ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SequentialSpace.isProperMap_iff_tendsto_cocompact
lemma SequentialSpace.isProperMap_iff_tendsto_cocompact [T2Space Y] [SequentialSpace Y] :
    IsProperMap f ↔ Continuous f ∧ Tendsto f (cocompact X) (cocompact Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [T2Space Y] [SequentialSpace Y]
Goal: IsProperMap f ↔ Continuous f ∧ Filter.Tendsto f (Filter.cocompact X) (Filter.cocompact Y)
-- IsProperMap.universally_closed
theorem IsProperMap.universally_closed (Z) [TopologicalSpace Z] (h : IsProperMap f) :
    IsClosedMap (Prod.map f id : X × Z → Y × Z) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (Z : Type u_6) [TopologicalSpace Z] (h : IsProperMap f)
Goal: IsClosedMap (Prod.map f id)
-- isProperMap_iff_isClosedMap_filter
theorem isProperMap_iff_isClosedMap_filter {X : Type u} {Y : Type v} [TopologicalSpace X]
    [TopologicalSpace Y] {f : X → Y} :
    IsProperMap f ↔ Continuous f ∧ IsClosedMap
      (Prod.map f id : X × Filter X → Y × Filter X) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: IsProperMap f ↔ Continuous f ∧ IsClosedMap (Prod.map f id)
-- tendsto_extendFrom
theorem tendsto_extendFrom {A : Set X} {f : X → Y} {x : X} (h : ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) :
    Tendsto f (𝓝[A] x) (𝓝 <| extendFrom A f x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {A : Set X} {f : X → Y} {x : X} (h : ∃ y, Filter.Tendsto f (𝓝[A] x) (𝓝 y))
Goal: Filter.Tendsto f (𝓝[A] x) (𝓝 (extendFrom A f x))
-- extendFrom_eq
theorem extendFrom_eq [T2Space Y] {A : Set X} {f : X → Y} {x : X} {y : Y} (hx : x ∈ closure A)
    (hf : Tendsto f (𝓝[A] x) (𝓝 y)) : extendFrom A f x = y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {A : Set X} {f : X → Y} {x : X} {y : Y} (hx : x ∈ closure A) (hf : Filter.Tendsto f (𝓝[A] x) (𝓝 y))
Goal: extendFrom A f x = y
-- extendFrom_extends
theorem extendFrom_extends [T2Space Y] {f : X → Y} {A : Set X} (hf : ContinuousOn f A) :
    ∀ x ∈ A, extendFrom A f x = f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} {A : Set X} (hf : ContinuousOn f A) (x : X)
Goal: x ∈ A → extendFrom A f x = f x
-- continuousOn_extendFrom
theorem continuousOn_extendFrom [RegularSpace Y] {f : X → Y} {A B : Set X} (hB : B ⊆ closure A)
    (hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) : ContinuousOn (extendFrom A f) B 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {f : X → Y} {A B : Set X} (hB : B ⊆ closure A) (hf : ∀ x ∈ B, ∃ y, Filter.Tendsto f (𝓝[A] x) (𝓝 y))
Goal: ContinuousOn (extendFrom A f) B
-- continuous_extendFrom
theorem continuous_extendFrom [RegularSpace Y] {f : X → Y} {A : Set X} (hA : Dense A)
    (hf : ∀ x, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) : Continuous (extendFrom A f) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {f : X → Y} {A : Set X} (hA : Dense A) (hf : ∀ (x : X), ∃ y, Filter.Tendsto f (𝓝[A] x) (𝓝 y))
Goal: Continuous (extendFrom A f)
-- ContinuousMap.HomotopyRel.pi_apply
@[simps!]
def HomotopyRel.pi (homotopies : ∀ i : I, HomotopyRel (f i) (g i) S) :
    HomotopyRel (pi f) (pi g) S 
Arguments: {I : Type u_1} {A : Type u_2} {X : I → Type u_3} [(i : I) → TopologicalSpace (X i)] [TopologicalSpace A] {f g : (i : I) → C(A, X i)} {S : Set A} (homotopies : (i : I) → (f i).HomotopyRel (g i) S) (a : ↑unitInterval × A) (i : I)
Goal: (ContinuousMap.HomotopyRel.pi homotopies) a i = (homotopies i) a
-- ContinuousMap.HomotopyRel.pi_toFun
@[simps!]
