ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousMap.continuous.comp'
@[deprecated _root_.Continuous.compCM (since := "2024-01-30")]
lemma continuous.comp' (hf : Continuous f) (hg : Continuous g) :
    Continuous fun x => (g x).comp (f x) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {X' : Type u_6} [TopologicalSpace X'] {g : X' → C(Y, Z)} {f : X' → C(X, Y)} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => (g x).comp (f x)
-- ContinuousMap.continuous_eval
@[continuity]
theorem continuous_eval [LocallyCompactPair X Y] : Continuous fun p : C(X, Y) × X => p.1 p.2 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactPair X Y]
Goal: Continuous fun p => p.1 p.2
-- ContinuousMap.continuous_eval_const
@[continuity]
theorem continuous_eval_const (a : X) : Continuous fun f : C(X, Y) => f a 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (a : X)
Goal: Continuous fun f => f a
-- ContinuousMap.continuous_coe
theorem continuous_coe : Continuous ((⇑) : C(X, Y) → (X → Y)) 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Continuous DFunLike.coe
-- ContinuousMap.isClosed_setOf_mapsTo
lemma isClosed_setOf_mapsTo {t : Set Y} (ht : IsClosed t) (s : Set X) :
    IsClosed {f : C(X, Y) | MapsTo f s t} 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {t : Set Y} (ht : IsClosed t) (s : Set X)
Goal: IsClosed {f | Set.MapsTo (⇑f) s t}
-- ContinuousMap.isClopen_setOf_mapsTo
lemma isClopen_setOf_mapsTo (hK : IsCompact K) (hU : IsClopen U) :
    IsClopen {f : C(X, Y) | MapsTo f K U} 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {K : Set X} {U : Set Y} (hK : IsCompact K) (hU : IsClopen U)
Goal: IsClopen {f | Set.MapsTo (⇑f) K U}
-- ContinuousMap.specializes_coe
@[norm_cast]
lemma specializes_coe {f g : C(X, Y)} : ⇑f ⤳ ⇑g ↔ f ⤳ g 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {f g : C(X, Y)}
Goal: ⇑f ⤳ ⇑g ↔ f ⤳ g
-- ContinuousMap.inseparable_coe
@[norm_cast]
lemma inseparable_coe {f g : C(X, Y)} : Inseparable (f : X → Y) g ↔ Inseparable f g 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {f g : C(X, Y)}
Goal: Inseparable ⇑f ⇑g ↔ Inseparable f g
-- ContinuousMap.continuous_restrict
theorem continuous_restrict (s : Set X) : Continuous fun F : C(X, Y) => F.restrict s 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (s : Set X)
Goal: Continuous fun F => ContinuousMap.restrict s F
-- ContinuousMap.compactOpen_le_induced
theorem compactOpen_le_induced (s : Set X) :
    (ContinuousMap.compactOpen : TopologicalSpace C(X, Y)) ≤
      .induced (restrict s) ContinuousMap.compactOpen 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (s : Set X)
Goal: ContinuousMap.compactOpen ≤ TopologicalSpace.induced (ContinuousMap.restrict s) ContinuousMap.compactOpen
-- ContinuousMap.compactOpen_eq_iInf_induced
theorem compactOpen_eq_iInf_induced :
    (ContinuousMap.compactOpen : TopologicalSpace C(X, Y)) =
      ⨅ (K : Set X) (_ : IsCompact K), .induced (.restrict K) ContinuousMap.compactOpen 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y]
Goal: ContinuousMap.compactOpen =
  ⨅ K, ⨅ (_ : IsCompact K), TopologicalSpace.induced (ContinuousMap.restrict K) ContinuousMap.compactOpen
-- ContinuousMap.nhds_compactOpen_eq_iInf_nhds_induced
