ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Path.Homotopy.trans_apply
theorem trans_apply (F : Homotopy p₀ p₁) (G : Homotopy p₁ p₂) (x : I × I) :
    (F.trans G) x =
      if h : (x.1 : ℝ) ≤ 1 / 2 then
        F (⟨2 * x.1, (unitInterval.mul_pos_mem_iff zero_lt_two).2 ⟨x.1.2.1, h⟩⟩, x.2)
      else
        G (⟨2 * x.1 - 1, unitInterval.two_mul_sub_one_mem_iff.2 ⟨(not_le.1 h).le, x.1.2.2⟩⟩, x.2) 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ p₂ : Path x₀ x₁} (F : p₀.Homotopy p₁) (G : p₁.Homotopy p₂) (x : ↑I × ↑I)
Goal: (F.trans G) x = if h : ↑x.1 ≤ 1 / 2 then F (⟨2 * ↑x.1, ⋯⟩, x.2) else G (⟨2 * ↑x.1 - 1, ⋯⟩, x.2)
-- Path.Homotopy.symm_trans
theorem symm_trans (F : Homotopy p₀ p₁) (G : Homotopy p₁ p₂) :
    (F.trans G).symm = G.symm.trans F.symm 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ p₂ : Path x₀ x₁} (F : p₀.Homotopy p₁) (G : p₁.Homotopy p₂)
Goal: (F.trans G).symm = G.symm.trans F.symm
-- Path.Homotopy.cast_toFun
@[simps!]
def cast {p₀ p₁ q₀ q₁ : Path x₀ x₁} (F : Homotopy p₀ p₁) (h₀ : p₀ = q₀) (h₁ : p₁ = q₁) :
    Homotopy q₀ q₁ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ q₀ q₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (h₀ : p₀ = q₀) (h₁ : p₁ = q₁) (a : ↑I × ↑I)
Goal: (F.cast h₀ h₁) a = F a
-- Path.Homotopy.cast_apply
@[simps!]
def cast {p₀ p₁ q₀ q₁ : Path x₀ x₁} (F : Homotopy p₀ p₁) (h₀ : p₀ = q₀) (h₁ : p₁ = q₁) :
    Homotopy q₀ q₁ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ q₀ q₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (h₀ : p₀ = q₀) (h₁ : p₁ = q₁) (a : ↑I × ↑I)
Goal: (F.cast h₀ h₁) a = F a
-- Path.Homotopy.hcomp_apply
theorem hcomp_apply (F : Homotopy p₀ q₀) (G : Homotopy p₁ q₁) (x : I × I) :
    F.hcomp G x =
      if h : (x.2 : ℝ) ≤ 1 / 2 then
        F.eval x.1 ⟨2 * x.2, (unitInterval.mul_pos_mem_iff zero_lt_two).2 ⟨x.2.2.1, h⟩⟩
      else
        G.eval x.1
          ⟨2 * x.2 - 1, unitInterval.two_mul_sub_one_mem_iff.2 ⟨(not_le.1 h).le, x.2.2.2⟩⟩ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ q₀ : Path x₀ x₁} {p₁ q₁ : Path x₁ x₂} (F : p₀.Homotopy q₀) (G : p₁.Homotopy q₁) (x : ↑I × ↑I)
Goal: (F.hcomp G) x = if h : ↑x.2 ≤ 1 / 2 then (F.eval x.1) ⟨2 * ↑x.2, ⋯⟩ else (G.eval x.1) ⟨2 * ↑x.2 - 1, ⋯⟩
-- Path.Homotopy.hcomp_half
theorem hcomp_half (F : Homotopy p₀ q₀) (G : Homotopy p₁ q₁) (t : I) :
    F.hcomp G (t, ⟨1 / 2, by norm_num, by norm_num⟩) = x₁ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ q₀ : Path x₀ x₁} {p₁ q₁ : Path x₁ x₂} (F : p₀.Homotopy q₀) (G : p₁.Homotopy q₁) (t : ↑I)
Goal: (F.hcomp G) (t, ⟨1 / 2, ⋯⟩) = x₁
-- Path.Homotopy.symm₂_toFun
@[simps]
def symm₂ {p q : Path x₀ x₁} (F : p.Homotopy q) : p.symm.Homotopy q.symm 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) (x : ↑I × ↑I)
Goal: F.symm₂ x = F (x.1, σ x.2)
-- Path.Homotopy.symm₂_apply
@[simps]
def symm₂ {p q : Path x₀ x₁} (F : p.Homotopy q) : p.symm.Homotopy q.symm 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) (x : ↑I × ↑I)
Goal: F.symm₂ x = F (x.1, σ x.2)