def HomotopyRel.pi (homotopies : ∀ i : I, HomotopyRel (f i) (g i) S) :
    HomotopyRel (pi f) (pi g) S 
Arguments: {I : Type u_1} {A : Type u_2} {X : I → Type u_3} [(i : I) → TopologicalSpace (X i)] [TopologicalSpace A] {f g : (i : I) → C(A, X i)} {S : Set A} (homotopies : (i : I) → (f i).HomotopyRel (g i) S) (a : ↑unitInterval × A) (i : I)
Goal: (ContinuousMap.HomotopyRel.pi homotopies) a i = (homotopies i) a
-- ContinuousMap.Homotopy.prod_apply
@[simps]
def Homotopy.prod (F : Homotopy f₀ f₁) (G : Homotopy g₀ g₁) :
    Homotopy (ContinuousMap.prodMk f₀ g₀) (ContinuousMap.prodMk f₁ g₁) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Type u_3} [TopologicalSpace A] {f₀ f₁ : C(A, α)} {g₀ g₁ : C(A, β)} (F : f₀.Homotopy f₁) (G : g₀.Homotopy g₁) (t : ↑unitInterval × A)
Goal: (F.prod G) t = (F t, G t)
-- ContinuousMap.HomotopyRel.prod_toFun
@[simps!]
def HomotopyRel.prod (F : HomotopyRel f₀ f₁ S) (G : HomotopyRel g₀ g₁ S) :
    HomotopyRel (prodMk f₀ g₀) (prodMk f₁ g₁) S 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Type u_3} [TopologicalSpace A] {f₀ f₁ : C(A, α)} {g₀ g₁ : C(A, β)} {S : Set A} (F : f₀.HomotopyRel f₁ S) (G : g₀.HomotopyRel g₁ S) (t : ↑unitInterval × A)
Goal: (F.prod G) t = (F t, G t)
-- ContinuousMap.HomotopyRel.prod_apply
@[simps!]
def HomotopyRel.prod (F : HomotopyRel f₀ f₁ S) (G : HomotopyRel g₀ g₁ S) :
    HomotopyRel (prodMk f₀ g₀) (prodMk f₁ g₁) S 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Type u_3} [TopologicalSpace A] {f₀ f₁ : C(A, α)} {g₀ g₁ : C(A, β)} {S : Set A} (F : f₀.HomotopyRel f₁ S) (G : g₀.HomotopyRel g₁ S) (t : ↑unitInterval × A)
Goal: (F.prod G) t = (F t, G t)
-- Path.Homotopic.pi_lift
theorem pi_lift (γ : ∀ i, Path (as i) (bs i)) :
    (Path.Homotopic.pi fun i => ⟦γ i⟧) = ⟦Path.pi γ⟧ 
Arguments: {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as bs : (i : ι) → X i} (γ : (i : ι) → Path (as i) (bs i))
Goal: (Path.Homotopic.pi fun i => ⟦γ i⟧) = ⟦Path.pi γ⟧
-- Path.Homotopic.comp_pi_eq_pi_comp
theorem comp_pi_eq_pi_comp (γ₀ : ∀ i, Path.Homotopic.Quotient (as i) (bs i))
    (γ₁ : ∀ i, Path.Homotopic.Quotient (bs i) (cs i)) : pi γ₀ ⬝ pi γ₁ = pi fun i ↦ γ₀ i ⬝ γ₁ i 
Arguments: {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as bs cs : (i : ι) → X i} (γ₀ : (i : ι) → Path.Homotopic.Quotient (as i) (bs i)) (γ₁ : (i : ι) → Path.Homotopic.Quotient (bs i) (cs i))
Goal: (Path.Homotopic.pi γ₀).comp (Path.Homotopic.pi γ₁) = Path.Homotopic.pi fun i => (γ₀ i).comp (γ₁ i)
-- Path.Homotopic.proj_pi
@[simp]
theorem proj_pi (i : ι) (paths : ∀ i, Path.Homotopic.Quotient (as i) (bs i)) :
    proj i (pi paths) = paths i 
Arguments: {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as bs : (i : ι) → X i} (i : ι) (paths : (i : ι) → Path.Homotopic.Quotient (as i) (bs i))
Goal: Path.Homotopic.proj i (Path.Homotopic.pi paths) = paths i
-- Path.Homotopic.pi_proj
@[simp]