theorem nhds_compactOpen_eq_iInf_nhds_induced (f : C(X, Y)) :
    𝓝 f = ⨅ (s) (hs : IsCompact s), (𝓝 (f.restrict s)).comap (ContinuousMap.restrict s) 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y))
Goal: 𝓝 f = ⨅ s, ⨅ (_ : IsCompact s), Filter.comap (ContinuousMap.restrict s) (𝓝 (ContinuousMap.restrict s f))
-- ContinuousMap.tendsto_compactOpen_restrict
theorem tendsto_compactOpen_restrict {ι : Type*} {l : Filter ι} {F : ι → C(X, Y)} {f : C(X, Y)}
    (hFf : Filter.Tendsto F l (𝓝 f)) (s : Set X) :
    Tendsto (fun i => (F i).restrict s) l (𝓝 (f.restrict s)) 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {ι : Type u_6} {l : Filter ι} {F : ι → C(X, Y)} {f : C(X, Y)} (hFf : Filter.Tendsto F l (𝓝 f)) (s : Set X)
Goal: Filter.Tendsto (fun i => ContinuousMap.restrict s (F i)) l (𝓝 (ContinuousMap.restrict s f))
-- ContinuousMap.tendsto_compactOpen_iff_forall
theorem tendsto_compactOpen_iff_forall {ι : Type*} {l : Filter ι} (F : ι → C(X, Y)) (f : C(X, Y)) :
    Tendsto F l (𝓝 f) ↔
      ∀ K, IsCompact K → Tendsto (fun i => (F i).restrict K) l (𝓝 (f.restrict K)) 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {ι : Type u_6} {l : Filter ι} (F : ι → C(X, Y)) (f : C(X, Y))
Goal: Filter.Tendsto F l (𝓝 f) ↔
  ∀ (K : Set X),
    IsCompact K → Filter.Tendsto (fun i => ContinuousMap.restrict K (F i)) l (𝓝 (ContinuousMap.restrict K f))
-- ContinuousMap.exists_tendsto_compactOpen_iff_forall
theorem exists_tendsto_compactOpen_iff_forall [WeaklyLocallyCompactSpace X] [T2Space Y]
    {ι : Type*} {l : Filter ι} [Filter.NeBot l] (F : ι → C(X, Y)) :
    (∃ f, Filter.Tendsto F l (𝓝 f)) ↔
      ∀ s : Set X, IsCompact s → ∃ f, Filter.Tendsto (fun i => (F i).restrict s) l (𝓝 f) 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [WeaklyLocallyCompactSpace X] [T2Space Y] {ι : Type u_6} {l : Filter ι} [l.NeBot] (F : ι → C(X, Y))
Goal: (∃ f, Filter.Tendsto F l (𝓝 f)) ↔
  ∀ (s : Set X), IsCompact s → ∃ f, Filter.Tendsto (fun i => ContinuousMap.restrict s (F i)) l (𝓝 f)
-- ContinuousMap.coev_apply
@[simps (config := .asFn)]
def coev (b : Y) : C(X, Y × X) 
Arguments: (X : Type u_2) (Y : Type u_3) [TopologicalSpace X] [TopologicalSpace Y] (b : Y)
Goal: ⇑(ContinuousMap.coev X Y b) = Prod.mk b
-- ContinuousMap.image_coev
theorem image_coev {y : Y} (s : Set X) : coev X Y y '' s = {y} ×ˢ s 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {y : Y} (s : Set X)
Goal: ⇑(ContinuousMap.coev X Y y) '' s = {y} ×ˢ s
-- ContinuousMap.continuous_coev
theorem continuous_coev : Continuous (coev X Y) 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Continuous (ContinuousMap.coev X Y)
-- ContinuousMap.curry_apply
@[simp]
theorem curry_apply (f : C(X × Y, Z)) (a : X) (b : Y) : f.curry a b = f (a, b) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : C(X × Y, Z)) (a : X) (b : Y)
Goal: (f.curry a) b = f (a, b)
-- ContinuousMap.continuous_curry'
@[deprecated ContinuousMap.curry (since := "2024-03-05")]
theorem continuous_curry' (f : C(X × Y, Z)) : Continuous (curry' f) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : C(X × Y, Z))
Goal: Continuous f.curry'
-- ContinuousMap.continuous_of_continuous_uncurry
