ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Specializes.fst
theorem Specializes.fst {a b : X × Y} (h : a ⤳ b) : a.1 ⤳ b.1 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {a b : X × Y} (h : a ⤳ b)
Goal: a.1 ⤳ b.1
-- Specializes.snd
theorem Specializes.snd {a b : X × Y} (h : a ⤳ b) : a.2 ⤳ b.2 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {a b : X × Y} (h : a ⤳ b)
Goal: a.2 ⤳ b.2
-- specializes_pi
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [(i : ι) → TopologicalSpace (π i)] {f g : (i : ι) → π i}
Goal: f ⤳ g ↔ ∀ (i : ι), f i ⤳ g i
-- not_specializes_iff_exists_open
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: ¬x ⤳ y ↔ ∃ S, IsOpen S ∧ y ∈ S ∧ x ∉ S
-- not_specializes_iff_exists_closed
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: ¬x ⤳ y ↔ ∃ S, IsClosed S ∧ x ∈ S ∧ y ∉ S
-- IsOpen.continuous_piecewise_of_specializes
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
    (hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
    Continuous (s.piecewise f g) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f g : X → Y} [DecidablePred fun x => x ∈ s] (hs : IsOpen s) (hf : Continuous f) (hg : Continuous g) (hspec : ∀ (x : X), f x ⤳ g x)
Goal: Continuous (s.piecewise f g)
-- IsClosed.continuous_piecewise_of_specializes
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
    (hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
    Continuous (s.piecewise f g) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f g : X → Y} [DecidablePred fun x => x ∈ s] (hs : IsClosed s) (hf : Continuous f) (hg : Continuous g) (hspec : ∀ (x : X), g x ⤳ f x)
Goal: Continuous (s.piecewise f g)
-- Continuous.specialization_monotone
theorem Continuous.specialization_monotone (hf : Continuous f) : Monotone f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f)
Goal: Monotone f
-- closure_singleton_eq_Iic
lemma closure_singleton_eq_Iic (x : X) : closure {x} = Iic x 
Arguments: {X : Type u_1} [TopologicalSpace X] (x : X)
Goal: closure {x} = Set.Iic x
-- IsClosed.stableUnderSpecialization
lemma IsClosed.stableUnderSpecialization {s : Set X} (hs : IsClosed s) :
    StableUnderSpecialization s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsClosed s)
Goal: StableUnderSpecialization s
-- IsOpen.stableUnderGeneralization
lemma IsOpen.stableUnderGeneralization {s : Set X} (hs : IsOpen s) :
    StableUnderGeneralization s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsOpen s)
Goal: StableUnderGeneralization s
-- stableUnderSpecialization_compl_iff
@[simp]
lemma stableUnderSpecialization_compl_iff {s : Set X} :
    StableUnderSpecialization sᶜ ↔ StableUnderGeneralization s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: StableUnderSpecialization sᶜ ↔ StableUnderGeneralization s
-- stableUnderGeneralization_compl_iff
@[simp]
lemma stableUnderGeneralization_compl_iff {s : Set X} :
    StableUnderGeneralization sᶜ ↔ StableUnderSpecialization s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: StableUnderGeneralization sᶜ ↔ StableUnderSpecialization s
-- stableUnderSpecialization_univ
lemma stableUnderSpecialization_univ : StableUnderSpecialization (univ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: StableUnderSpecialization Set.univ
-- stableUnderSpecialization_empty
lemma stableUnderSpecialization_empty : StableUnderSpecialization (∅ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: StableUnderSpecialization ∅
-- stableUnderGeneralization_univ
lemma stableUnderGeneralization_univ : StableUnderGeneralization (univ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: StableUnderGeneralization Set.univ
-- stableUnderGeneralization_empty
lemma stableUnderGeneralization_empty : StableUnderGeneralization (∅ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: StableUnderGeneralization ∅
-- stableUnderSpecialization_sUnion
lemma stableUnderSpecialization_sUnion (S : Set (Set X))
