ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- nhdsWithin_Iic_basis'
theorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}
    (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} (ha : ∃ l, l < a)
Goal: (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Set.Ioc l a
-- nhdsWithin_Ici_basis
theorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]
    (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] (a : α)
Goal: (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Set.Ico a u
-- nhdsWithin_Iic_basis
theorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]
    (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α] (a : α)
Goal: (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Set.Ioc l a
-- nhds_top_order
theorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :
    𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α]
Goal: 𝓝 ⊤ = ⨅ l, ⨅ (_ : l < ⊤), 𝓟 (Set.Ioi l)
-- nhds_bot_order
theorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :
    𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α]
Goal: 𝓝 ⊥ = ⨅ l, ⨅ (_ : ⊥ < l), 𝓟 (Set.Iio l)
-- nhds_top_basis
theorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]
    [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α] [Nontrivial α]
Goal: (𝓝 ⊤).HasBasis (fun a => a < ⊤) fun a => Set.Ioi a
-- nhds_bot_basis
theorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]
    [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α] [Nontrivial α]
Goal: (𝓝 ⊥).HasBasis (fun a => ⊥ < a) fun a => Set.Iio a
-- nhds_top_basis_Ici
theorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]
    [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α] [Nontrivial α] [DenselyOrdered α]
Goal: (𝓝 ⊤).HasBasis (fun a => a < ⊤) Set.Ici
-- nhds_bot_basis_Iic
theorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]
    [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α] [Nontrivial α] [DenselyOrdered α]
Goal: (𝓝 ⊥).HasBasis (fun a => ⊥ < a) Set.Iic
-- tendsto_nhds_top_mono
theorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]
    {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β] {l : Filter α} {f g : α → β} (hf : Filter.Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g)
Goal: Filter.Tendsto g l (𝓝 ⊤)
-- tendsto_nhds_bot_mono
theorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]
    {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β] {l : Filter α} {f g : α → β} (hf : Filter.Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f)
Goal: Filter.Tendsto g l (𝓝 ⊥)
-- tendsto_nhds_top_mono'
theorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]
    {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β] {l : Filter α} {f g : α → β} (hf : Filter.Tendsto f l (𝓝 ⊤)) (hg : f ≤ g)
Goal: Filter.Tendsto g l (𝓝 ⊤)
-- tendsto_nhds_bot_mono'
theorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]
    {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β] {l : Filter α} {f g : α → β} (hf : Filter.Tendsto f l (𝓝 ⊥)) (hg : g ≤ f)
Goal: Filter.Tendsto g l (𝓝 ⊥)
-- order_separated
theorem order_separated [OrderTopology α] {a₁ a₂ : α} (h : a₁ < a₂) :
    ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a₁ a₂ : α} (h : a₁ < a₂)
Goal: ∃ u v, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂
-- exists_Ioc_subset_of_mem_nhds
theorem exists_Ioc_subset_of_mem_nhds [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a)
    (h : ∃ l, l < a) : ∃ l < a, Ioc l a ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a)
Goal: ∃ l < a, Set.Ioc l a ⊆ s
-- exists_Ioc_subset_of_mem_nhds'
theorem exists_Ioc_subset_of_mem_nhds' [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α}
    (hl : l < a) : ∃ l' ∈ Ico l a, Ioc l' a ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a)
