ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Bornology.cobounded_prod
theorem cobounded_prod : cobounded (α × β) = (cobounded α).coprod (cobounded β) 
Arguments: {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β]
Goal: Bornology.cobounded (α × β) = (Bornology.cobounded α).coprod (Bornology.cobounded β)
-- Bornology.isBounded_image_fst_and_snd
theorem isBounded_image_fst_and_snd {s : Set (α × β)} :
    IsBounded (Prod.fst '' s) ∧ IsBounded (Prod.snd '' s) ↔ IsBounded s 
Arguments: {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set (α × β)}
Goal: Bornology.IsBounded (Prod.fst '' s) ∧ Bornology.IsBounded (Prod.snd '' s) ↔ Bornology.IsBounded s
-- Bornology.IsBounded.image_fst
lemma IsBounded.image_fst {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.fst '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set (α × β)} (hs : Bornology.IsBounded s)
Goal: Bornology.IsBounded (Prod.fst '' s)
-- Bornology.IsBounded.image_snd
lemma IsBounded.image_snd {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.snd '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set (α × β)} (hs : Bornology.IsBounded s)
Goal: Bornology.IsBounded (Prod.snd '' s)
-- Bornology.IsBounded.fst_of_prod
theorem IsBounded.fst_of_prod (h : IsBounded (s ×ˢ t)) (ht : t.Nonempty) : IsBounded s 
Arguments: {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (h : Bornology.IsBounded (s ×ˢ t)) (ht : t.Nonempty)
Goal: Bornology.IsBounded s
-- Bornology.IsBounded.snd_of_prod
theorem IsBounded.snd_of_prod (h : IsBounded (s ×ˢ t)) (hs : s.Nonempty) : IsBounded t 
Arguments: {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (h : Bornology.IsBounded (s ×ˢ t)) (hs : s.Nonempty)
Goal: Bornology.IsBounded t
-- Bornology.IsBounded.prod
theorem IsBounded.prod (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ×ˢ t) 
Arguments: {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
Goal: Bornology.IsBounded (s ×ˢ t)
-- Bornology.isBounded_prod_of_nonempty
theorem isBounded_prod_of_nonempty (hne : Set.Nonempty (s ×ˢ t)) :
    IsBounded (s ×ˢ t) ↔ IsBounded s ∧ IsBounded t 
Arguments: {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (hne : (s ×ˢ t).Nonempty)
Goal: Bornology.IsBounded (s ×ˢ t) ↔ Bornology.IsBounded s ∧ Bornology.IsBounded t
-- Bornology.isBounded_prod
theorem isBounded_prod : IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ IsBounded s ∧ IsBounded t 
Arguments: {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β}
Goal: Bornology.IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ Bornology.IsBounded s ∧ Bornology.IsBounded t
-- Bornology.isBounded_prod_self
theorem isBounded_prod_self : IsBounded (s ×ˢ s) ↔ IsBounded s 
Arguments: {α : Type u_1} [Bornology α] {s : Set α}
Goal: Bornology.IsBounded (s ×ˢ s) ↔ Bornology.IsBounded s
-- Bornology.cobounded_pi
theorem cobounded_pi : cobounded (∀ i, π i) = Filter.coprodᵢ fun i => cobounded (π i) 
Arguments: {ι : Type u_3} {π : ι → Type u_4} [(i : ι) → Bornology (π i)]
Goal: Bornology.cobounded ((i : ι) → π i) = Filter.coprodᵢ fun i => Bornology.cobounded (π i)
-- Bornology.forall_isBounded_image_eval_iff
theorem forall_isBounded_image_eval_iff {s : Set (∀ i, π i)} :
    (∀ i, IsBounded (eval i '' s)) ↔ IsBounded s 
Arguments: {ι : Type u_3} {π : ι → Type u_4} [(i : ι) → Bornology (π i)] {s : Set ((i : ι) → π i)}
Goal: (∀ (i : ι), Bornology.IsBounded (Function.eval i '' s)) ↔ Bornology.IsBounded s
-- Bornology.IsBounded.image_eval
lemma IsBounded.image_eval {s : Set (∀ i, π i)} (hs : IsBounded s) (i : ι) :
    IsBounded (eval i '' s) 
Arguments: {ι : Type u_3} {π : ι → Type u_4} [(i : ι) → Bornology (π i)] {s : Set ((i : ι) → π i)} (hs : Bornology.IsBounded s) (i : ι)
Goal: Bornology.IsBounded (Function.eval i '' s)
-- Bornology.IsBounded.pi
theorem IsBounded.pi (h : ∀ i, IsBounded (S i)) : IsBounded (pi univ S) 