    (H : ∀ s ∈ S, StableUnderSpecialization s) : StableUnderSpecialization (⋃₀ S) 
Arguments: {X : Type u_1} [TopologicalSpace X] (S : Set (Set X)) (H : ∀ s ∈ S, StableUnderSpecialization s)
Goal: StableUnderSpecialization (⋃₀ S)
-- stableUnderSpecialization_sInter
lemma stableUnderSpecialization_sInter (S : Set (Set X))
    (H : ∀ s ∈ S, StableUnderSpecialization s) : StableUnderSpecialization (⋂₀ S) 
Arguments: {X : Type u_1} [TopologicalSpace X] (S : Set (Set X)) (H : ∀ s ∈ S, StableUnderSpecialization s)
Goal: StableUnderSpecialization (⋂₀ S)
-- stableUnderGeneralization_sUnion
lemma stableUnderGeneralization_sUnion (S : Set (Set X))
    (H : ∀ s ∈ S, StableUnderGeneralization s) : StableUnderGeneralization (⋃₀ S) 
Arguments: {X : Type u_1} [TopologicalSpace X] (S : Set (Set X)) (H : ∀ s ∈ S, StableUnderGeneralization s)
Goal: StableUnderGeneralization (⋃₀ S)
-- stableUnderGeneralization_sInter
lemma stableUnderGeneralization_sInter (S : Set (Set X))
    (H : ∀ s ∈ S, StableUnderGeneralization s) : StableUnderGeneralization (⋂₀ S) 
Arguments: {X : Type u_1} [TopologicalSpace X] (S : Set (Set X)) (H : ∀ s ∈ S, StableUnderGeneralization s)
Goal: StableUnderGeneralization (⋂₀ S)
-- stableUnderSpecialization_iUnion
lemma stableUnderSpecialization_iUnion {ι : Sort*} (S : ι → Set X)
    (H : ∀ i, StableUnderSpecialization (S i)) : StableUnderSpecialization (⋃ i, S i) 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : Sort u_7} (S : ι → Set X) (H : ∀ (i : ι), StableUnderSpecialization (S i))
Goal: StableUnderSpecialization (⋃ i, S i)
-- stableUnderSpecialization_iInter
lemma stableUnderSpecialization_iInter {ι : Sort*} (S : ι → Set X)
    (H : ∀ i, StableUnderSpecialization (S i)) : StableUnderSpecialization (⋂ i, S i) 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : Sort u_7} (S : ι → Set X) (H : ∀ (i : ι), StableUnderSpecialization (S i))
Goal: StableUnderSpecialization (⋂ i, S i)
-- stableUnderGeneralization_iUnion
lemma stableUnderGeneralization_iUnion {ι : Sort*} (S : ι → Set X)
    (H : ∀ i, StableUnderGeneralization (S i)) : StableUnderGeneralization (⋃ i, S i) 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : Sort u_7} (S : ι → Set X) (H : ∀ (i : ι), StableUnderGeneralization (S i))
Goal: StableUnderGeneralization (⋃ i, S i)
-- stableUnderGeneralization_iInter
lemma stableUnderGeneralization_iInter {ι : Sort*} (S : ι → Set X)
    (H : ∀ i, StableUnderGeneralization (S i)) : StableUnderGeneralization (⋂ i, S i) 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : Sort u_7} (S : ι → Set X) (H : ∀ (i : ι), StableUnderGeneralization (S i))
Goal: StableUnderGeneralization (⋂ i, S i)
-- Union_closure_singleton_eq_iff
lemma Union_closure_singleton_eq_iff {s : Set X} :
    (⋃ x ∈ s, closure {x}) = s ↔ StableUnderSpecialization s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: ⋃ x ∈ s, closure {x} = s ↔ StableUnderSpecialization s
-- stableUnderSpecialization_iff_Union_eq
lemma stableUnderSpecialization_iff_Union_eq {s : Set X} :
    StableUnderSpecialization s ↔ (⋃ x ∈ s, closure {x}) = s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: StableUnderSpecialization s ↔ ⋃ x ∈ s, closure {x} = s
-- stableUnderSpecialization_iff_exists_sUnion_eq
lemma stableUnderSpecialization_iff_exists_sUnion_eq {s : Set X} :
    StableUnderSpecialization s ↔ ∃ (S : Set (Set X)), (∀ s ∈ S, IsClosed s) ∧ ⋃₀ S = s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: StableUnderSpecialization s ↔ ∃ S, (∀ s ∈ S, IsClosed s) ∧ ⋃₀ S = s
-- stableUnderGeneralization_iff_exists_sInter_eq
lemma stableUnderGeneralization_iff_exists_sInter_eq {s : Set X} :
    StableUnderGeneralization s ↔ ∃ (S : Set (Set X)), (∀ s ∈ S, IsOpen s) ∧ ⋂₀ S = s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: StableUnderGeneralization s ↔ ∃ S, (∀ s ∈ S, IsOpen s) ∧ ⋂₀ S = s
-- StableUnderSpecialization.preimage
lemma StableUnderSpecialization.preimage {s : Set Y}
    (hs : StableUnderSpecialization s) (hf : Continuous f) :
