ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- tsum_union_disjoint
@[to_additive]
theorem tprod_union_disjoint {s t : Set β} (hd : Disjoint s t) (hs : Multipliable (f ∘ (↑) : s → α))
    (ht : Multipliable (f ∘ (↑) : t → α)) :
    ∏' x : ↑(s ∪ t), f x = (∏' x : s, f x) * ∏' x : t, f x 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [T2Space α] [ContinuousAdd α] {s t : Set β} (hd : Disjoint s t) (hs : Summable (f ∘ Subtype.val)) (ht : Summable (f ∘ Subtype.val))
Goal: ∑' (x : ↑(s ∪ t)), f ↑x = ∑' (x : ↑s), f ↑x + ∑' (x : ↑t), f ↑x
-- tprod_finset_bUnion_disjoint
@[to_additive]
theorem tprod_finset_bUnion_disjoint {ι} {s : Finset ι} {t : ι → Set β}
    (hd : (s : Set ι).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, Multipliable (f ∘ (↑) : t i → α)) :
    ∏' x : ⋃ i ∈ s, t i, f x = ∏ i ∈ s, ∏' x : t i, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} [T2Space α] [ContinuousMul α] {ι : Type u_5} {s : Finset ι} {t : ι → Set β} (hd : (↑s).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, Multipliable (f ∘ Subtype.val))
Goal: ∏' (x : ↑(⋃ i ∈ s, t i)), f ↑x = ∏ i ∈ s, ∏' (x : ↑(t i)), f ↑x
-- tsum_finset_bUnion_disjoint
@[to_additive]
theorem tprod_finset_bUnion_disjoint {ι} {s : Finset ι} {t : ι → Set β}
    (hd : (s : Set ι).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, Multipliable (f ∘ (↑) : t i → α)) :
    ∏' x : ⋃ i ∈ s, t i, f x = ∏ i ∈ s, ∏' x : t i, f x 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [T2Space α] [ContinuousAdd α] {ι : Type u_5} {s : Finset ι} {t : ι → Set β} (hd : (↑s).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, Summable (f ∘ Subtype.val))
Goal: ∑' (x : ↑(⋃ i ∈ s, t i)), f ↑x = ∑ i ∈ s, ∑' (x : ↑(t i)), f ↑x
-- UniformFun.continuousSMul_induced_of_range_bounded
lemma UniformFun.continuousSMul_induced_of_range_bounded (φ : hom)
    (hφ : Inducing (ofFun ∘ φ)) (h : ∀ u : H, Bornology.IsVonNBounded 𝕜 (Set.range (φ u))) :
    ContinuousSMul 𝕜 H 
Arguments: (𝕜 : Type u_1) (α : Type u_2) (E : Type u_3) (H : Type u_4) {hom : Type u_5} [NormedField 𝕜] [AddCommGroup H] [Module 𝕜 H] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace H] [UniformSpace E] [UniformAddGroup E] [ContinuousSMul 𝕜 E] [FunLike hom H (α → E)] [LinearMapClass hom 𝕜 H (α → E)] (φ : hom) (hφ : Inducing (⇑UniformFun.ofFun ∘ ⇑φ)) (h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u)))
Goal: ContinuousSMul 𝕜 H
-- UniformOnFun.continuousSMul_induced_of_image_bounded
theorem UniformOnFun.continuousSMul_induced_of_image_bounded (φ : hom) (hφ : Inducing (ofFun 𝔖 ∘ φ))
    (h : ∀ u : H, ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 ((φ u : α → E) '' s)) :
    ContinuousSMul 𝕜 H 
Arguments: (𝕜 : Type u_1) (α : Type u_2) (E : Type u_3) (H : Type u_4) {hom : Type u_5} [NormedField 𝕜] [AddCommGroup H] [Module 𝕜 H] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace H] [UniformSpace E] [UniformAddGroup E] [ContinuousSMul 𝕜 E] {𝔖 : Set (Set α)} [FunLike hom H (α → E)] [LinearMapClass hom 𝕜 H (α → E)] (φ : hom) (hφ : Inducing (⇑(UniformOnFun.ofFun 𝔖) ∘ ⇑φ)) (h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s))
Goal: ContinuousSMul 𝕜 H
-- UniformOnFun.continuousSMul_submodule_of_image_bounded
theorem UniformOnFun.continuousSMul_submodule_of_image_bounded (H : Submodule 𝕜 (α →ᵤ[𝔖] E))
    (h : ∀ u ∈ H, ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (u '' s)) :
    @ContinuousSMul 𝕜 H _ _ ((UniformOnFun.topologicalSpace α E 𝔖).induced ((↑) : H → α →ᵤ[𝔖] E)) 