theorem pi_proj (p : Path.Homotopic.Quotient as bs) : (pi fun i => proj i p) = p 
Arguments: {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as bs : (i : ι) → X i} (p : Path.Homotopic.Quotient as bs)
Goal: (Path.Homotopic.pi fun i => Path.Homotopic.proj i p) = p
-- Path.Homotopic.prod_lift
theorem prod_lift : prod ⟦p₁⟧ ⟦p₂⟧ = ⟦p₁.prod p₂⟧ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ a₂ : α} {b₁ b₂ : β} (p₁ : Path a₁ a₂) (p₂ : Path b₁ b₂)
Goal: Path.Homotopic.prod ⟦p₁⟧ ⟦p₂⟧ = ⟦p₁.prod p₂⟧
-- Path.Homotopic.comp_prod_eq_prod_comp
theorem comp_prod_eq_prod_comp : prod q₁ q₂ ⬝ prod r₁ r₂ = prod (q₁ ⬝ r₁) (q₂ ⬝ r₂) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ a₂ a₃ : α} {b₁ b₂ b₃ : β} (q₁ : Path.Homotopic.Quotient a₁ a₂) (q₂ : Path.Homotopic.Quotient b₁ b₂) (r₁ : Path.Homotopic.Quotient a₂ a₃) (r₂ : Path.Homotopic.Quotient b₂ b₃)
Goal: (Path.Homotopic.prod q₁ q₂).comp (Path.Homotopic.prod r₁ r₂) = Path.Homotopic.prod (q₁.comp r₁) (q₂.comp r₂)
-- Path.Homotopic.projLeft_prod
@[simp]
theorem projLeft_prod : projLeft (prod q₁ q₂) = q₁ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ a₂ : α} {b₁ b₂ : β} (q₁ : Path.Homotopic.Quotient a₁ a₂) (q₂ : Path.Homotopic.Quotient b₁ b₂)
Goal: Path.Homotopic.projLeft (Path.Homotopic.prod q₁ q₂) = q₁
-- Path.Homotopic.projRight_prod
@[simp]
theorem projRight_prod : projRight (prod q₁ q₂) = q₂ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ a₂ : α} {b₁ b₂ : β} (q₁ : Path.Homotopic.Quotient a₁ a₂) (q₂ : Path.Homotopic.Quotient b₁ b₂)
Goal: Path.Homotopic.projRight (Path.Homotopic.prod q₁ q₂) = q₂
-- Path.Homotopic.prod_projLeft_projRight
@[simp]
theorem prod_projLeft_projRight (p : Path.Homotopic.Quotient (a₁, b₁) (a₂, b₂)) :
    prod (projLeft p) (projRight p) = p 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ a₂ : α} {b₁ b₂ : β} (p : Path.Homotopic.Quotient (a₁, b₁) (a₂, b₂))
Goal: Path.Homotopic.prod (Path.Homotopic.projLeft p) (Path.Homotopic.projRight p) = p
-- Int.dist_eq
theorem dist_eq (x y : ℤ) : dist x y = |(x : ℝ) - y| 
Arguments: (x y : ℤ)
Goal: dist x y = |↑x - ↑y|
-- Int.dist_eq'
theorem dist_eq' (m n : ℤ) : dist m n = |m - n| 
Arguments: (m n : ℤ)
Goal: dist m n = ↑|m - n|
-- Int.dist_cast_real
@[norm_cast, simp]
theorem dist_cast_real (x y : ℤ) : dist (x : ℝ) y = dist x y 
Arguments: (x y : ℤ)
Goal: dist ↑x ↑y = dist x y
-- Int.pairwise_one_le_dist
theorem pairwise_one_le_dist : Pairwise fun m n : ℤ => 1 ≤ dist m n 
Goal: Pairwise fun m n => 1 ≤ dist m n
-- Int.uniformEmbedding_coe_real
theorem uniformEmbedding_coe_real : UniformEmbedding ((↑) : ℤ → ℝ) 
Goal: UniformEmbedding Int.cast
-- Int.closedEmbedding_coe_real
theorem closedEmbedding_coe_real : ClosedEmbedding ((↑) : ℤ → ℝ) 
Goal: ClosedEmbedding Int.cast
-- Int.preimage_ball
theorem preimage_ball (x : ℤ) (r : ℝ) : (↑) ⁻¹' ball (x : ℝ) r = ball x r 
Arguments: (x : ℤ) (r : ℝ)
Goal: Int.cast ⁻¹' Metric.ball (↑x) r = Metric.ball x r
