x₂ y → p (x₁ + x₂) y) (Hadd_right : ∀ (x y₁ y₂ : A), p x y₁ → p x y₂ → p x (y₁ + y₂)) (Hmul_left : ∀ (x₁ x₂ y : A), p x₁ y → p x₂ y → p (x₁ * x₂) y) (Hmul_right : ∀ (x y₁ y₂ : A), p x y₁ → p x y₂ → p x (y₁ * y₂)) (Hstar : ∀ (x y : A), p x y → p (star x) (star y)) (Hstar_left : ∀ (x y : A), p x y → p (star x) y) (Hstar_right : ∀ (x y : A), p x y → p x (star y))
Goal: p a b
-- StarAlgebra.adjoin_induction'
@[elab_as_elim]
theorem adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)
    (mem : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin R s h⟩) (algebraMap : ∀ r, p (algebraMap R _ r))
    (add : ∀ x y, p x → p y → p (x + y)) (mul : ∀ x y, p x → p y → p (x * y))
    (star : ∀ x, p x → p (star x)) : p a 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {s : Set A} {p : ↥(StarAlgebra.adjoin R s) → Prop} (a : ↥(StarAlgebra.adjoin R s)) (mem : ∀ (x : A) (h : x ∈ s), p ⟨x, ⋯⟩) (algebraMap : ∀ (r : R), p ((algebraMap R ↥(StarAlgebra.adjoin R s)) r)) (add : ∀ (x y : ↥(StarAlgebra.adjoin R s)), p x → p y → p (x + y)) (mul : ∀ (x y : ↥(StarAlgebra.adjoin R s)), p x → p y → p (x * y)) (star : ∀ (x : ↥(StarAlgebra.adjoin R s)), p x → p (star x))
Goal: p a
-- StarSubalgebra.coe_top
@[simp]
theorem coe_top : (↑(⊤ : StarSubalgebra R A) : Set A) = Set.univ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A]
Goal: ↑⊤ = Set.univ
-- StarSubalgebra.mem_top
@[simp]
theorem mem_top {x : A} : x ∈ (⊤ : StarSubalgebra R A) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {x : A}
Goal: x ∈ ⊤
-- StarSubalgebra.top_toSubalgebra
@[simp]
theorem top_toSubalgebra : (⊤ : StarSubalgebra R A).toSubalgebra = ⊤ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A]
Goal: ⊤.toSubalgebra = ⊤
-- StarSubalgebra.toSubalgebra_eq_top
@[simp]
theorem toSubalgebra_eq_top {S : StarSubalgebra R A} : S.toSubalgebra = ⊤ ↔ S = ⊤ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S : StarSubalgebra R A}
Goal: S.toSubalgebra = ⊤ ↔ S = ⊤
-- StarSubalgebra.mem_sup_left
theorem mem_sup_left {S T : StarSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S T : StarSubalgebra R A} {x : A}
Goal: x ∈ S → x ∈ S ⊔ T
-- StarSubalgebra.mem_sup_right
theorem mem_sup_right {S T : StarSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S T : StarSubalgebra R A} {x : A}
Goal: x ∈ T → x ∈ S ⊔ T
-- StarSubalgebra.mul_mem_sup
theorem mul_mem_sup {S T : StarSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :
    x * y ∈ S ⊔ T 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S T : StarSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T)
Goal: x * y ∈ S ⊔ T
-- StarSubalgebra.map_sup
theorem map_sup (f : A →⋆ₐ[R] B) (S T : StarSubalgebra R A) : map f (S ⊔ T) = map f S ⊔ map f T 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] (f : A →⋆ₐ[R] B) (S T : StarSubalgebra R A)
Goal: StarSubalgebra.map f (S ⊔ T) = StarSubalgebra.map f S ⊔ StarSubalgebra.map f T
-- StarSubalgebra.coe_inf
@[simp, norm_cast]