Arguments: (𝕜 : Type u_1) (α : Type u_2) (E : Type u_3) [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [UniformSpace E] [UniformAddGroup E] [ContinuousSMul 𝕜 E] {𝔖 : Set (Set α)} (H : Submodule 𝕜 (α →ᵤ[𝔖] E)) (h : ∀ u ∈ H, ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (u '' s))
Goal: ContinuousSMul 𝕜 ↥H
-- TopCat.Presheaf.SheafCondition.pairwiseToOpensLeCover_map
@[simps]
def pairwiseToOpensLeCover : Pairwise ι ⥤ OpensLeCover U 
Arguments: {X : TopCat} {ι : Type w} (U : ι → TopologicalSpace.Opens ↑X) {V W : CategoryTheory.Pairwise ι} (i : V ⟶ W)
Goal: (TopCat.Presheaf.SheafCondition.pairwiseToOpensLeCover U).map i =
  TopCat.Presheaf.SheafCondition.pairwiseToOpensLeCoverMap U i
-- TopCat.Presheaf.SheafCondition.pairwiseToOpensLeCover_obj
@[simps]
def pairwiseToOpensLeCover : Pairwise ι ⥤ OpensLeCover U 
Arguments: {X : TopCat} {ι : Type w} (U : ι → TopologicalSpace.Opens ↑X)
Goal: ∀ (a : CategoryTheory.Pairwise ι),
  (TopCat.Presheaf.SheafCondition.pairwiseToOpensLeCover U).obj a =
    TopCat.Presheaf.SheafCondition.pairwiseToOpensLeCoverObj U a
-- TopCat.Presheaf.isSheafOpensLeCover_iff_isSheafPairwiseIntersections
theorem isSheafOpensLeCover_iff_isSheafPairwiseIntersections :
    F.IsSheafOpensLeCover ↔ F.IsSheafPairwiseIntersections 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (F : TopCat.Presheaf C X)
Goal: F.IsSheafOpensLeCover ↔ F.IsSheafPairwiseIntersections
-- TopCat.Presheaf.isSheaf_iff_isSheafPairwiseIntersections
theorem isSheaf_iff_isSheafPairwiseIntersections : F.IsSheaf ↔ F.IsSheafPairwiseIntersections 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (F : TopCat.Presheaf C X)
Goal: F.IsSheaf ↔ F.IsSheafPairwiseIntersections
-- TopCat.Presheaf.isSheaf_iff_isSheafPreservesLimitPairwiseIntersections
theorem isSheaf_iff_isSheafPreservesLimitPairwiseIntersections :
    F.IsSheaf ↔ F.IsSheafPreservesLimitPairwiseIntersections 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (F : TopCat.Presheaf C X)
Goal: F.IsSheaf ↔ F.IsSheafPreservesLimitPairwiseIntersections
-- TopCat.Sheaf.interUnionPullbackCone_pt
@[simp]
theorem interUnionPullbackCone_pt : (interUnionPullbackCone F U V).pt = F.1.obj (op <| U ⊔ V) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (F : TopCat.Sheaf C X) (U V : TopologicalSpace.Opens ↑X)
Goal: (F.interUnionPullbackCone U V).pt = F.val.obj (Opposite.op (U ⊔ V))
-- TopCat.Sheaf.interUnionPullbackCone_fst
@[simp]
theorem interUnionPullbackCone_fst :
    (interUnionPullbackCone F U V).fst = F.1.map (homOfLE le_sup_left).op 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (F : TopCat.Sheaf C X) (U V : TopologicalSpace.Opens ↑X)
Goal: (F.interUnionPullbackCone U V).fst = F.val.map (CategoryTheory.homOfLE ⋯).op
-- TopCat.Sheaf.interUnionPullbackCone_snd
@[simp]
theorem interUnionPullbackCone_snd :
    (interUnionPullbackCone F U V).snd = F.1.map (homOfLE le_sup_right).op 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (F : TopCat.Sheaf C X) (U V : TopologicalSpace.Opens ↑X)
Goal: (F.interUnionPullbackCone U V).snd = F.val.map (CategoryTheory.homOfLE ⋯).op
-- TopCat.Sheaf.interUnionPullbackConeLift_left
theorem interUnionPullbackConeLift_left :
    interUnionPullbackConeLift F U V s ≫ F.1.map (homOfLE le_sup_left).op = s.fst 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (F : TopCat.Sheaf C X) (U V : TopologicalSpace.Opens ↑X) (s :
  CategoryTheory.Limits.PullbackCone (F.val.map (CategoryTheory.homOfLE ⋯).op)
    (F.val.map (CategoryTheory.homOfLE ⋯).op))