-- Path.Homotopy.map_toFun
@[simps]
def map {p q : Path x₀ x₁} (F : p.Homotopy q) (f : C(X, Y)) :
    Homotopy (p.map f.continuous) (q.map f.continuous) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) (f : C(X, Y))
Goal: ∀ (a : ↑I × ↑I), (F.map f) a = (⇑f ∘ ⇑F) a
-- Path.Homotopy.map_apply
@[simps]
def map {p q : Path x₀ x₁} (F : p.Homotopy q) (f : C(X, Y)) :
    Homotopy (p.map f.continuous) (q.map f.continuous) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) (f : C(X, Y))
Goal: ∀ (a : ↑I × ↑I), (F.map f) a = (⇑f ∘ ⇑F) a
-- Path.Homotopic.refl
@[refl]
theorem refl (p : Path x₀ x₁) : p.Homotopic p 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁)
Goal: p.Homotopic p
-- Path.Homotopic.symm
@[symm]
theorem symm ⦃p₀ p₁ : Path x₀ x₁⦄ (h : p₀.Homotopic p₁) : p₁.Homotopic p₀ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} ⦃p₀ p₁ : Path x₀ x₁⦄ (h : p₀.Homotopic p₁)
Goal: p₁.Homotopic p₀
-- Path.Homotopic.trans
@[trans]
theorem trans ⦃p₀ p₁ p₂ : Path x₀ x₁⦄ (h₀ : p₀.Homotopic p₁) (h₁ : p₁.Homotopic p₂) :
    p₀.Homotopic p₂ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} ⦃p₀ p₁ p₂ : Path x₀ x₁⦄ (h₀ : p₀.Homotopic p₁) (h₁ : p₁.Homotopic p₂)
Goal: p₀.Homotopic p₂
-- Path.Homotopic.equivalence
theorem equivalence : Equivalence (@Homotopic X _ x₀ x₁) 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ : X}
Goal: Equivalence Path.Homotopic
-- Path.Homotopic.map
nonrec theorem map {p q : Path x₀ x₁} (h : p.Homotopic q) (f : C(X, Y)) :
    Homotopic (p.map f.continuous) (q.map f.continuous) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} {p q : Path x₀ x₁} (h : p.Homotopic q) (f : C(X, Y))
Goal: (p.map ⋯).Homotopic (q.map ⋯)
-- Path.Homotopic.hcomp
theorem hcomp {p₀ p₁ : Path x₀ x₁} {q₀ q₁ : Path x₁ x₂} (hp : p₀.Homotopic p₁)
    (hq : q₀.Homotopic q₁) : (p₀.trans q₀).Homotopic (p₁.trans q₁) 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ p₁ : Path x₀ x₁} {q₀ q₁ : Path x₁ x₂} (hp : p₀.Homotopic p₁) (hq : q₀.Homotopic q₁)
Goal: (p₀.trans q₀).Homotopic (p₁.trans q₁)
-- Path.Homotopic.comp_lift
theorem comp_lift (P₀ : Path x₀ x₁) (P₁ : Path x₁ x₂) : ⟦P₀.trans P₁⟧ = Quotient.comp ⟦P₀⟧ ⟦P₁⟧ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} (P₀ : Path x₀ x₁) (P₁ : Path x₁ x₂)
Goal: ⟦P₀.trans P₁⟧ = Path.Homotopic.Quotient.comp ⟦P₀⟧ ⟦P₁⟧
-- Path.Homotopic.map_lift
theorem map_lift (P₀ : Path x₀ x₁) (f : C(X, Y)) : ⟦P₀.map f.continuous⟧ = Quotient.mapFn ⟦P₀⟧ f 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} (P₀ : Path x₀ x₁) (f : C(X, Y))
Goal: ⟦P₀.map ⋯⟧ = Path.Homotopic.Quotient.mapFn ⟦P₀⟧ f
-- Path.Homotopic.hpath_hext
theorem hpath_hext {p₁ : Path x₀ x₁} {p₂ : Path x₂ x₃} (hp : ∀ t, p₁ t = p₂ t) :
    @HEq (Path.Homotopic.Quotient _ _) ⟦p₁⟧ (Path.Homotopic.Quotient _ _) ⟦p₂⟧ 
Arguments: {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ x₃ : X} {p₁ : Path x₀ x₁} {p₂ : Path x₂ x₃} (hp : ∀ (t : ↑I), p₁ t = p₂ t)
Goal: HEq ⟦p₁⟧ ⟦p₂⟧
-- Path.toHomotopyConst_apply
@[simps!]
