ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TopologicalSpace.Opens.coe_comap
@[simp]
theorem coe_comap (f : C(α, β)) (U : Opens β) : ↑(comap f U) = f ⁻¹' U 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (U : TopologicalSpace.Opens β)
Goal: ↑((TopologicalSpace.Opens.comap f) U) = ⇑f ⁻¹' ↑U
-- TopologicalSpace.Opens.comap_comp
protected theorem comap_comp (g : C(β, γ)) (f : C(α, β)) :
    comap (g.comp f) = (comap f).comp (comap g) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (g : C(β, γ)) (f : C(α, β))
Goal: TopologicalSpace.Opens.comap (g.comp f) = (TopologicalSpace.Opens.comap f).comp (TopologicalSpace.Opens.comap g)
-- TopologicalSpace.Opens.comap_comap
protected theorem comap_comap (g : C(β, γ)) (f : C(α, β)) (U : Opens γ) :
    comap f (comap g U) = comap (g.comp f) U 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (g : C(β, γ)) (f : C(α, β)) (U : TopologicalSpace.Opens γ)
Goal: (TopologicalSpace.Opens.comap f) ((TopologicalSpace.Opens.comap g) U) = (TopologicalSpace.Opens.comap (g.comp f)) U
-- TopologicalSpace.Opens.comap_injective
theorem comap_injective [T0Space β] : Injective (comap : C(α, β) → FrameHom (Opens β) (Opens α)) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [T0Space β]
Goal: Function.Injective TopologicalSpace.Opens.comap
-- Homeomorph.opensCongr_apply
@[simps (config := .asFn) apply]
def _root_.Homeomorph.opensCongr (f : α ≃ₜ β) : Opens α ≃o Opens β 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β)
Goal: ⇑f.opensCongr = ⇑(TopologicalSpace.Opens.comap f.symm.toContinuousMap)
-- Homeomorph.opensCongr_symm
@[simp]
theorem _root_.Homeomorph.opensCongr_symm (f : α ≃ₜ β) : f.opensCongr.symm = f.symm.opensCongr 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β)
Goal: f.opensCongr.symm = f.symm.opensCongr
-- TopologicalSpace.OpenNhdsOf.mem'
structure OpenNhdsOf (x : α) extends Opens α where
    mem' : x ∈ carrier
Arguments: {α : Type u_2} [TopologicalSpace α] {x : α} (self : TopologicalSpace.OpenNhdsOf x)
Goal: x ∈ self.carrier
-- TopologicalSpace.OpenNhdsOf.toOpens_injective
theorem toOpens_injective : Injective (toOpens : OpenNhdsOf x → Opens α)
  
Arguments: {α : Type u_2} [TopologicalSpace α] {x : α}
Goal: Function.Injective TopologicalSpace.OpenNhdsOf.toOpens
-- TopologicalSpace.OpenNhdsOf.mem
protected theorem mem (U : OpenNhdsOf x) : x ∈ U 
Arguments: {α : Type u_2} [TopologicalSpace α] {x : α} (U : TopologicalSpace.OpenNhdsOf x)
Goal: x ∈ U
-- TopologicalSpace.OpenNhdsOf.isOpen
protected theorem isOpen (U : OpenNhdsOf x) : IsOpen (U : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] {x : α} (U : TopologicalSpace.OpenNhdsOf x)
Goal: IsOpen ↑U
-- TopologicalSpace.OpenNhdsOf.basis_nhds
theorem basis_nhds : (𝓝 x).HasBasis (fun _ : OpenNhdsOf x => True) (↑) 
Arguments: {α : Type u_2} [TopologicalSpace α] {x : α}
Goal: (𝓝 x).HasBasis (fun x => True) SetLike.coe
-- subset_seqClosure
theorem subset_seqClosure {s : Set X} : s ⊆ seqClosure s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: s ⊆ seqClosure s
-- seqClosure_subset_closure
theorem seqClosure_subset_closure {s : Set X} : seqClosure s ⊆ closure s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: seqClosure s ⊆ closure s
-- IsSeqClosed.seqClosure_eq
theorem IsSeqClosed.seqClosure_eq {s : Set X} (hs : IsSeqClosed s) : seqClosure s = s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsSeqClosed s)
Goal: seqClosure s = s