Goal: F.interUnionPullbackConeLift U V s ≫ F.val.map (CategoryTheory.homOfLE ⋯).op = s.fst
-- TopCat.Sheaf.interUnionPullbackConeLift_right
theorem interUnionPullbackConeLift_right :
    interUnionPullbackConeLift F U V s ≫ F.1.map (homOfLE le_sup_right).op = s.snd 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (F : TopCat.Sheaf C X) (U V : TopologicalSpace.Opens ↑X) (s :
  CategoryTheory.Limits.PullbackCone (F.val.map (CategoryTheory.homOfLE ⋯).op)
    (F.val.map (CategoryTheory.homOfLE ⋯).op))
Goal: F.interUnionPullbackConeLift U V s ≫ F.val.map (CategoryTheory.homOfLE ⋯).op = s.snd
-- TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst
theorem objSupIsoProdEqLocus_hom_fst {X : TopCat} (F : X.Sheaf CommRingCat) (U V : Opens X) (x) :
    ((F.objSupIsoProdEqLocus U V).hom x).1.fst = F.1.map (homOfLE le_sup_left).op x 
Arguments: {X : TopCat} (F : TopCat.Sheaf CommRingCat X) (U V : TopologicalSpace.Opens ↑X) (x : ↑(F.val.obj (Opposite.op (U ⊔ V))))
Goal: (↑((F.objSupIsoProdEqLocus U V).hom x)).1 = (F.val.map (CategoryTheory.homOfLE ⋯).op) x
-- TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd
theorem objSupIsoProdEqLocus_hom_snd {X : TopCat} (F : X.Sheaf CommRingCat) (U V : Opens X) (x) :
    ((F.objSupIsoProdEqLocus U V).hom x).1.snd = F.1.map (homOfLE le_sup_right).op x 
Arguments: {X : TopCat} (F : TopCat.Sheaf CommRingCat X) (U V : TopologicalSpace.Opens ↑X) (x : ↑(F.val.obj (Opposite.op (U ⊔ V))))
Goal: (↑((F.objSupIsoProdEqLocus U V).hom x)).2 = (F.val.map (CategoryTheory.homOfLE ⋯).op) x
-- TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst
theorem objSupIsoProdEqLocus_inv_fst {X : TopCat} (F : X.Sheaf CommRingCat) (U V : Opens X) (x) :
    F.1.map (homOfLE le_sup_left).op ((F.objSupIsoProdEqLocus U V).inv x) = x.1.1 
Arguments: {X : TopCat} (F : TopCat.Sheaf CommRingCat X) (U V : TopologicalSpace.Opens ↑X) (x :
  ↥((RingHom.comp (F.val.map (CategoryTheory.homOfLE ⋯).op)
          (RingHom.fst ↑(F.val.obj (Opposite.op U)) ↑(F.val.obj (Opposite.op V)))).eqLocus
      (RingHom.comp (F.val.map (CategoryTheory.homOfLE ⋯).op)
        (RingHom.snd ↑(F.val.obj (Opposite.op U)) ↑(F.val.obj (Opposite.op V))))))
Goal: (F.val.map (CategoryTheory.homOfLE ⋯).op) ((F.objSupIsoProdEqLocus U V).inv x) = (↑x).1
-- TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd
theorem objSupIsoProdEqLocus_inv_snd {X : TopCat} (F : X.Sheaf CommRingCat) (U V : Opens X) (x) :
    F.1.map (homOfLE le_sup_right).op ((F.objSupIsoProdEqLocus U V).inv x) = x.1.2 
Arguments: {X : TopCat} (F : TopCat.Sheaf CommRingCat X) (U V : TopologicalSpace.Opens ↑X) (x :
  ↥((RingHom.comp (F.val.map (CategoryTheory.homOfLE ⋯).op)
          (RingHom.fst ↑(F.val.obj (Opposite.op U)) ↑(F.val.obj (Opposite.op V)))).eqLocus
      (RingHom.comp (F.val.map (CategoryTheory.homOfLE ⋯).op)
        (RingHom.snd ↑(F.val.obj (Opposite.op U)) ↑(F.val.obj (Opposite.op V))))))
Goal: (F.val.map (CategoryTheory.homOfLE ⋯).op) ((F.objSupIsoProdEqLocus U V).inv x) = (↑x).2
-- StarSubalgebra.embedding_inclusion
theorem embedding_inclusion {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂) : Embedding (inclusion h) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂)
Goal: Embedding ⇑(StarSubalgebra.inclusion h)
-- StarSubalgebra.closedEmbedding_inclusion
theorem closedEmbedding_inclusion {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂)
    (hS₁ : IsClosed (S₁ : Set A)) : ClosedEmbedding (inclusion h) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂) (hS₁ : IsClosed ↑S₁)