def toHomotopyConst (p : Path x₀ x₁) :
    (ContinuousMap.const Y x₀).Homotopy (ContinuousMap.const Y x₁) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} (p : Path x₀ x₁)
Goal: ∀ (a : ↑I × Y), p.toHomotopyConst a = p a.1
-- ContinuousMap.homotopic_const_iff
@[simp]
theorem ContinuousMap.homotopic_const_iff [Nonempty Y] :
    (ContinuousMap.const Y x₀).Homotopic (ContinuousMap.const Y x₁) ↔ Joined x₀ x₁ 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} [Nonempty Y]
Goal: (ContinuousMap.const Y x₀).Homotopic (ContinuousMap.const Y x₁) ↔ Joined x₀ x₁
-- UniformSpaceCat.coe_of
@[simp]
theorem coe_of (X : Type u) [UniformSpace X] : (of X : Type u) = X 
Arguments: (X : Type u) [UniformSpace X]
Goal: ↑(UniformSpaceCat.of X) = X
-- UniformSpaceCat.coe_comp
@[simp]
theorem coe_comp {X Y Z : UniformSpaceCat} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g : X → Z) = g ∘ f 
Arguments: {X Y Z : UniformSpaceCat} (f : X ⟶ Y) (g : Y ⟶ Z)
Goal: (CategoryTheory.forget UniformSpaceCat).map (f ≫ g) =
  (CategoryTheory.forget UniformSpaceCat).map g ∘ (CategoryTheory.forget UniformSpaceCat).map f
-- UniformSpaceCat.coe_id
@[simp]
theorem coe_id (X : UniformSpaceCat) : (𝟙 X : X → X) = id 
Arguments: (X : UniformSpaceCat)
Goal: (CategoryTheory.forget UniformSpaceCat).map (𝟙 X) = id
-- UniformSpaceCat.coe_mk
theorem coe_mk {X Y : UniformSpaceCat} (f : X → Y) (hf : UniformContinuous f) :
    ((⟨f, hf⟩ : X ⟶ Y) : X → Y) = f 
Arguments: {X Y : UniformSpaceCat} (f : ↑X → ↑Y) (hf : UniformContinuous f)
Goal: ↑⟨f, hf⟩ = f
-- UniformSpaceCat.hom_ext
theorem hom_ext {X Y : UniformSpaceCat} {f g : X ⟶ Y} : (f : X → Y) = g → f = g 
Arguments: {X Y : UniformSpaceCat} {f g : X ⟶ Y}
Goal: (CategoryTheory.forget UniformSpaceCat).map f = (CategoryTheory.forget UniformSpaceCat).map g → f = g
-- CpltSepUniformSpace.isCompleteSpace
structure CpltSepUniformSpace where
    α : Type u
  [isUniformSpace : UniformSpace α]
  [isCompleteSpace : CompleteSpace α]
  [isT0 : T0Space α]
Arguments: (self : CpltSepUniformSpace)
Goal: CompleteSpace self.α
-- CpltSepUniformSpace.isT0
structure CpltSepUniformSpace where
    α : Type u
  [isUniformSpace : UniformSpace α]
  [isCompleteSpace : CompleteSpace α]
  [isT0 : T0Space α]
Arguments: (self : CpltSepUniformSpace)
Goal: T0Space self.α
-- CpltSepUniformSpace.coe_of
@[simp]
theorem coe_of (X : Type u) [UniformSpace X] [CompleteSpace X] [T0Space X] :
    (of X : Type u) = X 
Arguments: (X : Type u) [UniformSpace X] [CompleteSpace X] [T0Space X]
Goal: (CpltSepUniformSpace.of X).α = X
-- UniformSpaceCat.completionHom_val
@[simp]
theorem completionHom_val (X : UniformSpaceCat) (x) : (completionHom X) x = (x : Completion X) 
Arguments: (X : UniformSpaceCat) (x : (CategoryTheory.forget UniformSpaceCat).obj X)
Goal: (CategoryTheory.forget UniformSpaceCat).map X.completionHom x = ↑((CategoryTheory.forget UniformSpaceCat).obj X) x
-- UniformSpaceCat.extensionHom_val
@[simp]