Goal: ∃ l' ∈ Set.Ico l a, Set.Ioc l' a ⊆ s
-- exists_Ico_subset_of_mem_nhds'
theorem exists_Ico_subset_of_mem_nhds' [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α}
    (hu : a < u) : ∃ u' ∈ Ioc a u, Ico a u' ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u)
Goal: ∃ u' ∈ Set.Ioc a u, Set.Ico a u' ⊆ s
-- exists_Ico_subset_of_mem_nhds
theorem exists_Ico_subset_of_mem_nhds [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a)
    (h : ∃ u, a < u) : ∃ u, a < u ∧ Ico a u ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u)
Goal: ∃ u, a < u ∧ Set.Ico a u ⊆ s
-- exists_Icc_mem_subset_of_mem_nhdsWithin_Ici
theorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici [OrderTopology α] {a : α} {s : Set α}
    (hs : s ∈ 𝓝[≥] a) : ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a)
Goal: ∃ b, a ≤ b ∧ Set.Icc a b ∈ 𝓝[≥] a ∧ Set.Icc a b ⊆ s
-- exists_Icc_mem_subset_of_mem_nhdsWithin_Iic
theorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic [OrderTopology α] {a : α} {s : Set α}
    (hs : s ∈ 𝓝[≤] a) : ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a)
Goal: ∃ b ≤ a, Set.Icc b a ∈ 𝓝[≤] a ∧ Set.Icc b a ⊆ s
-- exists_Icc_mem_subset_of_mem_nhds
theorem exists_Icc_mem_subset_of_mem_nhds [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :
    ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a)
Goal: ∃ b c, a ∈ Set.Icc b c ∧ Set.Icc b c ∈ 𝓝 a ∧ Set.Icc b c ⊆ s
-- IsOpen.exists_Ioo_subset
theorem IsOpen.exists_Ioo_subset [OrderTopology α] [Nontrivial α] {s : Set α} (hs : IsOpen s)
    (h : s.Nonempty) : ∃ a b, a < b ∧ Ioo a b ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty)
Goal: ∃ a b, a < b ∧ Set.Ioo a b ⊆ s
-- dense_of_exists_between
theorem dense_of_exists_between [OrderTopology α] [Nontrivial α] {s : Set α}
    (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [Nontrivial α] {s : Set α} (h : ∀ ⦃a b : α⦄, a < b → ∃ c ∈ s, a < c ∧ c < b)
Goal: Dense s
-- dense_iff_exists_between
theorem dense_iff_exists_between [OrderTopology α] [DenselyOrdered α] [Nontrivial α] {s : Set α} :
    Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [Nontrivial α] {s : Set α}
Goal: Dense s ↔ ∀ (a b : α), a < b → ∃ c ∈ s, a < c ∧ c < b
-- mem_nhds_iff_exists_Ioo_subset'
theorem mem_nhds_iff_exists_Ioo_subset' [OrderTopology α] {a : α} {s : Set α} (hl : ∃ l, l < a)
    (hu : ∃ u, a < u) : s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u)
Goal: s ∈ 𝓝 a ↔ ∃ l u, a ∈ Set.Ioo l u ∧ Set.Ioo l u ⊆ s
-- mem_nhds_iff_exists_Ioo_subset
theorem mem_nhds_iff_exists_Ioo_subset [OrderTopology α] [NoMaxOrder α] [NoMinOrder α] {a : α}
    {s : Set α} : s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α}
Goal: s ∈ 𝓝 a ↔ ∃ l u, a ∈ Set.Ioo l u ∧ Set.Ioo l u ⊆ s
-- nhds_basis_Ioo'
theorem nhds_basis_Ioo' [OrderTopology α] {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :
    (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u)
Goal: (𝓝 a).HasBasis (fun b => b.1 < a ∧ a < b.2) fun b => Set.Ioo b.1 b.2
-- nhds_basis_Ioo
theorem nhds_basis_Ioo [OrderTopology α] [NoMaxOrder α] [NoMinOrder α] (a : α) :
    (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] [NoMinOrder α] (a : α)
Goal: (𝓝 a).HasBasis (fun b => b.1 < a ∧ a < b.2) fun b => Set.Ioo b.1 b.2
-- Filter.Eventually.exists_Ioo_subset
theorem Filter.Eventually.exists_Ioo_subset [OrderTopology α] [NoMaxOrder α] [NoMinOrder α] {a : α}
    {p : α → Prop} (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop} (hp : ∀ᶠ (x : α) in 𝓝 a, p x)
Goal: ∃ l u, a ∈ Set.Ioo l u ∧ Set.Ioo l u ⊆ {x | p x}
-- Dense.topology_eq_generateFrom
theorem Dense.topology_eq_generateFrom [OrderTopology α] [DenselyOrdered α] {s : Set α}
    (hs : Dense s) : ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {s : Set α} (hs : Dense s)