-- Int.preimage_closedBall
theorem preimage_closedBall (x : ℤ) (r : ℝ) : (↑) ⁻¹' closedBall (x : ℝ) r = closedBall x r 
Arguments: (x : ℤ) (r : ℝ)
Goal: Int.cast ⁻¹' Metric.closedBall (↑x) r = Metric.closedBall x r
-- Int.ball_eq_Ioo
theorem ball_eq_Ioo (x : ℤ) (r : ℝ) : ball x r = Ioo ⌊↑x - r⌋ ⌈↑x + r⌉ 
Arguments: (x : ℤ) (r : ℝ)
Goal: Metric.ball x r = Set.Ioo ⌊↑x - r⌋ ⌈↑x + r⌉
-- Int.closedBall_eq_Icc
theorem closedBall_eq_Icc (x : ℤ) (r : ℝ) : closedBall x r = Icc ⌈↑x - r⌉ ⌊↑x + r⌋ 
Arguments: (x : ℤ) (r : ℝ)
Goal: Metric.closedBall x r = Set.Icc ⌈↑x - r⌉ ⌊↑x + r⌋
-- Int.cobounded_eq
@[simp]
theorem cobounded_eq : Bornology.cobounded ℤ = atBot ⊔ atTop 
Goal: Bornology.cobounded ℤ = Filter.atBot ⊔ Filter.atTop
-- Int.cofinite_eq
@[simp]
theorem cofinite_eq : (cofinite : Filter ℤ) = atBot ⊔ atTop 
Goal: Filter.cofinite = Filter.atBot ⊔ Filter.atTop
-- HSpace.hmul_e_e
class HSpace (X : Type u) [TopologicalSpace X] where
  hmul : C(X × X, X)
  e : X
  hmul_e_e : hmul (e, e) = e
  eHmul :
    (hmul.comp <| (const X e).prodMk <| ContinuousMap.id X).HomotopyRel (ContinuousMap.id X) {e}
  hmulE :
    (hmul.comp <| (ContinuousMap.id X).prodMk <| const X e).HomotopyRel (ContinuousMap.id X) {e}
Arguments: {X : Type u} [TopologicalSpace X] [self : HSpace X]
Goal: HSpace.hmul (HSpace.e, HSpace.e) = HSpace.e
-- TopologicalGroup.one_eq_hSpace_e
theorem one_eq_hSpace_e {G : Type u} [TopologicalSpace G] [Group G] [TopologicalGroup G] :
    (1 : G) = HSpace.e 
Arguments: {G : Type u} [TopologicalSpace G] [Group G] [TopologicalGroup G]
Goal: 1 = HSpace.e
-- unitInterval.continuous_qRight
theorem continuous_qRight : Continuous qRight 
Goal: Continuous unitInterval.qRight
-- unitInterval.qRight_zero_left
theorem qRight_zero_left (θ : I) : qRight (0, θ) = 0 
Arguments: (θ : ↑I)
Goal: unitInterval.qRight (0, θ) = 0
-- unitInterval.qRight_one_left
theorem qRight_one_left (θ : I) : qRight (1, θ) = 1 
Arguments: (θ : ↑I)
Goal: unitInterval.qRight (1, θ) = 1
-- unitInterval.qRight_zero_right
theorem qRight_zero_right (t : I) :
    (qRight (t, 0) : ℝ) = if (t : ℝ) ≤ 1 / 2 then (2 : ℝ) * t else 1 
Arguments: (t : ↑I)
Goal: ↑(unitInterval.qRight (t, 0)) = if ↑t ≤ 1 / 2 then 2 * ↑t else 1
-- unitInterval.qRight_one_right
theorem qRight_one_right (t : I) : qRight (t, 1) = t 
Arguments: (t : ↑I)
Goal: unitInterval.qRight (t, 1) = t
-- Path.continuous_delayReflRight
theorem continuous_delayReflRight : Continuous fun p : I × Path x y => delayReflRight p.1 p.2 
Arguments: {X : Type u} [TopologicalSpace X] {x y : X}
Goal: Continuous fun p => Path.delayReflRight p.1 p.2
-- Path.delayReflRight_zero
theorem delayReflRight_zero (γ : Path x y) : delayReflRight 0 γ = γ.trans (Path.refl y) 
Arguments: {X : Type u} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: Path.delayReflRight 0 γ = γ.trans (Path.refl y)
-- Path.delayReflRight_one