theorem extensionHom_val {X : UniformSpaceCat} {Y : CpltSepUniformSpace}
    (f : X ⟶ (forget₂ _ _).obj Y) (x) : (extensionHom f) x = Completion.extension f x 
Arguments: {X : UniformSpaceCat} {Y : CpltSepUniformSpace} (f : X ⟶ (CategoryTheory.forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y) (x : (CategoryTheory.forget CpltSepUniformSpace).obj (UniformSpaceCat.completionFunctor.obj X))
Goal: (UniformSpaceCat.extensionHom f) x = UniformSpace.Completion.extension ((CategoryTheory.forget UniformSpaceCat).map f) x
-- UniformSpaceCat.extension_comp_coe
@[simp]
theorem extension_comp_coe {X : UniformSpaceCat} {Y : CpltSepUniformSpace}
    (f : toUniformSpace (CpltSepUniformSpace.of (Completion X)) ⟶ toUniformSpace Y) :
    extensionHom (completionHom X ≫ f) = f 
Arguments: {X : UniformSpaceCat} {Y : CpltSepUniformSpace} (f : (CpltSepUniformSpace.of (UniformSpace.Completion ↑X)).toUniformSpace ⟶ Y.toUniformSpace)
Goal: UniformSpaceCat.extensionHom (X.completionHom ≫ f) = f
-- TopCat.effectiveEpi_iff_quotientMap
theorem effectiveEpi_iff_quotientMap {B X : TopCat.{u}} (π : X ⟶ B) :
    EffectiveEpi π ↔ QuotientMap π 
Arguments: {B X : TopCat} (π : X ⟶ B)
Goal: CategoryTheory.EffectiveEpi π ↔ QuotientMap ⇑π
-- isGenericPoint_def
theorem isGenericPoint_def {x : α} {S : Set α} : IsGenericPoint x S ↔ closure ({x} : Set α) = S 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {S : Set α}
Goal: IsGenericPoint x S ↔ closure {x} = S
-- IsGenericPoint.def
theorem IsGenericPoint.def {x : α} {S : Set α} (h : IsGenericPoint x S) :
    closure ({x} : Set α) = S 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {S : Set α} (h : IsGenericPoint x S)
Goal: closure {x} = S
-- isGenericPoint_closure
theorem isGenericPoint_closure {x : α} : IsGenericPoint x (closure ({x} : Set α)) 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α}
Goal: IsGenericPoint x (closure {x})
-- isGenericPoint_iff_specializes
theorem isGenericPoint_iff_specializes : IsGenericPoint x S ↔ ∀ y, x ⤳ y ↔ y ∈ S 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {S : Set α}
Goal: IsGenericPoint x S ↔ ∀ (y : α), x ⤳ y ↔ y ∈ S
-- IsGenericPoint.specializes_iff_mem
theorem specializes_iff_mem (h : IsGenericPoint x S) : x ⤳ y ↔ y ∈ S 
Arguments: {α : Type u_1} [TopologicalSpace α] {x y : α} {S : Set α} (h : IsGenericPoint x S)
Goal: x ⤳ y ↔ y ∈ S
-- IsGenericPoint.specializes
protected theorem specializes (h : IsGenericPoint x S) (h' : y ∈ S) : x ⤳ y 
Arguments: {α : Type u_1} [TopologicalSpace α] {x y : α} {S : Set α} (h : IsGenericPoint x S) (h' : y ∈ S)
Goal: x ⤳ y
-- IsGenericPoint.mem
protected theorem mem (h : IsGenericPoint x S) : x ∈ S 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {S : Set α} (h : IsGenericPoint x S)
Goal: x ∈ S
-- IsGenericPoint.isClosed
protected theorem isClosed (h : IsGenericPoint x S) : IsClosed S 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {S : Set α} (h : IsGenericPoint x S)