theorem continuous_of_continuous_uncurry (f : X → C(Y, Z))
    (h : Continuous (Function.uncurry fun x y => f x y)) : Continuous f 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : X → C(Y, Z)) (h : Continuous (Function.uncurry fun x y => (f x) y))
Goal: Continuous f
-- ContinuousMap.continuous_curry
theorem continuous_curry [LocallyCompactSpace (X × Y)] :
    Continuous (curry : C(X × Y, Z) → C(X, C(Y, Z))) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactSpace (X × Y)]
Goal: Continuous ContinuousMap.curry
-- ContinuousMap.continuous_uncurry_of_continuous
theorem continuous_uncurry_of_continuous [LocallyCompactSpace Y] (f : C(X, C(Y, Z))) :
    Continuous (Function.uncurry fun x y => f x y) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactSpace Y] (f : C(X, C(Y, Z)))
Goal: Continuous (Function.uncurry fun x y => (f x) y)
-- ContinuousMap.uncurry_apply
@[simps]
def uncurry [LocallyCompactSpace Y] (f : C(X, C(Y, Z))) : C(X × Y, Z) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactSpace Y] (f : C(X, C(Y, Z)))
Goal: ∀ (a : X × Y), f.uncurry a = Function.uncurry (fun x y => (f x) y) a
-- ContinuousMap.continuous_uncurry
theorem continuous_uncurry [LocallyCompactSpace X] [LocallyCompactSpace Y] :
    Continuous (uncurry : C(X, C(Y, Z)) → C(X × Y, Z)) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactSpace X] [LocallyCompactSpace Y]
Goal: Continuous ContinuousMap.uncurry
-- ContinuousMap.coe_const'
@[simp]
theorem coe_const' : (const' : Y → C(X, Y)) = const X 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y]
Goal: ⇑ContinuousMap.const' = ContinuousMap.const X
-- ContinuousMap.continuous_const'
theorem continuous_const' : Continuous (const X : Y → C(X, Y)) 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Continuous (ContinuousMap.const X)
-- Homeomorph.continuousMapOfUnique_apply
@[simp]
theorem continuousMapOfUnique_apply [Unique X] (y : Y) (x : X) : continuousMapOfUnique y x = y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Unique X] (y : Y) (x : X)
Goal: (Homeomorph.continuousMapOfUnique y) x = y
-- Homeomorph.continuousMapOfUnique_symm_apply
@[simp]
theorem continuousMapOfUnique_symm_apply [Unique X] (f : C(X, Y)) :
    continuousMapOfUnique.symm f = f default 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Unique X] (f : C(X, Y))
Goal: Homeomorph.continuousMapOfUnique.symm f = f default
-- QuotientMap.continuous_lift_prod_left
theorem QuotientMap.continuous_lift_prod_left (hf : QuotientMap f) {g : X × Y → Z}
    (hg : Continuous fun p : X₀ × Y => g (f p.1, p.2)) : Continuous g 
Arguments: {X₀ : Type u_1} {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X₀] [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactSpace Y] {f : X₀ → X} (hf : QuotientMap f) {g : X × Y → Z} (hg : Continuous fun p => g (f p.1, p.2))
Goal: Continuous g
-- QuotientMap.continuous_lift_prod_right
theorem QuotientMap.continuous_lift_prod_right (hf : QuotientMap f) {g : Y × X → Z}
    (hg : Continuous fun p : Y × X₀ => g (p.1, f p.2)) : Continuous g 
Arguments: {X₀ : Type u_1} {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X₀] [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactSpace Y] {f : X₀ → X} (hf : QuotientMap f) {g : Y × X → Z} (hg : Continuous fun p => g (p.1, f p.2))