theorem delayReflRight_one (γ : Path x y) : delayReflRight 1 γ = γ 
Arguments: {X : Type u} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: Path.delayReflRight 1 γ = γ
-- Path.continuous_delayReflLeft
theorem continuous_delayReflLeft : Continuous fun p : I × Path x y => delayReflLeft p.1 p.2 
Arguments: {X : Type u} [TopologicalSpace X] {x y : X}
Goal: Continuous fun p => Path.delayReflLeft p.1 p.2
-- Path.delayReflLeft_zero
theorem delayReflLeft_zero (γ : Path x y) : delayReflLeft 0 γ = (Path.refl x).trans γ 
Arguments: {X : Type u} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: Path.delayReflLeft 0 γ = (Path.refl x).trans γ
-- Path.delayReflLeft_one
theorem delayReflLeft_one (γ : Path x y) : delayReflLeft 1 γ = γ 
Arguments: {X : Type u} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: Path.delayReflLeft 1 γ = γ
-- IsIrreducible.nonempty
theorem IsIrreducible.nonempty (h : IsIrreducible s) : s.Nonempty 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (h : IsIrreducible s)
Goal: s.Nonempty
-- IsIrreducible.isPreirreducible
theorem IsIrreducible.isPreirreducible (h : IsIrreducible s) : IsPreirreducible s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (h : IsIrreducible s)
Goal: IsPreirreducible s
-- isPreirreducible_empty
theorem isPreirreducible_empty : IsPreirreducible (∅ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsPreirreducible ∅
-- Set.Subsingleton.isPreirreducible
theorem Set.Subsingleton.isPreirreducible (hs : s.Subsingleton) : IsPreirreducible s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : s.Subsingleton)
Goal: IsPreirreducible s
-- isPreirreducible_singleton
theorem isPreirreducible_singleton {x} : IsPreirreducible ({x} : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: IsPreirreducible {x}
-- isIrreducible_singleton
theorem isIrreducible_singleton {x} : IsIrreducible ({x} : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: IsIrreducible {x}
-- isPreirreducible_iff_closure
theorem isPreirreducible_iff_closure : IsPreirreducible (closure s) ↔ IsPreirreducible s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsPreirreducible (closure s) ↔ IsPreirreducible s
-- isIrreducible_iff_closure
theorem isIrreducible_iff_closure : IsIrreducible (closure s) ↔ IsIrreducible s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsIrreducible (closure s) ↔ IsIrreducible s
-- exists_preirreducible
theorem exists_preirreducible (s : Set X) (H : IsPreirreducible s) :
    ∃ t : Set X, IsPreirreducible t ∧ s ⊆ t ∧ ∀ u, IsPreirreducible u → t ⊆ u → u = t 
Arguments: {X : Type u_1} [TopologicalSpace X] (s : Set X) (H : IsPreirreducible s)
Goal: ∃ t, IsPreirreducible t ∧ s ⊆ t ∧ ∀ (u : Set X), IsPreirreducible u → t ⊆ u → u = t
-- isClosed_of_mem_irreducibleComponents
theorem isClosed_of_mem_irreducibleComponents (s) (H : s ∈ irreducibleComponents X) :
    IsClosed s 