Goal: ClosedEmbedding ⇑(StarSubalgebra.inclusion h)
-- StarSubalgebra.topologicalClosure_toSubalgebra_comm
theorem topologicalClosure_toSubalgebra_comm (s : StarSubalgebra R A) :
    s.topologicalClosure.toSubalgebra = s.toSubalgebra.topologicalClosure 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] (s : StarSubalgebra R A)
Goal: s.topologicalClosure.toSubalgebra = s.topologicalClosure
-- StarSubalgebra.topologicalClosure_coe
@[simp]
theorem topologicalClosure_coe (s : StarSubalgebra R A) :
    (s.topologicalClosure : Set A) = closure (s : Set A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] (s : StarSubalgebra R A)
Goal: ↑s.topologicalClosure = closure ↑s
-- StarSubalgebra.le_topologicalClosure
theorem le_topologicalClosure (s : StarSubalgebra R A) : s ≤ s.topologicalClosure 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] (s : StarSubalgebra R A)
Goal: s ≤ s.topologicalClosure
-- StarSubalgebra.isClosed_topologicalClosure
theorem isClosed_topologicalClosure (s : StarSubalgebra R A) :
    IsClosed (s.topologicalClosure : Set A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] (s : StarSubalgebra R A)
Goal: IsClosed ↑s.topologicalClosure
-- StarSubalgebra.topologicalClosure_minimal
theorem topologicalClosure_minimal {s t : StarSubalgebra R A} (h : s ≤ t)
    (ht : IsClosed (t : Set A)) : s.topologicalClosure ≤ t 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] {s t : StarSubalgebra R A} (h : s ≤ t) (ht : IsClosed ↑t)
Goal: s.topologicalClosure ≤ t
-- StarSubalgebra.topologicalClosure_mono
theorem topologicalClosure_mono : Monotone (topologicalClosure : _ → StarSubalgebra R A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A]
Goal: Monotone StarSubalgebra.topologicalClosure
-- StarSubalgebra.topologicalClosure_map_le
theorem topologicalClosure_map_le [StarModule R B] [TopologicalSemiring B] [ContinuousStar B]
    (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : IsClosedMap φ) :
    (map φ s).topologicalClosure ≤ map φ s.topologicalClosure 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] [TopologicalSpace B] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] [TopologicalSemiring B] [ContinuousStar B] (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : IsClosedMap ⇑φ)
Goal: (StarSubalgebra.map φ s).topologicalClosure ≤ StarSubalgebra.map φ s.topologicalClosure
-- StarSubalgebra.map_topologicalClosure_le
theorem map_topologicalClosure_le [StarModule R B] [TopologicalSemiring B] [ContinuousStar B]
    (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : Continuous φ) :
    map φ s.topologicalClosure ≤ (map φ s).topologicalClosure 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] [TopologicalSpace B] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] [TopologicalSemiring B] [ContinuousStar B] (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : Continuous ⇑φ)
Goal: StarSubalgebra.map φ s.topologicalClosure ≤ (StarSubalgebra.map φ s).topologicalClosure
-- StarSubalgebra.topologicalClosure_map
theorem topologicalClosure_map [StarModule R B] [TopologicalSemiring B] [ContinuousStar B]
    (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : ClosedEmbedding φ) :
    (map φ s).topologicalClosure = map φ s.topologicalClosure 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] [TopologicalSpace B] [Semiring B] [Algebra R B] [StarRing B] [StarModule R B] [TopologicalSemiring B] [ContinuousStar B] (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : ClosedEmbedding ⇑φ)