theorem coe_inf (S T : StarSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S T : StarSubalgebra R A)
Goal: ↑(S ⊓ T) = ↑S ∩ ↑T
-- StarSubalgebra.mem_inf
@[simp]
theorem mem_inf {S T : StarSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S T : StarSubalgebra R A} {x : A}
Goal: x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T
-- StarSubalgebra.inf_toSubalgebra
@[simp]
theorem inf_toSubalgebra (S T : StarSubalgebra R A) :
    (S ⊓ T).toSubalgebra = S.toSubalgebra ⊓ T.toSubalgebra 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S T : StarSubalgebra R A)
Goal: (S ⊓ T).toSubalgebra = S.toSubalgebra ⊓ T.toSubalgebra
-- StarSubalgebra.coe_sInf
@[simp, norm_cast]
theorem coe_sInf (S : Set (StarSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S : Set (StarSubalgebra R A))
Goal: ↑(sInf S) = ⋂ s ∈ S, ↑s
-- StarSubalgebra.mem_sInf
theorem mem_sInf {S : Set (StarSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S : Set (StarSubalgebra R A)} {x : A}
Goal: x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p
-- StarSubalgebra.sInf_toSubalgebra
@[simp]
theorem sInf_toSubalgebra (S : Set (StarSubalgebra R A)) :
    (sInf S).toSubalgebra = sInf (StarSubalgebra.toSubalgebra '' S) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S : Set (StarSubalgebra R A))
Goal: (sInf S).toSubalgebra = sInf (StarSubalgebra.toSubalgebra '' S)
-- StarSubalgebra.coe_iInf
@[simp, norm_cast]
theorem coe_iInf {ι : Sort*} {S : ι → StarSubalgebra R A} : (↑(⨅ i, S i) : Set A) = ⋂ i, S i 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {ι : Sort u_5} {S : ι → StarSubalgebra R A}
Goal: ↑(⨅ i, S i) = ⋂ i, ↑(S i)
-- StarSubalgebra.mem_iInf
theorem mem_iInf {ι : Sort*} {S : ι → StarSubalgebra R A} {x : A} :
    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {ι : Sort u_5} {S : ι → StarSubalgebra R A} {x : A}
Goal: x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i
-- StarSubalgebra.iInf_toSubalgebra
@[simp]
theorem iInf_toSubalgebra {ι : Sort*} (S : ι → StarSubalgebra R A) :
    (⨅ i, S i).toSubalgebra = ⨅ i, (S i).toSubalgebra 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {ι : Sort u_5} (S : ι → StarSubalgebra R A)
Goal: (⨅ i, S i).toSubalgebra = ⨅ i, (S i).toSubalgebra
-- StarSubalgebra.bot_toSubalgebra
theorem bot_toSubalgebra : (⊥ : StarSubalgebra R A).toSubalgebra = ⊥ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A]
Goal: ⊥.toSubalgebra = ⊥
-- StarSubalgebra.mem_bot
theorem mem_bot {x : A} : x ∈ (⊥ : StarSubalgebra R A) ↔ x ∈ Set.range (algebraMap R A) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {x : A}
Goal: x ∈ ⊥ ↔ x ∈ Set.range ⇑(algebraMap R A)
-- StarSubalgebra.coe_bot
@[simp]
theorem coe_bot : ((⊥ : StarSubalgebra R A) : Set A) = Set.range (algebraMap R A) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A]
Goal: ↑⊥ = Set.range ⇑(algebraMap R A)
-- StarSubalgebra.eq_top_iff
theorem eq_top_iff {S : StarSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S : StarSubalgebra R A}