Arguments: {X : Type u_1} [TopologicalSpace X] (s : Set X) (H : s ∈ irreducibleComponents X)
Goal: IsClosed s
-- irreducibleComponents_eq_maximals_closed
theorem irreducibleComponents_eq_maximals_closed (X : Type*) [TopologicalSpace X] :
    irreducibleComponents X = maximals (· ≤ ·) { s : Set X | IsClosed s ∧ IsIrreducible s } 
Arguments: (X : Type u_3) [TopologicalSpace X]
Goal: irreducibleComponents X = maximals (fun x x_1 => x ≤ x_1) {s | IsClosed s ∧ IsIrreducible s}
-- irreducibleComponent_property
theorem irreducibleComponent_property (x : X) :
    IsPreirreducible (irreducibleComponent x) ∧
      {x} ⊆ irreducibleComponent x ∧
        ∀ u, IsPreirreducible u → irreducibleComponent x ⊆ u → u = irreducibleComponent x 
Arguments: {X : Type u_1} [TopologicalSpace X] (x : X)
Goal: IsPreirreducible (irreducibleComponent x) ∧
  {x} ⊆ irreducibleComponent x ∧
    ∀ (u : Set X), IsPreirreducible u → irreducibleComponent x ⊆ u → u = irreducibleComponent x
-- mem_irreducibleComponent
theorem mem_irreducibleComponent {x : X} : x ∈ irreducibleComponent x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: x ∈ irreducibleComponent x
-- isIrreducible_irreducibleComponent
theorem isIrreducible_irreducibleComponent {x : X} : IsIrreducible (irreducibleComponent x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: IsIrreducible (irreducibleComponent x)
-- eq_irreducibleComponent
theorem eq_irreducibleComponent {x : X} :
    IsPreirreducible s → irreducibleComponent x ⊆ s → s = irreducibleComponent x 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} {x : X}
Goal: IsPreirreducible s → irreducibleComponent x ⊆ s → s = irreducibleComponent x
-- irreducibleComponent_mem_irreducibleComponents
theorem irreducibleComponent_mem_irreducibleComponents (x : X) :
    irreducibleComponent x ∈ irreducibleComponents X 
Arguments: {X : Type u_1} [TopologicalSpace X] (x : X)
Goal: irreducibleComponent x ∈ irreducibleComponents X
-- isClosed_irreducibleComponent
theorem isClosed_irreducibleComponent {x : X} : IsClosed (irreducibleComponent x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: IsClosed (irreducibleComponent x)
-- PreirreducibleSpace.isPreirreducible_univ
class PreirreducibleSpace (X : Type*) [TopologicalSpace X] : Prop where
    isPreirreducible_univ : IsPreirreducible (univ : Set X)
Arguments: {X : Type u_3} [TopologicalSpace X] [self : PreirreducibleSpace X]
Goal: IsPreirreducible Set.univ
-- IrreducibleSpace.toNonempty
class IrreducibleSpace (X : Type*) [TopologicalSpace X] extends PreirreducibleSpace X : Prop where
  toNonempty : Nonempty X
Arguments: {X : Type u_3} [TopologicalSpace X] [self : IrreducibleSpace X]
Goal: Nonempty X
-- IrreducibleSpace.isIrreducible_univ
theorem IrreducibleSpace.isIrreducible_univ (X : Type*) [TopologicalSpace X] [IrreducibleSpace X] :
    IsIrreducible (univ : Set X) 