Goal: (StarSubalgebra.map φ s).topologicalClosure = StarSubalgebra.map φ s.topologicalClosure
-- Subalgebra.topologicalClosure_star_comm
theorem _root_.Subalgebra.topologicalClosure_star_comm (s : Subalgebra R A) :
    (star s).topologicalClosure = star s.topologicalClosure 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] (s : Subalgebra R A)
Goal: (star s).topologicalClosure = star s.topologicalClosure
-- StarAlgHom.ext_topologicalClosure
theorem _root_.StarAlgHom.ext_topologicalClosure [T2Space B] {S : StarSubalgebra R A}
    {φ ψ : S.topologicalClosure →⋆ₐ[R] B} (hφ : Continuous φ) (hψ : Continuous ψ)
    (h :
      φ.comp (inclusion (le_topologicalClosure S)) = ψ.comp (inclusion (le_topologicalClosure S))) :
    φ = ψ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] [TopologicalSpace B] [Semiring B] [Algebra R B] [StarRing B] [T2Space B] {S : StarSubalgebra R A} {φ ψ : ↥S.topologicalClosure →⋆ₐ[R] B} (hφ : Continuous ⇑φ) (hψ : Continuous ⇑ψ) (h : φ.comp (StarSubalgebra.inclusion ⋯) = ψ.comp (StarSubalgebra.inclusion ⋯))
Goal: φ = ψ
-- StarAlgHomClass.ext_topologicalClosure
theorem _root_.StarAlgHomClass.ext_topologicalClosure [T2Space B] {F : Type*}
    {S : StarSubalgebra R A} [FunLike F S.topologicalClosure B]
    [AlgHomClass F R S.topologicalClosure B] [StarAlgHomClass F R S.topologicalClosure B] {φ ψ : F}
    (hφ : Continuous φ) (hψ : Continuous ψ) (h : ∀ x : S,
        φ (inclusion (le_topologicalClosure S) x) = ψ ((inclusion (le_topologicalClosure S)) x)) :
    φ = ψ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalSemiring A] [ContinuousStar A] [TopologicalSpace B] [Semiring B] [Algebra R B] [StarRing B] [T2Space B] {F : Type u_4} {S : StarSubalgebra R A} [FunLike F (↥S.topologicalClosure) B] [AlgHomClass F R (↥S.topologicalClosure) B] [StarAlgHomClass F R (↥S.topologicalClosure) B] {φ ψ : F} (hφ : Continuous ⇑φ) (hψ : Continuous ⇑ψ) (h : ∀ (x : ↥S), φ ((StarSubalgebra.inclusion ⋯) x) = ψ ((StarSubalgebra.inclusion ⋯) x))
Goal: φ = ψ
-- elementalStarAlgebra.self_mem
@[aesop safe apply (rule_sets := [SetLike])]
theorem self_mem (x : A) : x ∈ elementalStarAlgebra R x 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [StarRing A] [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A] (x : A)
Goal: x ∈ elementalStarAlgebra R x
-- elementalStarAlgebra.star_self_mem
theorem star_self_mem (x : A) : star x ∈ elementalStarAlgebra R x 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [StarRing A] [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A] (x : A)
Goal: star x ∈ elementalStarAlgebra R x
-- elementalStarAlgebra.isClosed
theorem isClosed (x : A) : IsClosed (elementalStarAlgebra R x : Set A) 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [StarRing A] [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A] (x : A)
Goal: IsClosed ↑(elementalStarAlgebra R x)
-- elementalStarAlgebra.le_of_isClosed_of_mem
theorem le_of_isClosed_of_mem {S : StarSubalgebra R A} (hS : IsClosed (S : Set A)) {x : A}
    (hx : x ∈ S) : elementalStarAlgebra R x ≤ S 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [StarRing A] [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A] {S : StarSubalgebra R A} (hS : IsClosed ↑S) {x : A} (hx : x ∈ S)
Goal: elementalStarAlgebra R x ≤ S
-- elementalStarAlgebra.closedEmbedding_coe
theorem closedEmbedding_coe (x : A) : ClosedEmbedding ((↑) : elementalStarAlgebra R x → A) 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [StarRing A] [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A] (x : A)