-- isSeqClosed_of_seqClosure_eq
theorem isSeqClosed_of_seqClosure_eq {s : Set X} (hs : seqClosure s = s) : IsSeqClosed s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : seqClosure s = s)
Goal: IsSeqClosed s
-- isSeqClosed_iff
theorem isSeqClosed_iff {s : Set X} : IsSeqClosed s ↔ seqClosure s = s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsSeqClosed s ↔ seqClosure s = s
-- IsClosed.isSeqClosed
protected theorem IsClosed.isSeqClosed {s : Set X} (hc : IsClosed s) : IsSeqClosed s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hc : IsClosed s)
Goal: IsSeqClosed s
-- seqClosure_eq_closure
theorem seqClosure_eq_closure [FrechetUrysohnSpace X] (s : Set X) : seqClosure s = closure s 
Arguments: {X : Type u_1} [TopologicalSpace X] [FrechetUrysohnSpace X] (s : Set X)
Goal: seqClosure s = closure s
-- mem_closure_iff_seq_limit
theorem mem_closure_iff_seq_limit [FrechetUrysohnSpace X] {s : Set X} {a : X} :
    a ∈ closure s ↔ ∃ x : ℕ → X, (∀ n : ℕ, x n ∈ s) ∧ Tendsto x atTop (𝓝 a) 
Arguments: {X : Type u_1} [TopologicalSpace X] [FrechetUrysohnSpace X] {s : Set X} {a : X}
Goal: a ∈ closure s ↔ ∃ x, (∀ (n : ℕ), x n ∈ s) ∧ Filter.Tendsto x Filter.atTop (𝓝 a)
-- tendsto_nhds_iff_seq_tendsto
theorem tendsto_nhds_iff_seq_tendsto [FrechetUrysohnSpace X] {f : X → Y} {a : X} {b : Y} :
    Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [FrechetUrysohnSpace X] {f : X → Y} {a : X} {b : Y}
Goal: Filter.Tendsto f (𝓝 a) (𝓝 b) ↔
  ∀ (u : ℕ → X), Filter.Tendsto u Filter.atTop (𝓝 a) → Filter.Tendsto (f ∘ u) Filter.atTop (𝓝 b)
-- FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
    (h : ∀ (f : X → Prop) (a : X),
      (∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a) :
    FrechetUrysohnSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] (h :
  ∀ (f : X → Prop) (a : X),
    (∀ (u : ℕ → X), Filter.Tendsto u Filter.atTop (𝓝 a) → Filter.Tendsto (f ∘ u) Filter.atTop (𝓝 (f a))) →
      ContinuousAt f a)
Goal: FrechetUrysohnSpace X
-- Inducing.frechetUrysohnSpace
theorem Inducing.frechetUrysohnSpace [FrechetUrysohnSpace Y] {f : X → Y} (hf : Inducing f) :
    FrechetUrysohnSpace X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [FrechetUrysohnSpace Y] {f : X → Y} (hf : Inducing f)
Goal: FrechetUrysohnSpace X
-- isSeqClosed_iff_isClosed
theorem isSeqClosed_iff_isClosed [SequentialSpace X] {M : Set X} : IsSeqClosed M ↔ IsClosed M 
Arguments: {X : Type u_1} [TopologicalSpace X] [SequentialSpace X] {M : Set X}
Goal: IsSeqClosed M ↔ IsClosed M
-- IsSeqClosed.preimage
theorem IsSeqClosed.preimage {f : X → Y} {s : Set Y} (hs : IsSeqClosed s) (hf : SeqContinuous f) :
    IsSeqClosed (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set Y} (hs : IsSeqClosed s) (hf : SeqContinuous f)
Goal: IsSeqClosed (f ⁻¹' s)
-- Continuous.seqContinuous
protected theorem Continuous.seqContinuous {f : X → Y} (hf : Continuous f) : SeqContinuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f)
Goal: SeqContinuous f
-- SeqContinuous.continuous
protected theorem SeqContinuous.continuous [SequentialSpace X] {f : X → Y} (hf : SeqContinuous f) :
    Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [SequentialSpace X] {f : X → Y} (hf : SeqContinuous f)
Goal: Continuous f
-- continuous_iff_seqContinuous
theorem continuous_iff_seqContinuous [SequentialSpace X] {f : X → Y} :
    Continuous f ↔ SeqContinuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [SequentialSpace X] {f : X → Y}