Goal: IsClosed S
-- IsGenericPoint.isIrreducible
protected theorem isIrreducible (h : IsGenericPoint x S) : IsIrreducible S 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {S : Set α} (h : IsGenericPoint x S)
Goal: IsIrreducible S
-- IsGenericPoint.inseparable
protected theorem inseparable (h : IsGenericPoint x S) (h' : IsGenericPoint y S) :
    Inseparable x y 
Arguments: {α : Type u_1} [TopologicalSpace α] {x y : α} {S : Set α} (h : IsGenericPoint x S) (h' : IsGenericPoint y S)
Goal: Inseparable x y
-- IsGenericPoint.eq
protected theorem eq [T0Space α] (h : IsGenericPoint x S) (h' : IsGenericPoint y S) : x = y 
Arguments: {α : Type u_1} [TopologicalSpace α] {x y : α} {S : Set α} [T0Space α] (h : IsGenericPoint x S) (h' : IsGenericPoint y S)
Goal: x = y
-- IsGenericPoint.mem_open_set_iff
theorem mem_open_set_iff (h : IsGenericPoint x S) (hU : IsOpen U) : x ∈ U ↔ (S ∩ U).Nonempty 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {S U : Set α} (h : IsGenericPoint x S) (hU : IsOpen U)
Goal: x ∈ U ↔ (S ∩ U).Nonempty
-- IsGenericPoint.disjoint_iff
theorem disjoint_iff (h : IsGenericPoint x S) (hU : IsOpen U) : Disjoint S U ↔ x ∉ U 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {S U : Set α} (h : IsGenericPoint x S) (hU : IsOpen U)
Goal: Disjoint S U ↔ x ∉ U
-- IsGenericPoint.mem_closed_set_iff
theorem mem_closed_set_iff (h : IsGenericPoint x S) (hZ : IsClosed Z) : x ∈ Z ↔ S ⊆ Z 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {S Z : Set α} (h : IsGenericPoint x S) (hZ : IsClosed Z)
Goal: x ∈ Z ↔ S ⊆ Z
-- IsGenericPoint.image
protected theorem image (h : IsGenericPoint x S) {f : α → β} (hf : Continuous f) :
    IsGenericPoint (f x) (closure (f '' S)) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {x : α} {S : Set α} (h : IsGenericPoint x S) {f : α → β} (hf : Continuous f)
Goal: IsGenericPoint (f x) (closure (f '' S))
-- isGenericPoint_iff_forall_closed
theorem isGenericPoint_iff_forall_closed (hS : IsClosed S) (hxS : x ∈ S) :
    IsGenericPoint x S ↔ ∀ Z : Set α, IsClosed Z → x ∈ Z → S ⊆ Z 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {S : Set α} (hS : IsClosed S) (hxS : x ∈ S)
Goal: IsGenericPoint x S ↔ ∀ (Z : Set α), IsClosed Z → x ∈ Z → S ⊆ Z
-- quasiSober_iff
@[mk_iff]
class QuasiSober (α : Type*) [TopologicalSpace α] : Prop where
  sober : ∀ {S : Set α}, IsIrreducible S → IsClosed S → ∃ x, IsGenericPoint x S
Arguments: (α : Type u_3) [TopologicalSpace α]
Goal: QuasiSober α ↔ ∀ {S : Set α}, IsIrreducible S → IsClosed S → ∃ x, IsGenericPoint x S
-- QuasiSober.sober
@[mk_iff]
class QuasiSober (α : Type*) [TopologicalSpace α] : Prop where
  sober : ∀ {S : Set α}, IsIrreducible S → IsClosed S → ∃ x, IsGenericPoint x S
Arguments: {α : Type u_3} [TopologicalSpace α] [self : QuasiSober α] {S : Set α}
Goal: IsIrreducible S → IsClosed S → ∃ x, IsGenericPoint x S
-- IsIrreducible.genericPoint_spec
theorem IsIrreducible.genericPoint_spec [QuasiSober α] {S : Set α} (hS : IsIrreducible S) :