Goal: Continuous g
-- nhds_bind_nhdsWithin
@[simp]
theorem nhds_bind_nhdsWithin {a : α} {s : Set α} : ((𝓝 a).bind fun x => 𝓝[s] x) = 𝓝[s] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α}
Goal: ((𝓝 a).bind fun x => 𝓝[s] x) = 𝓝[s] a
-- eventually_nhds_nhdsWithin
@[simp]
theorem eventually_nhds_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
    (∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α} {p : α → Prop}
Goal: (∀ᶠ (y : α) in 𝓝 a, ∀ᶠ (x : α) in 𝓝[s] y, p x) ↔ ∀ᶠ (x : α) in 𝓝[s] a, p x
-- eventually_nhdsWithin_iff
theorem eventually_nhdsWithin_iff {a : α} {s : Set α} {p : α → Prop} :
    (∀ᶠ x in 𝓝[s] a, p x) ↔ ∀ᶠ x in 𝓝 a, x ∈ s → p x 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α} {p : α → Prop}
Goal: (∀ᶠ (x : α) in 𝓝[s] a, p x) ↔ ∀ᶠ (x : α) in 𝓝 a, x ∈ s → p x
-- frequently_nhdsWithin_iff
theorem frequently_nhdsWithin_iff {z : α} {s : Set α} {p : α → Prop} :
    (∃ᶠ x in 𝓝[s] z, p x) ↔ ∃ᶠ x in 𝓝 z, p x ∧ x ∈ s 
Arguments: {α : Type u_1} [TopologicalSpace α] {z : α} {s : Set α} {p : α → Prop}
Goal: (∃ᶠ (x : α) in 𝓝[s] z, p x) ↔ ∃ᶠ (x : α) in 𝓝 z, p x ∧ x ∈ s
-- mem_closure_ne_iff_frequently_within
theorem mem_closure_ne_iff_frequently_within {z : α} {s : Set α} :
    z ∈ closure (s \ {z}) ↔ ∃ᶠ x in 𝓝[≠] z, x ∈ s 
Arguments: {α : Type u_1} [TopologicalSpace α] {z : α} {s : Set α}
Goal: z ∈ closure (s \ {z}) ↔ ∃ᶠ (x : α) in 𝓝[≠] z, x ∈ s
-- eventually_nhdsWithin_nhdsWithin
@[simp]
theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
    (∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α} {p : α → Prop}
Goal: (∀ᶠ (y : α) in 𝓝[s] a, ∀ᶠ (x : α) in 𝓝[s] y, p x) ↔ ∀ᶠ (x : α) in 𝓝[s] a, p x
-- nhdsWithin_eq
theorem nhdsWithin_eq (a : α) (s : Set α) :
    𝓝[s] a = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (t ∩ s) 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α) (s : Set α)
Goal: 𝓝[s] a = ⨅ t ∈ {t | a ∈ t ∧ IsOpen t}, 𝓟 (t ∩ s)
-- nhdsWithin_univ
@[simp] lemma nhdsWithin_univ (a : α) : 𝓝[Set.univ] a = 𝓝 a 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α)
Goal: 𝓝[Set.univ] a = 𝓝 a
-- nhdsWithin_hasBasis
theorem nhdsWithin_hasBasis {p : β → Prop} {s : β → Set α} {a : α} (h : (𝓝 a).HasBasis p s)
    (t : Set α) : (𝓝[t] a).HasBasis p fun i => s i ∩ t 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {p : β → Prop} {s : β → Set α} {a : α} (h : (𝓝 a).HasBasis p s) (t : Set α)
Goal: (𝓝[t] a).HasBasis p fun i => s i ∩ t
-- nhdsWithin_basis_open
theorem nhdsWithin_basis_open (a : α) (t : Set α) :
    (𝓝[t] a).HasBasis (fun u => a ∈ u ∧ IsOpen u) fun u => u ∩ t 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α) (t : Set α)
Goal: (𝓝[t] a).HasBasis (fun u => a ∈ u ∧ IsOpen u) fun u => u ∩ t
-- mem_nhdsWithin
theorem mem_nhdsWithin {t : Set α} {a : α} {s : Set α} :
    t ∈ 𝓝[s] a ↔ ∃ u, IsOpen u ∧ a ∈ u ∧ u ∩ s ⊆ t 
Arguments: {α : Type u_1} [TopologicalSpace α] {t : Set α} {a : α} {s : Set α}
Goal: t ∈ 𝓝[s] a ↔ ∃ u, IsOpen u ∧ a ∈ u ∧ u ∩ s ⊆ t
-- mem_nhdsWithin_iff_exists_mem_nhds_inter
theorem mem_nhdsWithin_iff_exists_mem_nhds_inter {t : Set α} {a : α} {s : Set α} :
    t ∈ 𝓝[s] a ↔ ∃ u ∈ 𝓝 a, u ∩ s ⊆ t 