Goal: S = ⊤ ↔ ∀ (x : A), x ∈ S
-- StarAlgHom.mem_equalizer
@[simp]
theorem mem_equalizer (x : A) : x ∈ StarAlgHom.equalizer f g ↔ f x = g x 
Arguments: {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [Semiring B] [Algebra R B] [StarRing B] [FunLike F A B] [AlgHomClass F R A B] [StarAlgHomClass F R A B] (f g : F) (x : A)
Goal: x ∈ StarAlgHom.equalizer f g ↔ f x = g x
-- StarAlgHom.adjoin_le_equalizer
theorem adjoin_le_equalizer {s : Set A} (h : s.EqOn f g) : adjoin R s ≤ StarAlgHom.equalizer f g 
Arguments: {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [Semiring B] [Algebra R B] [StarRing B] [FunLike F A B] [AlgHomClass F R A B] [StarAlgHomClass F R A B] (f g : F) {s : Set A} (h : Set.EqOn (⇑f) (⇑g) s)
Goal: StarAlgebra.adjoin R s ≤ StarAlgHom.equalizer f g
-- StarAlgHom.ext_of_adjoin_eq_top
theorem ext_of_adjoin_eq_top {s : Set A} (h : adjoin R s = ⊤) ⦃f g : F⦄ (hs : s.EqOn f g) : f = g 
Arguments: {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [Semiring B] [Algebra R B] [StarRing B] [FunLike F A B] [AlgHomClass F R A B] [StarAlgHomClass F R A B] {s : Set A} (h : StarAlgebra.adjoin R s = ⊤) ⦃f g : F⦄ (hs : Set.EqOn (⇑f) (⇑g) s)
Goal: f = g
-- StarAlgHom.map_adjoin
theorem map_adjoin (f : A →⋆ₐ[R] B) (s : Set A) :
    map f (adjoin R s) = adjoin R (f '' s) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] (f : A →⋆ₐ[R] B) (s : Set A)
Goal: StarSubalgebra.map f (StarAlgebra.adjoin R s) = StarAlgebra.adjoin R (⇑f '' s)
-- StarAlgHom.ext_adjoin
theorem ext_adjoin {s : Set A} [FunLike F (adjoin R s) B]
    [AlgHomClass F R (adjoin R s) B] [StarAlgHomClass F R (adjoin R s) B] {f g : F}
    (h : ∀ x : adjoin R s, (x : A) ∈ s → f x = g x) : f = g 
Arguments: {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [Semiring B] [Algebra R B] [StarRing B] {s : Set A} [FunLike F (↥(StarAlgebra.adjoin R s)) B] [AlgHomClass F R (↥(StarAlgebra.adjoin R s)) B] [StarAlgHomClass F R (↥(StarAlgebra.adjoin R s)) B] {f g : F} (h : ∀ (x : ↥(StarAlgebra.adjoin R s)), ↑x ∈ s → f x = g x)
Goal: f = g
-- StarAlgHom.ext_adjoin_singleton
theorem ext_adjoin_singleton {a : A} [FunLike F (adjoin R ({a} : Set A)) B]
    [AlgHomClass F R (adjoin R ({a} : Set A)) B] [StarAlgHomClass F R (adjoin R ({a} : Set A)) B]
    {f g : F} (h : f ⟨a, self_mem_adjoin_singleton R a⟩ = g ⟨a, self_mem_adjoin_singleton R a⟩) :
    f = g 
Arguments: {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [Semiring B] [Algebra R B] [StarRing B] {a : A} [FunLike F (↥(StarAlgebra.adjoin R {a})) B] [AlgHomClass F R (↥(StarAlgebra.adjoin R {a})) B] [StarAlgHomClass F R (↥(StarAlgebra.adjoin R {a})) B] {f g : F} (h : f ⟨a, ⋯⟩ = g ⟨a, ⋯⟩)
Goal: f = g
-- StarAlgHom.range_eq_map_top
theorem range_eq_map_top (φ : A →⋆ₐ[R] B) : φ.range = (⊤ : StarSubalgebra R A).map φ 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] (φ : A →⋆ₐ[R] B)
Goal: φ.range = StarSubalgebra.map φ ⊤
-- StarAlgHom.coe_codRestrict