Goal: Continuous f ↔ SeqContinuous f
-- SequentialSpace.coinduced
theorem SequentialSpace.coinduced [SequentialSpace X] (f : X → Y) :
    @SequentialSpace Y (.coinduced f ‹_›) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [SequentialSpace X] (f : X → Y)
Goal: SequentialSpace Y
-- SequentialSpace.iSup
protected theorem SequentialSpace.iSup {ι : Sort*} {t : ι → TopologicalSpace X}
    (h : ∀ i, @SequentialSpace X (t i)) : @SequentialSpace X (⨆ i, t i) 
Arguments: {X : Type u_1} {ι : Sort u_3} {t : ι → TopologicalSpace X} (h : ∀ (i : ι), SequentialSpace X)
Goal: SequentialSpace X
-- SequentialSpace.sup
protected theorem SequentialSpace.sup {t₁ t₂ : TopologicalSpace X}
    (h₁ : @SequentialSpace X t₁) (h₂ : @SequentialSpace X t₂) :
    @SequentialSpace X (t₁ ⊔ t₂) 
Arguments: {X : Type u_1} {t₁ t₂ : TopologicalSpace X} (h₁ : SequentialSpace X) (h₂ : SequentialSpace X)
Goal: SequentialSpace X
-- QuotientMap.sequentialSpace
theorem QuotientMap.sequentialSpace [SequentialSpace X] {f : X → Y} (hf : QuotientMap f) :
    SequentialSpace Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [SequentialSpace X] {f : X → Y} (hf : QuotientMap f)
Goal: SequentialSpace Y
-- IsSeqCompact.subseq_of_frequently_in
theorem IsSeqCompact.subseq_of_frequently_in {s : Set X} (hs : IsSeqCompact s) {x : ℕ → X}
    (hx : ∃ᶠ n in atTop, x n ∈ s) :
    ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsSeqCompact s) {x : ℕ → X} (hx : ∃ᶠ (n : ℕ) in Filter.atTop, x n ∈ s)
Goal: ∃ a ∈ s, ∃ φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (𝓝 a)
-- SeqCompactSpace.tendsto_subseq
theorem SeqCompactSpace.tendsto_subseq [SeqCompactSpace X] (x : ℕ → X) :
    ∃ (a : X) (φ : ℕ → ℕ), StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) 
Arguments: {X : Type u_1} [TopologicalSpace X] [SeqCompactSpace X] (x : ℕ → X)
Goal: ∃ a φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (𝓝 a)
-- IsCompact.isSeqCompact
protected theorem IsCompact.isSeqCompact {s : Set X} (hs : IsCompact s) : IsSeqCompact s 
Arguments: {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] {s : Set X} (hs : IsCompact s)
Goal: IsSeqCompact s
-- IsCompact.tendsto_subseq'
theorem IsCompact.tendsto_subseq' {s : Set X} {x : ℕ → X} (hs : IsCompact s)
    (hx : ∃ᶠ n in atTop, x n ∈ s) :
    ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) 
Arguments: {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] {s : Set X} {x : ℕ → X} (hs : IsCompact s) (hx : ∃ᶠ (n : ℕ) in Filter.atTop, x n ∈ s)
Goal: ∃ a ∈ s, ∃ φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (𝓝 a)
-- IsCompact.tendsto_subseq
theorem IsCompact.tendsto_subseq {s : Set X} {x : ℕ → X} (hs : IsCompact s) (hx : ∀ n, x n ∈ s) :
    ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) 
Arguments: {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] {s : Set X} {x : ℕ → X} (hs : IsCompact s) (hx : ∀ (n : ℕ), x n ∈ s)
Goal: ∃ a ∈ s, ∃ φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (𝓝 a)
-- CompactSpace.tendsto_subseq
theorem CompactSpace.tendsto_subseq [CompactSpace X] (x : ℕ → X) :
    ∃ (a : _) (φ : ℕ → ℕ), StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) 
Arguments: {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] [CompactSpace X] (x : ℕ → X)
Goal: ∃ a φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (𝓝 a)
-- IsSeqCompact.image
theorem IsSeqCompact.image (f_cont : SeqContinuous f) {K : Set X} (K_cpt : IsSeqCompact K) :