    StableUnderSpecialization (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set Y} (hs : StableUnderSpecialization s) (hf : Continuous f)
Goal: StableUnderSpecialization (f ⁻¹' s)
-- StableUnderGeneralization.preimage
lemma StableUnderGeneralization.preimage {s : Set Y}
    (hs : StableUnderGeneralization s) (hf : Continuous f) :
    StableUnderGeneralization (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set Y} (hs : StableUnderGeneralization s) (hf : Continuous f)
Goal: StableUnderGeneralization (f ⁻¹' s)
-- specializingMap_iff_closure_singleton_subset
lemma specializingMap_iff_closure_singleton_subset :
    SpecializingMap f ↔ ∀ x, closure {f x} ⊆ f '' closure {x} 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: SpecializingMap f ↔ ∀ (x : X), closure {f x} ⊆ f '' closure {x}
-- SpecializingMap.stableUnderSpecialization_image
lemma SpecializingMap.stableUnderSpecialization_image (hf : SpecializingMap f)
    {s : Set X} (hs : StableUnderSpecialization s) : StableUnderSpecialization (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : SpecializingMap f) {s : Set X} (hs : StableUnderSpecialization s)
Goal: StableUnderSpecialization (f '' s)
-- specializingMap_iff_stableUnderSpecialization_image_singleton
lemma specializingMap_iff_stableUnderSpecialization_image_singleton :
    SpecializingMap f ↔ ∀ x, StableUnderSpecialization (f '' closure {x}) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: SpecializingMap f ↔ ∀ (x : X), StableUnderSpecialization (f '' closure {x})
-- specializingMap_iff_stableUnderSpecialization_image
lemma specializingMap_iff_stableUnderSpecialization_image :
    SpecializingMap f ↔ ∀ s, StableUnderSpecialization s → StableUnderSpecialization (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: SpecializingMap f ↔ ∀ (s : Set X), StableUnderSpecialization s → StableUnderSpecialization (f '' s)
-- specializingMap_iff_closure_singleton
lemma specializingMap_iff_closure_singleton (hf : Continuous f) :
    SpecializingMap f ↔ ∀ x, f '' closure {x} = closure {f x} 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f)
Goal: SpecializingMap f ↔ ∀ (x : X), f '' closure {x} = closure {f x}
-- specializingMap_iff_isClosed_image_closure_singleton
lemma specializingMap_iff_isClosed_image_closure_singleton (hf : Continuous f) :
    SpecializingMap f ↔ ∀ x, IsClosed (f '' closure {x}) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f)
Goal: SpecializingMap f ↔ ∀ (x : X), IsClosed (f '' closure {x})
-- IsClosedMap.specializingMap
lemma IsClosedMap.specializingMap (hf : IsClosedMap f) : SpecializingMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsClosedMap f)
Goal: SpecializingMap f
-- Inducing.specializingMap
lemma Inducing.specializingMap (hf : Inducing f) (h : StableUnderSpecialization (range f)) :
    SpecializingMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Inducing f) (h : StableUnderSpecialization (Set.range f))
Goal: SpecializingMap f
-- Inducing.generalizingMap
lemma Inducing.generalizingMap (hf : Inducing f) (h : StableUnderGeneralization (range f)) :
    GeneralizingMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Inducing f) (h : StableUnderGeneralization (Set.range f))
Goal: GeneralizingMap f
-- OpenEmbedding.generalizingMap
lemma OpenEmbedding.generalizingMap (hf : OpenEmbedding f) : GeneralizingMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : OpenEmbedding f)
Goal: GeneralizingMap f
-- SpecializingMap.stableUnderSpecialization_range
lemma SpecializingMap.stableUnderSpecialization_range (h : SpecializingMap f) :
    StableUnderSpecialization (range f) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (h : SpecializingMap f)
Goal: StableUnderSpecialization (Set.range f)
-- GeneralizingMap.stableUnderGeneralization_image
lemma GeneralizingMap.stableUnderGeneralization_image (hf : GeneralizingMap f) {s : Set X}