    IsGenericPoint hS.genericPoint (closure S) 
Arguments: {α : Type u_1} [TopologicalSpace α] [QuasiSober α] {S : Set α} (hS : IsIrreducible S)
Goal: IsGenericPoint hS.genericPoint (closure S)
-- IsIrreducible.genericPoint_closure_eq
@[simp]
theorem IsIrreducible.genericPoint_closure_eq [QuasiSober α] {S : Set α} (hS : IsIrreducible S) :
    closure ({hS.genericPoint} : Set α) = closure S 
Arguments: {α : Type u_1} [TopologicalSpace α] [QuasiSober α] {S : Set α} (hS : IsIrreducible S)
Goal: closure {hS.genericPoint} = closure S
-- genericPoint_spec
theorem genericPoint_spec [QuasiSober α] [IrreducibleSpace α] :
    IsGenericPoint (genericPoint α) ⊤ 
Arguments: (α : Type u_1) [TopologicalSpace α] [QuasiSober α] [IrreducibleSpace α]
Goal: IsGenericPoint (genericPoint α) ⊤
-- genericPoint_closure
@[simp]
theorem genericPoint_closure [QuasiSober α] [IrreducibleSpace α] :
    closure ({genericPoint α} : Set α) = ⊤ 
Arguments: (α : Type u_1) [TopologicalSpace α] [QuasiSober α] [IrreducibleSpace α]
Goal: closure {genericPoint α} = ⊤
-- genericPoint_specializes
theorem genericPoint_specializes [QuasiSober α] [IrreducibleSpace α] (x : α) : genericPoint α ⤳ x 
Arguments: {α : Type u_1} [TopologicalSpace α] [QuasiSober α] [IrreducibleSpace α] (x : α)
Goal: genericPoint α ⤳ x
-- ClosedEmbedding.quasiSober
theorem ClosedEmbedding.quasiSober {f : α → β} (hf : ClosedEmbedding f) [QuasiSober β] :
    QuasiSober α 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : ClosedEmbedding f) [QuasiSober β]
Goal: QuasiSober α
-- OpenEmbedding.quasiSober
theorem OpenEmbedding.quasiSober {f : α → β} (hf : OpenEmbedding f) [QuasiSober β] :
    QuasiSober α 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : OpenEmbedding f) [QuasiSober β]
Goal: QuasiSober α
-- quasiSober_of_open_cover
theorem quasiSober_of_open_cover (S : Set (Set α)) (hS : ∀ s : S, IsOpen (s : Set α))
    [hS' : ∀ s : S, QuasiSober s] (hS'' : ⋃₀ S = ⊤) : QuasiSober α 
Arguments: {α : Type u_1} [TopologicalSpace α] (S : Set (Set α)) (hS : ∀ (s : ↑S), IsOpen ↑s) [hS' : ∀ (s : ↑S), QuasiSober ↑↑s] (hS'' : ⋃₀ S = ⊤)
Goal: QuasiSober α
-- TopologicalSpace.IsTopologicalBasis.exists_subset_inter
structure IsTopologicalBasis (s : Set (Set α)) : Prop where
    exists_subset_inter : ∀ t₁ ∈ s, ∀ t₂ ∈ s, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
    sUnion_eq : ⋃₀ s = univ
    eq_generateFrom : t = generateFrom s
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (self : TopologicalSpace.IsTopologicalBasis s) (t₁ : Set α)
Goal: t₁ ∈ s → ∀ t₂ ∈ s, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
-- TopologicalSpace.IsTopologicalBasis.sUnion_eq
structure IsTopologicalBasis (s : Set (Set α)) : Prop where
    exists_subset_inter : ∀ t₁ ∈ s, ∀ t₂ ∈ s, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
    sUnion_eq : ⋃₀ s = univ
    eq_generateFrom : t = generateFrom s