Arguments: {ι : Type u_3} {π : ι → Type u_4} [(i : ι) → Bornology (π i)] {S : (i : ι) → Set (π i)} (h : ∀ (i : ι), Bornology.IsBounded (S i))
Goal: Bornology.IsBounded (Set.univ.pi S)
-- Bornology.isBounded_pi_of_nonempty
theorem isBounded_pi_of_nonempty (hne : (pi univ S).Nonempty) :
    IsBounded (pi univ S) ↔ ∀ i, IsBounded (S i) 
Arguments: {ι : Type u_3} {π : ι → Type u_4} [(i : ι) → Bornology (π i)] {S : (i : ι) → Set (π i)} (hne : (Set.univ.pi S).Nonempty)
Goal: Bornology.IsBounded (Set.univ.pi S) ↔ ∀ (i : ι), Bornology.IsBounded (S i)
-- Bornology.isBounded_pi
theorem isBounded_pi : IsBounded (pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ i, IsBounded (S i) 
Arguments: {ι : Type u_3} {π : ι → Type u_4} [(i : ι) → Bornology (π i)] {S : (i : ι) → Set (π i)}
Goal: Bornology.IsBounded (Set.univ.pi S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), Bornology.IsBounded (S i)
-- Bornology.isBounded_induced
theorem isBounded_induced {α β : Type*} [Bornology β] {f : α → β} {s : Set α} :
    @IsBounded α (Bornology.induced f) s ↔ IsBounded (f '' s) 
Arguments: {α : Type u_5} {β : Type u_6} [Bornology β] {f : α → β} {s : Set α}
Goal: Bornology.IsBounded s ↔ Bornology.IsBounded (f '' s)
-- Bornology.isBounded_image_subtype_val
theorem isBounded_image_subtype_val {p : α → Prop} {s : Set { x // p x }} :
    IsBounded (Subtype.val '' s) ↔ IsBounded s 
Arguments: {α : Type u_1} [Bornology α] {p : α → Prop} {s : Set { x // p x }}
Goal: Bornology.IsBounded (Subtype.val '' s) ↔ Bornology.IsBounded s
-- boundedSpace_induced_iff
theorem boundedSpace_induced_iff {α β : Type*} [Bornology β] {f : α → β} :
    @BoundedSpace α (Bornology.induced f) ↔ IsBounded (range f) 
Arguments: {α : Type u_5} {β : Type u_6} [Bornology β] {f : α → β}
Goal: BoundedSpace α ↔ Bornology.IsBounded (Set.range f)
-- boundedSpace_subtype_iff
theorem boundedSpace_subtype_iff {p : α → Prop} :
    BoundedSpace (Subtype p) ↔ IsBounded { x | p x } 
Arguments: {α : Type u_1} [Bornology α] {p : α → Prop}
Goal: BoundedSpace (Subtype p) ↔ Bornology.IsBounded {x | p x}
-- boundedSpace_val_set_iff
theorem boundedSpace_val_set_iff {s : Set α} : BoundedSpace s ↔ IsBounded s 
Arguments: {α : Type u_1} [Bornology α] {s : Set α}
Goal: BoundedSpace ↑s ↔ Bornology.IsBounded s
-- ClosedIicTopology.isClosed_Iic
class ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
    isClosed_Iic (a : α) : IsClosed (Iic a)
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [self : ClosedIicTopology α] (a : α)
Goal: IsClosed (Set.Iic a)
-- ClosedIciTopology.isClosed_Ici
class ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
    isClosed_Ici (a : α) : IsClosed (Ici a)
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [self : ClosedIciTopology α] (a : α)
Goal: IsClosed (Set.Ici a)
-- OrderClosedTopology.isClosed_le'
class OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
    isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [self : OrderClosedTopology α]
Goal: IsClosed {p | p.1 ≤ p.2}
-- Dense.orderDual
theorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :
    Dense (OrderDual.ofDual ⁻¹' s) 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (hs : Dense s)
Goal: Dense (⇑OrderDual.ofDual ⁻¹' s)
-- BddAbove.of_closure
protected lemma BddAbove.of_closure : BddAbove (closure s) → BddAbove s 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] {s : Set α}
Goal: BddAbove (closure s) → BddAbove s
-- BddBelow.of_closure
protected lemma BddBelow.of_closure : BddBelow (closure s) → BddBelow s 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] {s : Set α}
Goal: BddBelow (closure s) → BddBelow s
-- isClosed_Iic
theorem isClosed_Iic : IsClosed (Iic a) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] {a : α}
Goal: IsClosed (Set.Iic a)
-- ClosedIicTopology.isClosed_le'
@[deprecated isClosed_Iic (since := "2024-02-15")]