    (hs : StableUnderGeneralization s) : StableUnderGeneralization (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : GeneralizingMap f) {s : Set X} (hs : StableUnderGeneralization s)
Goal: StableUnderGeneralization (f '' s)
-- GeneralizingMap_iff_stableUnderGeneralization_image
lemma GeneralizingMap_iff_stableUnderGeneralization_image :
    GeneralizingMap f ↔ ∀ s, StableUnderGeneralization s → StableUnderGeneralization (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: GeneralizingMap f ↔ ∀ (s : Set X), StableUnderGeneralization s → StableUnderGeneralization (f '' s)
-- GeneralizingMap.stableUnderGeneralization_range
lemma GeneralizingMap.stableUnderGeneralization_range (h : GeneralizingMap f) :
    StableUnderGeneralization (range f) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (h : GeneralizingMap f)
Goal: StableUnderGeneralization (Set.range f)
-- inseparable_def
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y
-- inseparable_iff_specializes_and
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x
-- Inseparable.specializes
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (h : x ~ᵢ y)
Goal: x ⤳ y
-- Inseparable.specializes'
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (h : x ~ᵢ y)
Goal: y ⤳ x
-- Specializes.antisymm
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (h₁ : x ⤳ y) (h₂ : y ⤳ x)
Goal: x ~ᵢ y
-- inseparable_iff_forall_open
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: (x ~ᵢ y) ↔ ∀ (s : Set X), IsOpen s → (x ∈ s ↔ y ∈ s)
-- not_inseparable_iff_exists_open
theorem not_inseparable_iff_exists_open :
    ¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: ¬(x ~ᵢ y) ↔ ∃ s, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s)
-- inseparable_iff_forall_closed
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: (x ~ᵢ y) ↔ ∀ (s : Set X), IsClosed s → (x ∈ s ↔ y ∈ s)
-- inseparable_iff_mem_closure
theorem inseparable_iff_mem_closure :
    (x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: (x ~ᵢ y) ↔ x ∈ closure {y} ∧ y ∈ closure {x}
-- inseparable_iff_closure_eq
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: (x ~ᵢ y) ↔ closure {x} = closure {y}
-- inseparable_of_nhdsWithin_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {s : Set X} (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y)
Goal: x ~ᵢ y
-- Inducing.inseparable_iff
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {f : X → Y} (hf : Inducing f)
Goal: (f x ~ᵢ f y) ↔ (x ~ᵢ y)
-- subtype_inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) 
Arguments: {X : Type u_1} [TopologicalSpace X] {p : X → Prop} (x y : Subtype p)
Goal: (x ~ᵢ y) ↔ (↑x ~ᵢ ↑y)
-- inseparable_prod
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
    ((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x₁ x₂ : X} {y₁ y₂ : Y}
Goal: ((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂)
-- Inseparable.prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
    (x₁, y₁) ~ᵢ (x₂, y₂) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂)
Goal: (x₁, y₁) ~ᵢ (x₂, y₂)
-- inseparable_pi
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [(i : ι) → TopologicalSpace (π i)] {f g : (i : ι) → π i}
Goal: (f ~ᵢ g) ↔ ∀ (i : ι), f i ~ᵢ g i
-- Inseparable.refl
@[refl]
theorem refl (x : X) : x ~ᵢ x 
Arguments: {X : Type u_1} [TopologicalSpace X] (x : X)
Goal: x ~ᵢ x
-- Inseparable.rfl
theorem rfl : x ~ᵢ x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: x ~ᵢ x
-- Inseparable.of_eq
theorem of_eq (e : x = y) : Inseparable x y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (e : x = y)
Goal: x ~ᵢ y
-- Inseparable.symm
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (h : x ~ᵢ y)
Goal: y ~ᵢ x
-- Inseparable.trans
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y z : X} (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z)
Goal: x ~ᵢ z
-- Inseparable.nhds_eq
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (h : x ~ᵢ y)
Goal: 𝓝 x = 𝓝 y
-- Inseparable.mem_open_iff