Arguments: (X : Type u_3) [TopologicalSpace X] [IrreducibleSpace X]
Goal: IsIrreducible Set.univ
-- irreducibleSpace_def
theorem irreducibleSpace_def (X : Type*) [TopologicalSpace X] :
    IrreducibleSpace X ↔ IsIrreducible (⊤ : Set X) 
Arguments: (X : Type u_3) [TopologicalSpace X]
Goal: IrreducibleSpace X ↔ IsIrreducible ⊤
-- nonempty_preirreducible_inter
theorem nonempty_preirreducible_inter [PreirreducibleSpace X] :
    IsOpen s → IsOpen t → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty 
Arguments: {X : Type u_1} [TopologicalSpace X] {s t : Set X} [PreirreducibleSpace X]
Goal: IsOpen s → IsOpen t → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty
-- IsOpen.dense
protected theorem IsOpen.dense [PreirreducibleSpace X] (ho : IsOpen s) (hne : s.Nonempty) :
    Dense s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} [PreirreducibleSpace X] (ho : IsOpen s) (hne : s.Nonempty)
Goal: Dense s
-- IsPreirreducible.image
theorem IsPreirreducible.image (H : IsPreirreducible s) (f : X → Y) (hf : ContinuousOn f s) :
    IsPreirreducible (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} (H : IsPreirreducible s) (f : X → Y) (hf : ContinuousOn f s)
Goal: IsPreirreducible (f '' s)
-- IsIrreducible.image
theorem IsIrreducible.image (H : IsIrreducible s) (f : X → Y) (hf : ContinuousOn f s) :
    IsIrreducible (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} (H : IsIrreducible s) (f : X → Y) (hf : ContinuousOn f s)
Goal: IsIrreducible (f '' s)
-- Subtype.preirreducibleSpace
theorem Subtype.preirreducibleSpace (h : IsPreirreducible s) : PreirreducibleSpace s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (h : IsPreirreducible s)
Goal: PreirreducibleSpace ↑s
-- Subtype.irreducibleSpace
theorem Subtype.irreducibleSpace (h : IsIrreducible s) : IrreducibleSpace s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (h : IsIrreducible s)
Goal: IrreducibleSpace ↑s
-- isIrreducible_iff_sInter
theorem isIrreducible_iff_sInter :
    IsIrreducible s ↔
      ∀ (U : Finset (Set X)), (∀ u ∈ U, IsOpen u) → (∀ u ∈ U, (s ∩ u).Nonempty) →
        (s ∩ ⋂₀ ↑U).Nonempty 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsIrreducible s ↔ ∀ (U : Finset (Set X)), (∀ u ∈ U, IsOpen u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty
-- isPreirreducible_iff_closed_union_closed
theorem isPreirreducible_iff_closed_union_closed :
    IsPreirreducible s ↔
      ∀ z₁ z₂ : Set X, IsClosed z₁ → IsClosed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂ 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsPreirreducible s ↔ ∀ (z₁ z₂ : Set X), IsClosed z₁ → IsClosed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂
-- isIrreducible_iff_sUnion_closed
theorem isIrreducible_iff_sUnion_closed :
    IsIrreducible s ↔
      ∀ t : Finset (Set X), (∀ z ∈ t, IsClosed z) → (s ⊆ ⋃₀ ↑t) → ∃ z ∈ t, s ⊆ z 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsIrreducible s ↔ ∀ (t : Finset (Set X)), (∀ z ∈ t, IsClosed z) → s ⊆ ⋃₀ ↑t → ∃ z ∈ t, s ⊆ z