lemma ClosedIicTopology.isClosed_le' (a : α) : IsClosed {x | x ≤ a} 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] (a : α)
Goal: IsClosed {x | x ≤ a}
-- closure_Iic
@[simp]
theorem closure_Iic (a : α) : closure (Iic a) = Iic a 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] (a : α)
Goal: closure (Set.Iic a) = Set.Iic a
-- le_of_tendsto_of_frequently
theorem le_of_tendsto_of_frequently {x : Filter β} (lim : Tendsto f x (𝓝 a))
    (h : ∃ᶠ c in x, f c ≤ b) : a ≤ b 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] {f : β → α} {a b : α} {x : Filter β} (lim : Filter.Tendsto f x (𝓝 a)) (h : ∃ᶠ (c : β) in x, f c ≤ b)
Goal: a ≤ b
-- le_of_tendsto
theorem le_of_tendsto {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))
    (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] {f : β → α} {a b : α} {x : Filter β} [x.NeBot] (lim : Filter.Tendsto f x (𝓝 a)) (h : ∀ᶠ (c : β) in x, f c ≤ b)
Goal: a ≤ b
-- le_of_tendsto'
theorem le_of_tendsto' {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))
    (h : ∀ c, f c ≤ b) : a ≤ b 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] {f : β → α} {a b : α} {x : Filter β} [x.NeBot] (lim : Filter.Tendsto f x (𝓝 a)) (h : ∀ (c : β), f c ≤ b)
Goal: a ≤ b
-- upperBounds_closure
@[simp] lemma upperBounds_closure (s : Set α) : upperBounds (closure s : Set α) = upperBounds s 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] (s : Set α)
Goal: upperBounds (closure s) = upperBounds s
-- bddAbove_closure
@[simp] lemma bddAbove_closure : BddAbove (closure s) ↔ BddAbove s 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] {s : Set α}
Goal: BddAbove (closure s) ↔ BddAbove s
-- disjoint_nhds_atBot_iff
@[simp]
theorem disjoint_nhds_atBot_iff : Disjoint (𝓝 a) atBot ↔ ¬IsBot a 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] {a : α}
Goal: Disjoint (𝓝 a) Filter.atBot ↔ ¬IsBot a
-- disjoint_nhds_atBot
theorem disjoint_nhds_atBot (a : α) : Disjoint (𝓝 a) atBot 
Arguments: {α : Type u} [Preorder α] [NoBotOrder α] [TopologicalSpace α] [ClosedIicTopology α] (a : α)
Goal: Disjoint (𝓝 a) Filter.atBot
-- inf_nhds_atBot
@[simp]
theorem inf_nhds_atBot (a : α) : 𝓝 a ⊓ atBot = ⊥ 
Arguments: {α : Type u} [Preorder α] [NoBotOrder α] [TopologicalSpace α] [ClosedIicTopology α] (a : α)
Goal: 𝓝 a ⊓ Filter.atBot = ⊥
-- not_tendsto_nhds_of_tendsto_atBot
theorem not_tendsto_nhds_of_tendsto_atBot (hf : Tendsto f l atBot) (a : α) : ¬Tendsto f l (𝓝 a) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [NoBotOrder α] [TopologicalSpace α] [ClosedIicTopology α] {l : Filter β} [l.NeBot] {f : β → α} (hf : Filter.Tendsto f l Filter.atBot) (a : α)
Goal: ¬Filter.Tendsto f l (𝓝 a)
-- not_tendsto_atBot_of_tendsto_nhds
theorem not_tendsto_atBot_of_tendsto_nhds (hf : Tendsto f l (𝓝 a)) : ¬Tendsto f l atBot 
Arguments: {α : Type u} {β : Type v} [Preorder α] [NoBotOrder α] [TopologicalSpace α] [ClosedIicTopology α] {a : α} {l : Filter β} [l.NeBot] {f : β → α} (hf : Filter.Tendsto f l (𝓝 a))
Goal: ¬Filter.Tendsto f l Filter.atBot
-- isOpen_Ioi
theorem isOpen_Ioi : IsOpen (Ioi a) 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a : α}
Goal: IsOpen (Set.Ioi a)
-- interior_Ioi
@[simp]
theorem interior_Ioi : interior (Ioi a) = Ioi a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a : α}
Goal: interior (Set.Ioi a) = Set.Ioi a
-- Ioi_mem_nhds
theorem Ioi_mem_nhds (h : a < b) : Ioi a ∈ 𝓝 b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (h : a < b)
Goal: Set.Ioi a ∈ 𝓝 b
-- eventually_gt_nhds
theorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (hab : b < a)
Goal: ∀ᶠ (x : α) in 𝓝 a, b < x
-- Ici_mem_nhds
theorem Ici_mem_nhds (h : a < b) : Ici a ∈ 𝓝 b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (h : a < b)
Goal: Set.Ici a ∈ 𝓝 b
-- eventually_ge_nhds
theorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (hab : b < a)
Goal: ∀ᶠ (x : α) in 𝓝 a, b ≤ x