Arguments: {α : Type u_1} [TopologicalSpace α] {t : Set α} {a : α} {s : Set α}
Goal: t ∈ 𝓝[s] a ↔ ∃ u ∈ 𝓝 a, u ∩ s ⊆ t
-- diff_mem_nhdsWithin_compl
theorem diff_mem_nhdsWithin_compl {x : α} {s : Set α} (hs : s ∈ 𝓝 x) (t : Set α) :
    s \ t ∈ 𝓝[tᶜ] x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} (hs : s ∈ 𝓝 x) (t : Set α)
Goal: s \ t ∈ 𝓝[tᶜ] x
-- diff_mem_nhdsWithin_diff
theorem diff_mem_nhdsWithin_diff {x : α} {s t : Set α} (hs : s ∈ 𝓝[t] x) (t' : Set α) :
    s \ t' ∈ 𝓝[t \ t'] x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s t : Set α} (hs : s ∈ 𝓝[t] x) (t' : Set α)
Goal: s \ t' ∈ 𝓝[t \ t'] x
-- nhds_of_nhdsWithin_of_nhds
theorem nhds_of_nhdsWithin_of_nhds {s t : Set α} {a : α} (h1 : s ∈ 𝓝 a) (h2 : t ∈ 𝓝[s] a) :
    t ∈ 𝓝 a 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : Set α} {a : α} (h1 : s ∈ 𝓝 a) (h2 : t ∈ 𝓝[s] a)
Goal: t ∈ 𝓝 a
-- mem_nhdsWithin_iff_eventually
theorem mem_nhdsWithin_iff_eventually {s t : Set α} {x : α} :
    t ∈ 𝓝[s] x ↔ ∀ᶠ y in 𝓝 x, y ∈ s → y ∈ t 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : Set α} {x : α}
Goal: t ∈ 𝓝[s] x ↔ ∀ᶠ (y : α) in 𝓝 x, y ∈ s → y ∈ t
-- mem_nhdsWithin_iff_eventuallyEq
theorem mem_nhdsWithin_iff_eventuallyEq {s t : Set α} {x : α} :
    t ∈ 𝓝[s] x ↔ s =ᶠ[𝓝 x] (s ∩ t : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : Set α} {x : α}
Goal: t ∈ 𝓝[s] x ↔ s =ᶠ[𝓝 x] s ∩ t
-- nhdsWithin_eq_iff_eventuallyEq
theorem nhdsWithin_eq_iff_eventuallyEq {s t : Set α} {x : α} : 𝓝[s] x = 𝓝[t] x ↔ s =ᶠ[𝓝 x] t 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : Set α} {x : α}
Goal: 𝓝[s] x = 𝓝[t] x ↔ s =ᶠ[𝓝 x] t
-- nhdsWithin_le_iff
theorem nhdsWithin_le_iff {s t : Set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔ t ∈ 𝓝[s] x 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : Set α} {x : α}
Goal: 𝓝[s] x ≤ 𝓝[t] x ↔ t ∈ 𝓝[s] x
-- preimage_nhdsWithin_coinduced'
theorem preimage_nhdsWithin_coinduced' {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t)
    (hs : s ∈ @nhds β (.coinduced (fun x : t => π x) inferInstance) (π a)) :
    π ⁻¹' s ∈ 𝓝[t] a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t) (hs : s ∈ 𝓝 (π a))
Goal: π ⁻¹' s ∈ 𝓝[t] a
-- mem_nhdsWithin_of_mem_nhds
theorem mem_nhdsWithin_of_mem_nhds {s t : Set α} {a : α} (h : s ∈ 𝓝 a) : s ∈ 𝓝[t] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : Set α} {a : α} (h : s ∈ 𝓝 a)
Goal: s ∈ 𝓝[t] a
-- self_mem_nhdsWithin
theorem self_mem_nhdsWithin {a : α} {s : Set α} : s ∈ 𝓝[s] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α}
Goal: s ∈ 𝓝[s] a
-- eventually_mem_nhdsWithin
theorem eventually_mem_nhdsWithin {a : α} {s : Set α} : ∀ᶠ x in 𝓝[s] a, x ∈ s 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α}
Goal: ∀ᶠ (x : α) in 𝓝[s] a, x ∈ s
-- inter_mem_nhdsWithin
theorem inter_mem_nhdsWithin (s : Set α) {t : Set α} {a : α} (h : t ∈ 𝓝 a) : s ∩ t ∈ 𝓝[s] a 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : Set α) {t : Set α} {a : α} (h : t ∈ 𝓝 a)
Goal: s ∩ t ∈ 𝓝[s] a
-- pure_le_nhdsWithin
theorem pure_le_nhdsWithin {a : α} {s : Set α} (ha : a ∈ s) : pure a ≤ 𝓝[s] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α} (ha : a ∈ s)
Goal: pure a ≤ 𝓝[s] a
-- mem_of_mem_nhdsWithin