    IsSeqCompact (f '' K) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (f_cont : SeqContinuous f) {K : Set X} (K_cpt : IsSeqCompact K)
Goal: IsSeqCompact (f '' K)
-- IsSeqCompact.range
theorem IsSeqCompact.range [SeqCompactSpace X] (f_cont : SeqContinuous f) :
    IsSeqCompact (Set.range f) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [SeqCompactSpace X] (f_cont : SeqContinuous f)
Goal: IsSeqCompact (Set.range f)
-- IsSeqCompact.exists_tendsto_of_frequently_mem
theorem IsSeqCompact.exists_tendsto_of_frequently_mem (hs : IsSeqCompact s) {u : ℕ → X}
    (hu : ∃ᶠ n in atTop, u n ∈ s) (huc : CauchySeq u) : ∃ x ∈ s, Tendsto u atTop (𝓝 x) 
Arguments: {X : Type u_1} [UniformSpace X] {s : Set X} (hs : IsSeqCompact s) {u : ℕ → X} (hu : ∃ᶠ (n : ℕ) in Filter.atTop, u n ∈ s) (huc : CauchySeq u)
Goal: ∃ x ∈ s, Filter.Tendsto u Filter.atTop (𝓝 x)
-- IsSeqCompact.exists_tendsto
theorem IsSeqCompact.exists_tendsto (hs : IsSeqCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s)
    (huc : CauchySeq u) : ∃ x ∈ s, Tendsto u atTop (𝓝 x) 
Arguments: {X : Type u_1} [UniformSpace X] {s : Set X} (hs : IsSeqCompact s) {u : ℕ → X} (hu : ∀ (n : ℕ), u n ∈ s) (huc : CauchySeq u)
Goal: ∃ x ∈ s, Filter.Tendsto u Filter.atTop (𝓝 x)
-- IsSeqCompact.totallyBounded
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s 
Arguments: {X : Type u_1} [UniformSpace X] {s : Set X} (h : IsSeqCompact s)
Goal: TotallyBounded s
-- IsSeqCompact.isComplete
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s 
Arguments: {X : Type u_1} [UniformSpace X] {s : Set X} [(𝓤 X).IsCountablyGenerated] (hs : IsSeqCompact s)
Goal: IsComplete s
-- IsSeqCompact.isCompact
protected theorem IsSeqCompact.isCompact (hs : IsSeqCompact s) : IsCompact s 
Arguments: {X : Type u_1} [UniformSpace X] {s : Set X} [(𝓤 X).IsCountablyGenerated] (hs : IsSeqCompact s)
Goal: IsCompact s
-- UniformSpace.isCompact_iff_isSeqCompact
protected theorem UniformSpace.isCompact_iff_isSeqCompact : IsCompact s ↔ IsSeqCompact s 
Arguments: {X : Type u_1} [UniformSpace X] {s : Set X} [(𝓤 X).IsCountablyGenerated]
Goal: IsCompact s ↔ IsSeqCompact s
-- UniformSpace.compactSpace_iff_seqCompactSpace
theorem UniformSpace.compactSpace_iff_seqCompactSpace : CompactSpace X ↔ SeqCompactSpace X 
Arguments: {X : Type u_1} [UniformSpace X] [(𝓤 X).IsCountablyGenerated]
Goal: CompactSpace X ↔ SeqCompactSpace X
-- TFAE_mem_nhdsWithin_Ioi
theorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :
    TFAE [s ∈ 𝓝[>] a,
      s ∈ 𝓝[Ioc a b] a,
      s ∈ 𝓝[Ioo a b] a,
      ∃ u ∈ Ioc a b, Ioo a u ⊆ s,
      ∃ u ∈ Ioi a, Ioo a u ⊆ s] 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a b : α} (hab : a < b) (s : Set α)
Goal: [s ∈ 𝓝[>] a, s ∈ 𝓝[Set.Ioc a b] a, s ∈ 𝓝[Set.Ioo a b] a, ∃ u ∈ Set.Ioc a b, Set.Ioo a u ⊆ s,
    ∃ u ∈ Set.Ioi a, Set.Ioo a u ⊆ s].TFAE
-- mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset
theorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :
    s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a u' : α} {s : Set α} (hu' : a < u')
Goal: s ∈ 𝓝[>] a ↔ ∃ u ∈ Set.Ioc a u', Set.Ioo a u ⊆ s
-- mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'
theorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :
    s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a u' : α} {s : Set α} (hu' : a < u')
Goal: s ∈ 𝓝[>] a ↔ ∃ u ∈ Set.Ioi a, Set.Ioo a u ⊆ s
-- nhdsWithin_Ioi_basis'
theorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} (h : ∃ b, a < b)
Goal: (𝓝[>] a).HasBasis (fun x => a < x) (Set.Ioo a)
-- nhdsWithin_Ioi_basis
lemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] (a : α)
Goal: (𝓝[>] a).HasBasis (fun x => a < x) (Set.Ioo a)
-- nhdsWithin_Ioi_eq_bot_iff
theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}
Goal: 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b
-- mem_nhdsWithin_Ioi_iff_exists_Ioo_subset
theorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :
    s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] {a : α} {s : Set α}
Goal: s ∈ 𝓝[>] a ↔ ∃ u ∈ Set.Ioi a, Set.Ioo a u ⊆ s
-- countable_setOf_isolated_right
theorem countable_setOf_isolated_right [SecondCountableTopology α] :
    { x : α | 𝓝[>] x = ⊥ }.Countable 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α]
Goal: {x | 𝓝[>] x = ⊥}.Countable
-- countable_setOf_isolated_left
theorem countable_setOf_isolated_left [SecondCountableTopology α] :
    { x : α | 𝓝[<] x = ⊥ }.Countable 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α]
Goal: {x | 𝓝[<] x = ⊥}.Countable
-- mem_nhdsWithin_Ioi_iff_exists_Ioc_subset
theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
    {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] [DenselyOrdered α] {a : α} {s : Set α}
Goal: s ∈ 𝓝[>] a ↔ ∃ u ∈ Set.Ioi a, Set.Ioc a u ⊆ s
-- TFAE_mem_nhdsWithin_Iio
theorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :
    TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`
        s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`
        s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`
        ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`
        ∃ l ∈ Iio b, Ioo l b ⊆ s] 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a b : α} (h : a < b) (s : Set α)
Goal: [s ∈ 𝓝[<] b, s ∈ 𝓝[Set.Ico a b] b, s ∈ 𝓝[Set.Ioo a b] b, ∃ l ∈ Set.Ico a b, Set.Ioo l b ⊆ s,
    ∃ l ∈ Set.Iio b, Set.Ioo l b ⊆ s].TFAE
-- mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset
theorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :
    s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a l' : α} {s : Set α} (hl' : l' < a)
Goal: s ∈ 𝓝[<] a ↔ ∃ l ∈ Set.Ico l' a, Set.Ioo l a ⊆ s
-- mem_nhdsWithin_Iio_iff_exists_Ioo_subset'
theorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :
    s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a l' : α} {s : Set α} (hl' : l' < a)
Goal: s ∈ 𝓝[<] a ↔ ∃ l ∈ Set.Iio a, Set.Ioo l a ⊆ s
-- mem_nhdsWithin_Iio_iff_exists_Ioo_subset
theorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :
    s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α] {a : α} {s : Set α}
Goal: s ∈ 𝓝[<] a ↔ ∃ l ∈ Set.Iio a, Set.Ioo l a ⊆ s
-- mem_nhdsWithin_Iio_iff_exists_Ico_subset
theorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}
    {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α] [DenselyOrdered α] {a : α} {s : Set α}
Goal: s ∈ 𝓝[<] a ↔ ∃ l ∈ Set.Iio a, Set.Ico l a ⊆ s
-- nhdsWithin_Iio_basis'
theorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} (h : ∃ b, b < a)
Goal: (𝓝[<] a).HasBasis (fun x => x < a) fun x => Set.Ioo x a
-- nhdsWithin_Iio_eq_bot_iff
theorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}
Goal: 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a
-- TFAE_mem_nhdsWithin_Ici
theorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :
    TFAE [s ∈ 𝓝[≥] a,
      s ∈ 𝓝[Icc a b] a,
      s ∈ 𝓝[Ico a b] a,