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {s : Set X} (h : x ~ᵢ y) (hs : IsOpen s)
Goal: x ∈ s ↔ y ∈ s
-- Inseparable.mem_closed_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {s : Set X} (h : x ~ᵢ y) (hs : IsClosed s)
Goal: x ∈ s ↔ y ∈ s
-- Inseparable.map_of_continuousAt
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
    f x ~ᵢ f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {f : X → Y} (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y)
Goal: f x ~ᵢ f y
-- Inseparable.map
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {f : X → Y} (h : x ~ᵢ y) (hf : Continuous f)
Goal: f x ~ᵢ f y
-- IsClosed.not_inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {s : Set X} (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s)
Goal: ¬(x ~ᵢ y)
-- IsOpen.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {s : Set X} (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s)
Goal: ¬(x ~ᵢ y)
-- SeparationQuotient.quotientMap_mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: QuotientMap SeparationQuotient.mk
-- SeparationQuotient.continuous_mk
@[fun_prop, continuity]
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Continuous SeparationQuotient.mk
-- SeparationQuotient.mk_eq_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: SeparationQuotient.mk x = SeparationQuotient.mk y ↔ (x ~ᵢ y)
-- SeparationQuotient.surjective_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Function.Surjective SeparationQuotient.mk
-- SeparationQuotient.range_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Set.range SeparationQuotient.mk = Set.univ
-- SeparationQuotient.preimage_image_mk_open
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsOpen s)
Goal: SeparationQuotient.mk ⁻¹' (SeparationQuotient.mk '' s) = s
-- SeparationQuotient.isOpenMap_mk
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsOpenMap SeparationQuotient.mk
-- SeparationQuotient.preimage_image_mk_closed
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsClosed s)
Goal: SeparationQuotient.mk ⁻¹' (SeparationQuotient.mk '' s) = s
-- SeparationQuotient.inducing_mk
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Inducing SeparationQuotient.mk
-- SeparationQuotient.isClosedMap_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsClosedMap SeparationQuotient.mk
-- SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: Filter.comap SeparationQuotient.mk (𝓝 (SeparationQuotient.mk x)) = 𝓝 x
-- SeparationQuotient.comap_mk_nhdsSet_image
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: Filter.comap SeparationQuotient.mk (𝓝ˢ (SeparationQuotient.mk '' s)) = 𝓝ˢ s
-- SeparationQuotient.map_mk_nhds
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: Filter.map SeparationQuotient.mk (𝓝 x) = 𝓝 (SeparationQuotient.mk x)
-- SeparationQuotient.map_mk_nhdsSet
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: Filter.map SeparationQuotient.mk (𝓝ˢ s) = 𝓝ˢ (SeparationQuotient.mk '' s)
-- SeparationQuotient.comap_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) 
Arguments: {X : Type u_1} [TopologicalSpace X] {t : Set (SeparationQuotient X)}
Goal: Filter.comap SeparationQuotient.mk (𝓝ˢ t) = 𝓝ˢ (SeparationQuotient.mk ⁻¹' t)
-- SeparationQuotient.preimage_mk_closure
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) 
Arguments: {X : Type u_1} [TopologicalSpace X] {t : Set (SeparationQuotient X)}
Goal: SeparationQuotient.mk ⁻¹' closure t = closure (SeparationQuotient.mk ⁻¹' t)
-- SeparationQuotient.preimage_mk_interior
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) 
Arguments: {X : Type u_1} [TopologicalSpace X] {t : Set (SeparationQuotient X)}
Goal: SeparationQuotient.mk ⁻¹' interior t = interior (SeparationQuotient.mk ⁻¹' t)
-- SeparationQuotient.preimage_mk_frontier
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) 
Arguments: {X : Type u_1} [TopologicalSpace X] {t : Set (SeparationQuotient X)}
Goal: SeparationQuotient.mk ⁻¹' frontier t = frontier (SeparationQuotient.mk ⁻¹' t)