Goal: ClosedEmbedding Subtype.val
-- elementalStarAlgebra.induction_on
@[elab_as_elim]
theorem induction_on {x y : A}
    (hy : y ∈ elementalStarAlgebra R x) {P : (u : A) → u ∈ elementalStarAlgebra R x → Prop}
    (self : P x (self_mem R x)) (star_self : P (star x) (star_self_mem R x))
    (algebraMap : ∀ r, P (algebraMap R A r) (_root_.algebraMap_mem _ r))
    (add : ∀ u hu v hv, P u hu → P v hv → P (u + v) (add_mem hu hv))
    (mul : ∀ u hu v hv, P u hu → P v hv → P (u * v) (mul_mem hu hv))
    (closure : ∀ s : Set A, (hs : s ⊆ elementalStarAlgebra R x) → (∀ u, (hu : u ∈ s) →
      P u (hs hu)) → ∀ v, (hv : v ∈ closure s) → P v (closure_minimal hs (isClosed R x) hv)) :
    P y hy 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [StarRing A] [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A] {x y : A} (hy : y ∈ elementalStarAlgebra R x) {P : (u : A) → u ∈ elementalStarAlgebra R x → Prop} (self : P x ⋯) (star_self : P (star x) ⋯) (algebraMap : ∀ (r : R), P ((algebraMap R A) r) ⋯) (add :
  ∀ (u : A) (hu : u ∈ elementalStarAlgebra R x) (v : A) (hv : v ∈ elementalStarAlgebra R x),
    P u hu → P v hv → P (u + v) ⋯) (mul :
  ∀ (u : A) (hu : u ∈ elementalStarAlgebra R x) (v : A) (hv : v ∈ elementalStarAlgebra R x),
    P u hu → P v hv → P (u * v) ⋯) (closure :
  ∀ (s : Set A) (hs : s ⊆ ↑(elementalStarAlgebra R x)),
    (∀ (u : A) (hu : u ∈ s), P u ⋯) → ∀ (v : A) (hv : v ∈ closure s), P v ⋯)
Goal: P y hy
-- elementalStarAlgebra.starAlgHomClass_ext
theorem starAlgHomClass_ext [T2Space B] {F : Type*} {a : A}
    [FunLike F (elementalStarAlgebra R a) B] [AlgHomClass F R _ B] [StarAlgHomClass F R _ B]
    {φ ψ : F} (hφ : Continuous φ)
    (hψ : Continuous ψ) (h : φ ⟨a, self_mem R a⟩ = ψ ⟨a, self_mem R a⟩) : φ = ψ 
Arguments: (R : Type u_1) {A : Type u_2} {B : Type u_3} [CommSemiring R] [StarRing R] [TopologicalSpace A] [Semiring A] [StarRing A] [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A] [TopologicalSpace B] [Semiring B] [StarRing B] [Algebra R B] [T2Space B] {F : Type u_4} {a : A} [FunLike F (↥(elementalStarAlgebra R a)) B] [AlgHomClass F R (↥(elementalStarAlgebra R a)) B] [StarAlgHomClass F R (↥(elementalStarAlgebra R a)) B] {φ ψ : F} (hφ : Continuous ⇑φ) (hψ : Continuous ⇑ψ) (h : φ ⟨a, ⋯⟩ = ψ ⟨a, ⋯⟩)
Goal: φ = ψ
-- starL_apply
@[simps!]
def starL (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [AddCommMonoid A]
    [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
    A ≃L⋆[R] A 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A]
Goal: ∀ (a : A), (starL R) a = star a
-- starL_symm_apply
@[simps!]
def starL (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [AddCommMonoid A]
    [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
    A ≃L⋆[R] A 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A]
Goal: ∀ (a : A), (starL R).symm a = starAddEquiv.symm a
-- starL'_symm_apply
@[simps!]
def starL' (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A]
    [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
    A ≃L[R] A 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A]
Goal: ∀ (a : A),
  (starL' R).symm a =
    (starL R).symm ({ toFun := id, map_add' := ⋯, map_smul' := ⋯, invFun := id, left_inv := ⋯, right_inv := ⋯ }.symm a)
-- starL'_apply
@[simps!]
def starL' (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A]
    [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
    A ≃L[R] A 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A]
Goal: ∀ (a : A), (starL' R) a = star a