Goal: inst✝ = TopologicalSpace.generateFrom (Set.Ioi '' s ∪ Set.Iio '' s)
-- atBot_le_nhds_bot
@[deprecated OrderBot.atBot_eq (since := "2024-02-14")]
theorem atBot_le_nhds_bot [OrderBot α] : (atBot : Filter α) ≤ 𝓝 ⊥ 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderBot α]
Goal: Filter.atBot ≤ 𝓝 ⊥
-- atTop_le_nhds_top
@[deprecated OrderTop.atTop_eq (since := "2024-02-14")]
theorem atTop_le_nhds_top [OrderTop α] : (atTop : Filter α) ≤ 𝓝 ⊤ 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTop α]
Goal: Filter.atTop ≤ 𝓝 ⊤
-- SecondCountableTopology.of_separableSpace_orderTopology
theorem SecondCountableTopology.of_separableSpace_orderTopology [OrderTopology α] [DenselyOrdered α]
    [SeparableSpace α] : SecondCountableTopology α 
Arguments: (α : Type u) [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [TopologicalSpace.SeparableSpace α]
Goal: SecondCountableTopology α
-- countable_setOf_covBy_right
theorem countable_setOf_covBy_right [OrderTopology α] [SecondCountableTopology α] :
    Set.Countable { x : α | ∃ y, x ⋖ y } 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α]
Goal: {x | ∃ y, x ⋖ y}.Countable
-- countable_setOf_covBy_left
theorem countable_setOf_covBy_left [OrderTopology α] [SecondCountableTopology α] :
    Set.Countable { x : α | ∃ y, y ⋖ x } 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α]
Goal: {x | ∃ y, y ⋖ x}.Countable
-- countable_of_isolated_left'
theorem countable_of_isolated_left' [OrderTopology α] [SecondCountableTopology α] :
    Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α]
Goal: {x | ∃ y < x, Set.Ioo y x = ∅}.Countable
-- Set.PairwiseDisjoint.countable_of_Ioo
theorem Set.PairwiseDisjoint.countable_of_Ioo [OrderTopology α] [SecondCountableTopology α]
    {y : α → α} {s : Set α} (h : PairwiseDisjoint s fun x => Ioo x (y x))
    (h' : ∀ x ∈ s, x < y x) : s.Countable 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {y : α → α} {s : Set α} (h : s.PairwiseDisjoint fun x => Set.Ioo x (y x)) (h' : ∀ x ∈ s, x < y x)
Goal: s.Countable
-- countable_image_lt_image_Ioi
theorem countable_image_lt_image_Ioi [OrderTopology α] [LinearOrder β] (f : β → α)
    [SecondCountableTopology α] : Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [LinearOrder β] (f : β → α) [SecondCountableTopology α]
Goal: {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}.Countable
-- countable_image_gt_image_Ioi
theorem countable_image_gt_image_Ioi [OrderTopology α] [LinearOrder β] (f : β → α)
    [SecondCountableTopology α] : Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [LinearOrder β] (f : β → α) [SecondCountableTopology α]
Goal: {x | ∃ z < f x, ∀ (y : β), x < y → f y ≤ z}.Countable
-- countable_image_lt_image_Iio
theorem countable_image_lt_image_Iio [OrderTopology α] [LinearOrder β] (f : β → α)
    [SecondCountableTopology α] : Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [LinearOrder β] (f : β → α) [SecondCountableTopology α]
Goal: {x | ∃ z, f x < z ∧ ∀ y < x, z ≤ f y}.Countable
-- countable_image_gt_image_Iio
theorem countable_image_gt_image_Iio [OrderTopology α] [LinearOrder β] (f : β → α)
    [SecondCountableTopology α] : Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [LinearOrder β] (f : β → α) [SecondCountableTopology α]
Goal: {x | ∃ z < f x, ∀ y < x, f y ≤ z}.Countable
-- pi_Iic_mem_nhds
theorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [Finite ι] [(i : ι) → LinearOrder (π i)] [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), OrderTopology (π i)] {a x : (i : ι) → π i} (ha : ∀ (i : ι), x i < a i)
Goal: Set.Iic a ∈ 𝓝 x
-- pi_Iic_mem_nhds'
theorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {ι : Type u_1} [Finite ι] {a' x' : ι → α} (ha : ∀ (i : ι), x' i < a' i)
Goal: Set.Iic a' ∈ 𝓝 x'
-- pi_Ici_mem_nhds
theorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [Finite ι] [(i : ι) → LinearOrder (π i)] [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), OrderTopology (π i)] {a x : (i : ι) → π i} (ha : ∀ (i : ι), a i < x i)
Goal: Set.Ici a ∈ 𝓝 x
-- pi_Ici_mem_nhds'
theorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {ι : Type u_1} [Finite ι] {a' x' : ι → α} (ha : ∀ (i : ι), a' i < x' i)
Goal: Set.Ici a' ∈ 𝓝 x'
-- pi_Icc_mem_nhds
theorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [Finite ι] [(i : ι) → LinearOrder (π i)] [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), OrderTopology (π i)] {a b x : (i : ι) → π i} (ha : ∀ (i : ι), a i < x i) (hb : ∀ (i : ι), x i < b i)
Goal: Set.Icc a b ∈ 𝓝 x
-- pi_Icc_mem_nhds'
theorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {ι : Type u_1} [Finite ι] {a' b' x' : ι → α} (ha : ∀ (i : ι), a' i < x' i) (hb : ∀ (i : ι), x' i < b' i)
Goal: Set.Icc a' b' ∈ 𝓝 x'
-- pi_Iio_mem_nhds
theorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [Finite ι] [(i : ι) → LinearOrder (π i)] [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), OrderTopology (π i)] {a x : (i : ι) → π i} [Nonempty ι] (ha : ∀ (i : ι), x i < a i)
Goal: Set.Iio a ∈ 𝓝 x
-- pi_Iio_mem_nhds'
theorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {ι : Type u_1} [Finite ι] {a' x' : ι → α} [Nonempty ι] (ha : ∀ (i : ι), x' i < a' i)
Goal: Set.Iio a' ∈ 𝓝 x'
-- pi_Ioi_mem_nhds
theorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [Finite ι] [(i : ι) → LinearOrder (π i)] [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), OrderTopology (π i)] {a x : (i : ι) → π i} [Nonempty ι] (ha : ∀ (i : ι), a i < x i)
Goal: Set.Ioi a ∈ 𝓝 x
-- pi_Ioi_mem_nhds'
theorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {ι : Type u_1} [Finite ι] {a' x' : ι → α} [Nonempty ι] (ha : ∀ (i : ι), a' i < x' i)
Goal: Set.Ioi a' ∈ 𝓝 x'
-- pi_Ioc_mem_nhds
theorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [Finite ι] [(i : ι) → LinearOrder (π i)] [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), OrderTopology (π i)] {a b x : (i : ι) → π i} [Nonempty ι] (ha : ∀ (i : ι), a i < x i) (hb : ∀ (i : ι), x i < b i)
Goal: Set.Ioc a b ∈ 𝓝 x
-- pi_Ioc_mem_nhds'
theorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {ι : Type u_1} [Finite ι] {a' b' x' : ι → α} [Nonempty ι] (ha : ∀ (i : ι), a' i < x' i) (hb : ∀ (i : ι), x' i < b' i)
Goal: Set.Ioc a' b' ∈ 𝓝 x'
-- pi_Ico_mem_nhds
theorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [Finite ι] [(i : ι) → LinearOrder (π i)] [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), OrderTopology (π i)] {a b x : (i : ι) → π i} [Nonempty ι] (ha : ∀ (i : ι), a i < x i) (hb : ∀ (i : ι), x i < b i)
Goal: Set.Ico a b ∈ 𝓝 x
-- pi_Ico_mem_nhds'
theorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {ι : Type u_1} [Finite ι] {a' b' x' : ι → α} [Nonempty ι] (ha : ∀ (i : ι), a' i < x' i) (hb : ∀ (i : ι), x' i < b' i)
Goal: Set.Ico a' b' ∈ 𝓝 x'
-- pi_Ioo_mem_nhds
theorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [Finite ι] [(i : ι) → LinearOrder (π i)] [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), OrderTopology (π i)] {a b x : (i : ι) → π i} [Nonempty ι] (ha : ∀ (i : ι), a i < x i) (hb : ∀ (i : ι), x i < b i)
Goal: Set.Ioo a b ∈ 𝓝 x
-- pi_Ioo_mem_nhds'
theorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {ι : Type u_1} [Finite ι] {a' b' x' : ι → α} [Nonempty ι] (ha : ∀ (i : ι), a' i < x' i) (hb : ∀ (i : ι), x' i < b' i)
Goal: Set.Ioo a' b' ∈ 𝓝 x'
-- AccPt.nhds_inter
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
    AccPt x (𝓟 (U ∩ C)) 
Arguments: {α : Type u_1} [TopologicalSpace α] {C : Set α} {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x)
Goal: AccPt x (𝓟 (U ∩ C))
-- perfect_def
@[mk_iff perfect_def]