@[simp]
theorem coe_codRestrict (f : A →⋆ₐ[R] B) (S : StarSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :
    ↑(f.codRestrict S hf x) = f x 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] (f : A →⋆ₐ[R] B) (S : StarSubalgebra R B) (hf : ∀ (x : A), f x ∈ S) (x : A)
Goal: ↑((f.codRestrict S hf) x) = f x
-- StarAlgHom.subtype_comp_codRestrict
@[simp]
theorem subtype_comp_codRestrict (f : A →⋆ₐ[R] B) (S : StarSubalgebra R B)
    (hf : ∀ x : A, f x ∈ S) : S.subtype.comp (f.codRestrict S hf) = f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] (f : A →⋆ₐ[R] B) (S : StarSubalgebra R B) (hf : ∀ (x : A), f x ∈ S)
Goal: S.subtype.comp (f.codRestrict S hf) = f
-- StarAlgHom.injective_codRestrict
theorem injective_codRestrict (f : A →⋆ₐ[R] B) (S : StarSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :
    Function.Injective (StarAlgHom.codRestrict f S hf) ↔ Function.Injective f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] (f : A →⋆ₐ[R] B) (S : StarSubalgebra R B) (hf : ∀ (x : A), f x ∈ S)
Goal: Function.Injective ⇑(f.codRestrict S hf) ↔ Function.Injective ⇑f
-- StarAlgEquiv.ofInjective_apply
@[simps]
noncomputable def _root_.StarAlgEquiv.ofInjective (f : A →⋆ₐ[R] B)
    (hf : Function.Injective f) : A ≃⋆ₐ[R] f.range 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] (f : A →⋆ₐ[R] B) (hf : Function.Injective ⇑f) (a : A)
Goal: (StarAlgEquiv.ofInjective f hf) a = f.rangeRestrict a
-- StarAlgEquiv.ofInjective_symm_apply
@[simps]
noncomputable def _root_.StarAlgEquiv.ofInjective (f : A →⋆ₐ[R] B)
    (hf : Function.Injective f) : A ≃⋆ₐ[R] f.range 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] (f : A →⋆ₐ[R] B) (hf : Function.Injective ⇑f)
Goal: ∀ (a : ↥(↑f).range), (StarAlgEquiv.ofInjective f hf).symm a = (AlgEquiv.ofInjective (↑f) hf).invFun a
-- StarAlgHom.restrictScalars_apply
@[simps!]
def StarAlgHom.restrictScalars (f : A →⋆ₐ[S] B) : A →⋆ₐ[R] B 
Arguments: (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] [Star A] [Star B] (f : A →⋆ₐ[S] B)
Goal: ∀ (a : A), (StarAlgHom.restrictScalars R f) a = f a
-- StarAlgHom.restrictScalars_injective
theorem StarAlgHom.restrictScalars_injective :
    Function.Injective (StarAlgHom.restrictScalars R : (A →⋆ₐ[S] B) → A →⋆ₐ[R] B) 
Arguments: (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] [Star A] [Star B]
Goal: Function.Injective (StarAlgHom.restrictScalars R)
-- StarAlgEquiv.restrictScalars_apply
@[simps]
def StarAlgEquiv.restrictScalars (f : A ≃⋆ₐ[S] B) : A ≃⋆ₐ[R] B 
Arguments: (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] [Star A] [Star B] (f : A ≃⋆ₐ[S] B) (a : A)
Goal: (StarAlgEquiv.restrictScalars R f) a = f a
-- StarAlgEquiv.restrictScalars_symm_apply
@[simps]
def StarAlgEquiv.restrictScalars (f : A ≃⋆ₐ[S] B) : A ≃⋆ₐ[R] B 
Arguments: (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] [Star A] [Star B] (f : A ≃⋆ₐ[S] B)
Goal: ∀ (a : B), (StarAlgEquiv.restrictScalars R f).symm a = f.invFun a