-- subset_closure_inter_of_isPreirreducible_of_isOpen
theorem subset_closure_inter_of_isPreirreducible_of_isOpen {S U : Set X} (hS : IsPreirreducible S)
    (hU : IsOpen U) (h : (S ∩ U).Nonempty) : S ⊆ closure (S ∩ U) 
Arguments: {X : Type u_1} [TopologicalSpace X] {S U : Set X} (hS : IsPreirreducible S) (hU : IsOpen U) (h : (S ∩ U).Nonempty)
Goal: S ⊆ closure (S ∩ U)
-- IsPreirreducible.subset_irreducible
theorem IsPreirreducible.subset_irreducible {S U : Set X} (ht : IsPreirreducible t)
    (hU : U.Nonempty) (hU' : IsOpen U) (h₁ : U ⊆ S) (h₂ : S ⊆ t) : IsIrreducible S 
Arguments: {X : Type u_1} [TopologicalSpace X] {t S U : Set X} (ht : IsPreirreducible t) (hU : U.Nonempty) (hU' : IsOpen U) (h₁ : U ⊆ S) (h₂ : S ⊆ t)
Goal: IsIrreducible S
-- IsPreirreducible.open_subset
theorem IsPreirreducible.open_subset {U : Set X} (ht : IsPreirreducible t) (hU : IsOpen U)
    (hU' : U ⊆ t) : IsPreirreducible U 
Arguments: {X : Type u_1} [TopologicalSpace X] {t U : Set X} (ht : IsPreirreducible t) (hU : IsOpen U) (hU' : U ⊆ t)
Goal: IsPreirreducible U
-- IsPreirreducible.interior
theorem IsPreirreducible.interior (ht : IsPreirreducible t) : IsPreirreducible (interior t) 
Arguments: {X : Type u_1} [TopologicalSpace X] {t : Set X} (ht : IsPreirreducible t)
Goal: IsPreirreducible (interior t)
-- IsPreirreducible.preimage
theorem IsPreirreducible.preimage (ht : IsPreirreducible t) {f : Y → X}
    (hf : OpenEmbedding f) : IsPreirreducible (f ⁻¹' t) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {t : Set X} (ht : IsPreirreducible t) {f : Y → X} (hf : OpenEmbedding f)
Goal: IsPreirreducible (f ⁻¹' t)
-- Mathlib.Tactic.Ring.add_overlap_pf
theorem add_overlap_pf (x : R) (e) (pq_pf : a + b = c) :
    x ^ e * a + x ^ e * b = x ^ e * c 
Arguments: {R : Type u_1} [CommSemiring R] {a b c : R} (x : R) (e : ℕ) (pq_pf : a + b = c)
Goal: x ^ e * a + x ^ e * b = x ^ e * c
-- Mathlib.Tactic.Ring.add_overlap_pf_zero
theorem add_overlap_pf_zero (x : R) (e) :
    IsNat (a + b) (nat_lit 0) → IsNat (x ^ e * a + x ^ e * b) (nat_lit 0)
  
Arguments: {R : Type u_1} [CommSemiring R] {a b : R} (x : R) (e : ℕ)
Goal: Mathlib.Meta.NormNum.IsNat (a + b) 0 → Mathlib.Meta.NormNum.IsNat (x ^ e * a + x ^ e * b) 0
-- Mathlib.Tactic.Ring.add_pf_zero_add
theorem add_pf_zero_add (b : R) : 0 + b = b 
Arguments: {R : Type u_1} [CommSemiring R] (b : R)
Goal: 0 + b = b
-- Mathlib.Tactic.Ring.add_pf_add_zero
theorem add_pf_add_zero (a : R) : a + 0 = a 
Arguments: {R : Type u_1} [CommSemiring R] (a : R)
Goal: a + 0 = a
-- Mathlib.Tactic.Ring.add_pf_add_overlap
theorem add_pf_add_overlap
    (_ : a₁ + b₁ = c₁) (_ : a₂ + b₂ = c₂) : (a₁ + a₂ : R) + (b₁ + b₂) = c₁ + c₂ 
Arguments: {R : Type u_1} [CommSemiring R] {a₁ a₂ b₁ b₂ c₁ c₂ : R}
Goal: a₁ + b₁ = c₁ → a₂ + b₂ = c₂ → a₁ + a₂ + (b₁ + b₂) = c₁ + c₂