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (self : TopologicalSpace.IsTopologicalBasis s)
Goal: ⋃₀ s = Set.univ
-- TopologicalSpace.IsTopologicalBasis.eq_generateFrom
structure IsTopologicalBasis (s : Set (Set α)) : Prop where
    exists_subset_inter : ∀ t₁ ∈ s, ∀ t₂ ∈ s, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
    sUnion_eq : ⋃₀ s = univ
    eq_generateFrom : t = generateFrom s
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (self : TopologicalSpace.IsTopologicalBasis s)
Goal: t = TopologicalSpace.generateFrom s
-- TopologicalSpace.IsTopologicalBasis.insert_empty
theorem IsTopologicalBasis.insert_empty {s : Set (Set α)} (h : IsTopologicalBasis s) :
    IsTopologicalBasis (insert ∅ s) 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (h : TopologicalSpace.IsTopologicalBasis s)
Goal: TopologicalSpace.IsTopologicalBasis (insert ∅ s)
-- TopologicalSpace.IsTopologicalBasis.diff_empty
theorem IsTopologicalBasis.diff_empty {s : Set (Set α)} (h : IsTopologicalBasis s) :
    IsTopologicalBasis (s \ {∅}) 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (h : TopologicalSpace.IsTopologicalBasis s)
Goal: TopologicalSpace.IsTopologicalBasis (s \ {∅})
-- TopologicalSpace.isTopologicalBasis_of_subbasis
theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom s) :
    IsTopologicalBasis ((fun f => ⋂₀ f) '' { f : Set (Set α) | f.Finite ∧ f ⊆ s }) 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (hs : t = TopologicalSpace.generateFrom s)
Goal: TopologicalSpace.IsTopologicalBasis ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s})
-- TopologicalSpace.isTopologicalBasis_of_subbasis_of_finiteInter
theorem isTopologicalBasis_of_subbasis_of_finiteInter {s : Set (Set α)} (hsg : t = generateFrom s)
    (hsi : FiniteInter s) : IsTopologicalBasis s 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (hsg : t = TopologicalSpace.generateFrom s) (hsi : FiniteInter s)
Goal: TopologicalSpace.IsTopologicalBasis s
-- TopologicalSpace.isTopologicalBasis_of_subbasis_of_inter
theorem isTopologicalBasis_of_subbasis_of_inter {r : Set (Set α)} (hsg : t = generateFrom r)
    (hsi : ∀ ⦃s⦄, s ∈ r → ∀ ⦃t⦄, t ∈ r → s ∩ t ∈ r) : IsTopologicalBasis (insert univ r) 
Arguments: {α : Type u} [t : TopologicalSpace α] {r : Set (Set α)} (hsg : t = TopologicalSpace.generateFrom r) (hsi : ∀ ⦃s : Set α⦄, s ∈ r → ∀ ⦃t : Set α⦄, t ∈ r → s ∩ t ∈ r)
Goal: TopologicalSpace.IsTopologicalBasis (insert Set.univ r)
-- TopologicalSpace.IsTopologicalBasis.of_hasBasis_nhds
theorem IsTopologicalBasis.of_hasBasis_nhds {s : Set (Set α)}
    (h_nhds : ∀ a, (𝓝 a).HasBasis (fun t ↦ t ∈ s ∧ a ∈ t) id) : IsTopologicalBasis s 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (h_nhds : ∀ (a : α), (𝓝 a).HasBasis (fun t => t ∈ s ∧ a ∈ t) id)
Goal: TopologicalSpace.IsTopologicalBasis s
-- TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
theorem isTopologicalBasis_of_isOpen_of_nhds {s : Set (Set α)} (h_open : ∀ u ∈ s, IsOpen u)