      ∃ u ∈ Ioc a b, Ico a u ⊆ s,
      ∃ u ∈ Ioi a , Ico a u ⊆ s] 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a b : α} (hab : a < b) (s : Set α)
Goal: [s ∈ 𝓝[≥] a, s ∈ 𝓝[Set.Icc a b] a, s ∈ 𝓝[Set.Ico a b] a, ∃ u ∈ Set.Ioc a b, Set.Ico a u ⊆ s,
    ∃ u ∈ Set.Ioi a, Set.Ico a u ⊆ s].TFAE
-- mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset
theorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :
    s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a u' : α} {s : Set α} (hu' : a < u')
Goal: s ∈ 𝓝[≥] a ↔ ∃ u ∈ Set.Ioc a u', Set.Ico a u ⊆ s
-- mem_nhdsWithin_Ici_iff_exists_Ico_subset'
theorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :
    s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a u' : α} {s : Set α} (hu' : a < u')
Goal: s ∈ 𝓝[≥] a ↔ ∃ u ∈ Set.Ioi a, Set.Ico a u ⊆ s
-- mem_nhdsWithin_Ici_iff_exists_Ico_subset
theorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :
    s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] {a : α} {s : Set α}
Goal: s ∈ 𝓝[≥] a ↔ ∃ u ∈ Set.Ioi a, Set.Ico a u ⊆ s
-- nhdsWithin_Ici_basis_Ico
theorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :
    (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] (a : α)
Goal: (𝓝[≥] a).HasBasis (fun u => a < u) (Set.Ico a)
-- nhdsWithin_Ici_basis_Icc
theorem nhdsWithin_Ici_basis_Icc [NoMaxOrder α] [DenselyOrdered α] {a : α} :
    (𝓝[≥] a).HasBasis (a < ·) (Icc a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] [DenselyOrdered α] {a : α}
Goal: (𝓝[≥] a).HasBasis (fun x => a < x) (Set.Icc a)
-- mem_nhdsWithin_Ici_iff_exists_Icc_subset
theorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
    {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] [DenselyOrdered α] {a : α} {s : Set α}
Goal: s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Set.Icc a u ⊆ s
-- TFAE_mem_nhdsWithin_Iic
theorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :
    TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`
      s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`
      s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`
      ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`
      ∃ l ∈ Iio b, Ioc l b ⊆ s] 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a b : α} (h : a < b) (s : Set α)
Goal: [s ∈ 𝓝[≤] b, s ∈ 𝓝[Set.Icc a b] b, s ∈ 𝓝[Set.Ioc a b] b, ∃ l ∈ Set.Ico a b, Set.Ioc l b ⊆ s,
    ∃ l ∈ Set.Iio b, Set.Ioc l b ⊆ s].TFAE
-- mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset
theorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :
    s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a l' : α} {s : Set α} (hl' : l' < a)
Goal: s ∈ 𝓝[≤] a ↔ ∃ l ∈ Set.Ico l' a, Set.Ioc l a ⊆ s
-- mem_nhdsWithin_Iic_iff_exists_Ioc_subset'
theorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :
    s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a l' : α} {s : Set α} (hl' : l' < a)
Goal: s ∈ 𝓝[≤] a ↔ ∃ l ∈ Set.Iio a, Set.Ioc l a ⊆ s
-- mem_nhdsWithin_Iic_iff_exists_Ioc_subset
theorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :
    s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α] {a : α} {s : Set α}
Goal: s ∈ 𝓝[≤] a ↔ ∃ l ∈ Set.Iio a, Set.Ioc l a ⊆ s
-- mem_nhdsWithin_Iic_iff_exists_Icc_subset
theorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}
    {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α] [DenselyOrdered α] {a : α} {s : Set α}
Goal: s ∈ 𝓝[≤] a ↔ ∃ l < a, Set.Icc l a ⊆ s