theorem mem_of_mem_nhdsWithin {a : α} {s t : Set α} (ha : a ∈ s) (ht : t ∈ 𝓝[s] a) : a ∈ t 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s t : Set α} (ha : a ∈ s) (ht : t ∈ 𝓝[s] a)
Goal: a ∈ t
-- Filter.Eventually.self_of_nhdsWithin
theorem Filter.Eventually.self_of_nhdsWithin {p : α → Prop} {s : Set α} {x : α}
    (h : ∀ᶠ y in 𝓝[s] x, p y) (hx : x ∈ s) : p x 
Arguments: {α : Type u_1} [TopologicalSpace α] {p : α → Prop} {s : Set α} {x : α} (h : ∀ᶠ (y : α) in 𝓝[s] x, p y) (hx : x ∈ s)
Goal: p x
-- tendsto_const_nhdsWithin
theorem tendsto_const_nhdsWithin {l : Filter β} {s : Set α} {a : α} (ha : a ∈ s) :
    Tendsto (fun _ : β => a) l (𝓝[s] a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {l : Filter β} {s : Set α} {a : α} (ha : a ∈ s)
Goal: Filter.Tendsto (fun x => a) l (𝓝[s] a)
-- nhdsWithin_restrict''
theorem nhdsWithin_restrict'' {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝[s] a) :
    𝓝[s] a = 𝓝[s ∩ t] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝[s] a)
Goal: 𝓝[s] a = 𝓝[s ∩ t] a
-- nhdsWithin_restrict'
theorem nhdsWithin_restrict' {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝 a) : 𝓝[s] a = 𝓝[s ∩ t] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝 a)
Goal: 𝓝[s] a = 𝓝[s ∩ t] a
-- nhdsWithin_restrict
theorem nhdsWithin_restrict {a : α} (s : Set α) {t : Set α} (h₀ : a ∈ t) (h₁ : IsOpen t) :
    𝓝[s] a = 𝓝[s ∩ t] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} (s : Set α) {t : Set α} (h₀ : a ∈ t) (h₁ : IsOpen t)
Goal: 𝓝[s] a = 𝓝[s ∩ t] a
-- nhdsWithin_le_of_mem
theorem nhdsWithin_le_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[t] a ≤ 𝓝[s] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a)
Goal: 𝓝[t] a ≤ 𝓝[s] a
-- nhdsWithin_le_nhds
theorem nhdsWithin_le_nhds {a : α} {s : Set α} : 𝓝[s] a ≤ 𝓝 a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α}
Goal: 𝓝[s] a ≤ 𝓝 a
-- nhdsWithin_eq_nhdsWithin'
theorem nhdsWithin_eq_nhdsWithin' {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) :
    𝓝[t] a = 𝓝[u] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s)
Goal: 𝓝[t] a = 𝓝[u] a
-- nhdsWithin_eq_nhdsWithin
theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h₁ : IsOpen s)
    (h₂ : t ∩ s = u ∩ s) : 𝓝[t] a = 𝓝[u] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s t u : Set α} (h₀ : a ∈ s) (h₁ : IsOpen s) (h₂ : t ∩ s = u ∩ s)
Goal: 𝓝[t] a = 𝓝[u] a
-- nhdsWithin_eq_nhds
@[simp] theorem nhdsWithin_eq_nhds {a : α} {s : Set α} : 𝓝[s] a = 𝓝 a ↔ s ∈ 𝓝 a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α}
Goal: 𝓝[s] a = 𝓝 a ↔ s ∈ 𝓝 a
-- IsOpen.nhdsWithin_eq
theorem IsOpen.nhdsWithin_eq {a : α} {s : Set α} (h : IsOpen s) (ha : a ∈ s) : 𝓝[s] a = 𝓝 a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α} (h : IsOpen s) (ha : a ∈ s)
Goal: 𝓝[s] a = 𝓝 a
-- preimage_nhds_within_coinduced
theorem preimage_nhds_within_coinduced {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t)
    (ht : IsOpen t)
    (hs : s ∈ @nhds β (.coinduced (fun x : t => π x) inferInstance) (π a)) :
    π ⁻¹' s ∈ 𝓝 a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t) (ht : IsOpen t) (hs : s ∈ 𝓝 (π a))
Goal: π ⁻¹' s ∈ 𝓝 a
-- nhdsWithin_empty
@[simp]