    (h_nhds : ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ s, a ∈ v ∧ v ⊆ u) :
    IsTopologicalBasis s 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (h_open : ∀ u ∈ s, IsOpen u) (h_nhds : ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ s, a ∈ v ∧ v ⊆ u)
Goal: TopologicalSpace.IsTopologicalBasis s
-- TopologicalSpace.IsTopologicalBasis.mem_nhds_iff
theorem IsTopologicalBasis.mem_nhds_iff {a : α} {s : Set α} {b : Set (Set α)}
    (hb : IsTopologicalBasis b) : s ∈ 𝓝 a ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s 
Arguments: {α : Type u} [t : TopologicalSpace α] {a : α} {s : Set α} {b : Set (Set α)} (hb : TopologicalSpace.IsTopologicalBasis b)
Goal: s ∈ 𝓝 a ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s
-- TopologicalSpace.IsTopologicalBasis.isOpen_iff
theorem IsTopologicalBasis.isOpen_iff {s : Set α} {b : Set (Set α)} (hb : IsTopologicalBasis b) :
    IsOpen s ↔ ∀ a ∈ s, ∃ t ∈ b, a ∈ t ∧ t ⊆ s 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set α} {b : Set (Set α)} (hb : TopologicalSpace.IsTopologicalBasis b)
Goal: IsOpen s ↔ ∀ a ∈ s, ∃ t ∈ b, a ∈ t ∧ t ⊆ s
-- TopologicalSpace.IsTopologicalBasis.nhds_hasBasis
theorem IsTopologicalBasis.nhds_hasBasis {b : Set (Set α)} (hb : IsTopologicalBasis b) {a : α} :
    (𝓝 a).HasBasis (fun t : Set α => t ∈ b ∧ a ∈ t) fun t => t 
Arguments: {α : Type u} [t : TopologicalSpace α] {b : Set (Set α)} (hb : TopologicalSpace.IsTopologicalBasis b) {a : α}
Goal: (𝓝 a).HasBasis (fun t => t ∈ b ∧ a ∈ t) fun t => t
-- TopologicalSpace.IsTopologicalBasis.isOpen
protected theorem IsTopologicalBasis.isOpen {s : Set α} {b : Set (Set α)}
    (hb : IsTopologicalBasis b) (hs : s ∈ b) : IsOpen s 
Arguments: {α : Type u} [t : TopologicalSpace α] {s : Set α} {b : Set (Set α)} (hb : TopologicalSpace.IsTopologicalBasis b) (hs : s ∈ b)
Goal: IsOpen s
-- TopologicalSpace.IsTopologicalBasis.mem_nhds
protected theorem IsTopologicalBasis.mem_nhds {a : α} {s : Set α} {b : Set (Set α)}
    (hb : IsTopologicalBasis b) (hs : s ∈ b) (ha : a ∈ s) : s ∈ 𝓝 a 
Arguments: {α : Type u} [t : TopologicalSpace α] {a : α} {s : Set α} {b : Set (Set α)} (hb : TopologicalSpace.IsTopologicalBasis b) (hs : s ∈ b) (ha : a ∈ s)
Goal: s ∈ 𝓝 a
-- TopologicalSpace.IsTopologicalBasis.exists_subset_of_mem_open
theorem IsTopologicalBasis.exists_subset_of_mem_open {b : Set (Set α)} (hb : IsTopologicalBasis b)
    {a : α} {u : Set α} (au : a ∈ u) (ou : IsOpen u) : ∃ v ∈ b, a ∈ v ∧ v ⊆ u 
Arguments: {α : Type u} [t : TopologicalSpace α] {b : Set (Set α)} (hb : TopologicalSpace.IsTopologicalBasis b) {a : α} {u : Set α} (au : a ∈ u) (ou : IsOpen u)
Goal: ∃ v ∈ b, a ∈ v ∧ v ⊆ u
-- TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
    (ou : IsOpen u) : u = ⋃₀ { s ∈ B | s ⊆ u } 
Arguments: {α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} (hB : TopologicalSpace.IsTopologicalBasis B) {u : Set α} (ou : IsOpen u)
Goal: u = ⋃₀ {s | s ∈ B ∧ s ⊆ u}