structure Perfect (C : Set α) : Prop where
  closed : IsClosed C
  acc : Preperfect C
Arguments: {α : Type u_1} [TopologicalSpace α] (C : Set α)
Goal: Perfect C ↔ IsClosed C ∧ Preperfect C
-- Perfect.closed
@[mk_iff perfect_def]
structure Perfect (C : Set α) : Prop where
  closed : IsClosed C
  acc : Preperfect C
Arguments: {α : Type u_1} [TopologicalSpace α] {C : Set α} (self : Perfect C)
Goal: IsClosed C
-- Perfect.acc
@[mk_iff perfect_def]
structure Perfect (C : Set α) : Prop where
  closed : IsClosed C
  acc : Preperfect C
Arguments: {α : Type u_1} [TopologicalSpace α] {C : Set α} (self : Perfect C)
Goal: Preperfect C
-- preperfect_iff_nhds
theorem preperfect_iff_nhds : Preperfect C ↔ ∀ x ∈ C, ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x 
Arguments: {α : Type u_1} [TopologicalSpace α] {C : Set α}
Goal: Preperfect C ↔ ∀ x ∈ C, ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x
-- perfectSpace_def
@[mk_iff perfectSpace_def]
class PerfectSpace : Prop :=
  univ_preperfect : Preperfect (Set.univ : Set α)
Arguments: (α : Type u_1) [TopologicalSpace α]
Goal: PerfectSpace α ↔ Preperfect Set.univ
-- PerfectSpace.univ_preperfect
@[mk_iff perfectSpace_def]
class PerfectSpace : Prop :=
  univ_preperfect : Preperfect (Set.univ : Set α)
Arguments: {α : Type u_1} [TopologicalSpace α] [self : PerfectSpace α]
Goal: Preperfect Set.univ
-- PerfectSpace.univ_perfect
theorem PerfectSpace.univ_perfect [PerfectSpace α] : Perfect (Set.univ : Set α) 
Arguments: (α : Type u_1) [TopologicalSpace α] [PerfectSpace α]
Goal: Perfect Set.univ
-- Preperfect.open_inter
theorem Preperfect.open_inter {U : Set α} (hC : Preperfect C) (hU : IsOpen U) :
    Preperfect (U ∩ C) 
Arguments: {α : Type u_1} [TopologicalSpace α] {C U : Set α} (hC : Preperfect C) (hU : IsOpen U)
Goal: Preperfect (U ∩ C)
-- Preperfect.perfect_closure
theorem Preperfect.perfect_closure (hC : Preperfect C) : Perfect (closure C) 
Arguments: {α : Type u_1} [TopologicalSpace α] {C : Set α} (hC : Preperfect C)
Goal: Perfect (closure C)
-- preperfect_iff_perfect_closure
theorem preperfect_iff_perfect_closure [T1Space α] : Preperfect C ↔ Perfect (closure C) 
Arguments: {α : Type u_1} [TopologicalSpace α] {C : Set α} [T1Space α]
Goal: Preperfect C ↔ Perfect (closure C)
-- Perfect.closure_nhds_inter
theorem Perfect.closure_nhds_inter {U : Set α} (hC : Perfect C) (x : α) (xC : x ∈ C) (xU : x ∈ U)
    (Uop : IsOpen U) : Perfect (closure (U ∩ C)) ∧ (closure (U ∩ C)).Nonempty 
Arguments: {α : Type u_1} [TopologicalSpace α] {C U : Set α} (hC : Perfect C) (x : α) (xC : x ∈ C) (xU : x ∈ U) (Uop : IsOpen U)
Goal: Perfect (closure (U ∩ C)) ∧ (closure (U ∩ C)).Nonempty
-- Perfect.splitting
theorem Perfect.splitting [T25Space α] (hC : Perfect C) (hnonempty : C.Nonempty) :
    ∃ C₀ C₁ : Set α,
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C) ∧ (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C) ∧ Disjoint C₀ C₁ 
Arguments: {α : Type u_1} [TopologicalSpace α] {C : Set α} [T25Space α] (hC : Perfect C) (hnonempty : C.Nonempty)
Goal: ∃ C₀ C₁, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C) ∧ (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C) ∧ Disjoint C₀ C₁
-- IsPreconnected.preperfect_of_nontrivial
lemma IsPreconnected.preperfect_of_nontrivial [T1Space α] {U : Set α} (hu : U.Nontrivial)
    (h : IsPreconnected U) : Preperfect U 
Arguments: {α : Type u_1} [TopologicalSpace α] [T1Space α] {U : Set α} (hu : U.Nontrivial) (h : IsPreconnected U)
Goal: Preperfect U
-- exists_countable_union_perfect_of_isClosed
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D 
Arguments: {α : Type u_1} [TopologicalSpace α] {C : Set α} [SecondCountableTopology α] (hclosed : IsClosed C)
Goal: ∃ V D, V.Countable ∧ Perfect D ∧ C = V ∪ D