theorem nhdsWithin_empty (a : α) : 𝓝[∅] a = ⊥ 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α)
Goal: 𝓝[∅] a = ⊥
-- nhdsWithin_union
theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a ⊔ 𝓝[t] a 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α) (s t : Set α)
Goal: 𝓝[s ∪ t] a = 𝓝[s] a ⊔ 𝓝[t] a
-- nhdsWithin_biUnion
theorem nhdsWithin_biUnion {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
    𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Type u_5} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α)
Goal: 𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a
-- nhdsWithin_sUnion
theorem nhdsWithin_sUnion {S : Set (Set α)} (hS : S.Finite) (a : α) :
    𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {S : Set (Set α)} (hS : S.Finite) (a : α)
Goal: 𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a
-- nhdsWithin_iUnion
theorem nhdsWithin_iUnion {ι} [Finite ι] (s : ι → Set α) (a : α) :
    𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Sort u_5} [Finite ι] (s : ι → Set α) (a : α)
Goal: 𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a
-- nhdsWithin_inter
theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α) (s t : Set α)
Goal: 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a
-- nhdsWithin_inter'
theorem nhdsWithin_inter' (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓟 t 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α) (s t : Set α)
Goal: 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓟 t
-- nhdsWithin_inter_of_mem
theorem nhdsWithin_inter_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[s ∩ t] a = 𝓝[t] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a)
Goal: 𝓝[s ∩ t] a = 𝓝[t] a
-- nhdsWithin_inter_of_mem'
theorem nhdsWithin_inter_of_mem' {a : α} {s t : Set α} (h : t ∈ 𝓝[s] a) : 𝓝[s ∩ t] a = 𝓝[s] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s t : Set α} (h : t ∈ 𝓝[s] a)
Goal: 𝓝[s ∩ t] a = 𝓝[s] a
-- nhdsWithin_singleton
@[simp]
theorem nhdsWithin_singleton (a : α) : 𝓝[{a}] a = pure a 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α)
Goal: 𝓝[{a}] a = pure a
-- nhdsWithin_insert
@[simp]
theorem nhdsWithin_insert (a : α) (s : Set α) : 𝓝[insert a s] a = pure a ⊔ 𝓝[s] a 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α) (s : Set α)
Goal: 𝓝[insert a s] a = pure a ⊔ 𝓝[s] a
-- mem_nhdsWithin_insert
theorem mem_nhdsWithin_insert {a : α} {s t : Set α} : t ∈ 𝓝[insert a s] a ↔ a ∈ t ∧ t ∈ 𝓝[s] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s t : Set α}
Goal: t ∈ 𝓝[insert a s] a ↔ a ∈ t ∧ t ∈ 𝓝[s] a
-- insert_mem_nhdsWithin_insert
theorem insert_mem_nhdsWithin_insert {a : α} {s t : Set α} (h : t ∈ 𝓝[s] a) :
    insert a t ∈ 𝓝[insert a s] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s t : Set α} (h : t ∈ 𝓝[s] a)
Goal: insert a t ∈ 𝓝[insert a s] a
-- insert_mem_nhds_iff
theorem insert_mem_nhds_iff {a : α} {s : Set α} : insert a s ∈ 𝓝 a ↔ s ∈ 𝓝[≠] a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a : α} {s : Set α}
Goal: insert a s ∈ 𝓝 a ↔ s ∈ 𝓝[≠] a
-- nhdsWithin_compl_singleton_sup_pure
@[simp]
theorem nhdsWithin_compl_singleton_sup_pure (a : α) : 𝓝[≠] a ⊔ pure a = 𝓝 a 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α)
Goal: 𝓝[≠] a ⊔ pure a = 𝓝 a