-- eventually_gt_of_tendsto_gt
theorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)
    (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a 
Arguments: {α : Type u} {γ : Type w} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v) (h : Filter.Tendsto f l (𝓝 v))
Goal: ∀ᶠ (a : γ) in l, u < f a
-- eventually_ge_of_tendsto_gt
theorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)
    (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a 
Arguments: {α : Type u} {γ : Type w} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v) (h : Filter.Tendsto f l (𝓝 v))
Goal: ∀ᶠ (a : γ) in l, u ≤ f a
-- Dense.exists_gt
protected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :
    ∃ y ∈ s, x < y 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α)
Goal: ∃ y ∈ s, x < y
-- Dense.exists_ge
protected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :
    ∃ y ∈ s, x ≤ y 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α)
Goal: ∃ y ∈ s, x ≤ y
-- Dense.exists_ge'
theorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :
    ∃ y ∈ s, x ≤ y 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {s : Set α} (hs : Dense s) (htop : ∀ (x : α), IsTop x → x ∈ s) (x : α)
Goal: ∃ y ∈ s, x ≤ y
-- Ioo_mem_nhdsWithin_Iio'
theorem Ioo_mem_nhdsWithin_Iio' (H : a < b) : Ioo a b ∈ 𝓝[<] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (H : a < b)
Goal: Set.Ioo a b ∈ 𝓝[<] b
-- Ioo_mem_nhdsWithin_Iio
theorem Ioo_mem_nhdsWithin_Iio (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b c : α} (H : b ∈ Set.Ioc a c)
Goal: Set.Ioo a c ∈ 𝓝[<] b
-- CovBy.nhdsWithin_Iio
theorem CovBy.nhdsWithin_Iio (h : a ⋖ b) : 𝓝[<] b = ⊥ 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (h : a ⋖ b)
Goal: 𝓝[<] b = ⊥
-- Ico_mem_nhdsWithin_Iio
theorem Ico_mem_nhdsWithin_Iio (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b c : α} (H : b ∈ Set.Ioc a c)
Goal: Set.Ico a c ∈ 𝓝[<] b
-- Ico_mem_nhdsWithin_Iio'
theorem Ico_mem_nhdsWithin_Iio' (H : a < b) : Ico a b ∈ 𝓝[<] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (H : a < b)
Goal: Set.Ico a b ∈ 𝓝[<] b
-- Ioc_mem_nhdsWithin_Iio
theorem Ioc_mem_nhdsWithin_Iio (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b c : α} (H : b ∈ Set.Ioc a c)
Goal: Set.Ioc a c ∈ 𝓝[<] b
-- Ioc_mem_nhdsWithin_Iio'
theorem Ioc_mem_nhdsWithin_Iio' (H : a < b) : Ioc a b ∈ 𝓝[<] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (H : a < b)
Goal: Set.Ioc a b ∈ 𝓝[<] b
-- Icc_mem_nhdsWithin_Iio
theorem Icc_mem_nhdsWithin_Iio (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b c : α} (H : b ∈ Set.Ioc a c)
Goal: Set.Icc a c ∈ 𝓝[<] b
-- Icc_mem_nhdsWithin_Iio'
theorem Icc_mem_nhdsWithin_Iio' (H : a < b) : Icc a b ∈ 𝓝[<] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (H : a < b)
Goal: Set.Icc a b ∈ 𝓝[<] b
-- nhdsWithin_Ico_eq_nhdsWithin_Iio
@[simp]
theorem nhdsWithin_Ico_eq_nhdsWithin_Iio (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (h : a < b)
Goal: 𝓝[Set.Ico a b] b = 𝓝[<] b
-- nhdsWithin_Ioo_eq_nhdsWithin_Iio
@[simp]
theorem nhdsWithin_Ioo_eq_nhdsWithin_Iio (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (h : a < b)
Goal: 𝓝[Set.Ioo a b] b = 𝓝[<] b
-- continuousWithinAt_Ico_iff_Iio
@[simp]
theorem continuousWithinAt_Ico_iff_Iio (h : a < b) :
    ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b)
Goal: ContinuousWithinAt f (Set.Ico a b) b ↔ ContinuousWithinAt f (Set.Iio b) b
-- continuousWithinAt_Ioo_iff_Iio
@[simp]
theorem continuousWithinAt_Ioo_iff_Iio (h : a < b) :
    ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b)
Goal: ContinuousWithinAt f (Set.Ioo a b) b ↔ ContinuousWithinAt f (Set.Iio b) b
-- Ioc_mem_nhdsWithin_Iic'
theorem Ioc_mem_nhdsWithin_Iic' (H : a < b) : Ioc a b ∈ 𝓝[≤] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (H : a < b)
Goal: Set.Ioc a b ∈ 𝓝[≤] b
-- Ioo_mem_nhdsWithin_Iic
theorem Ioo_mem_nhdsWithin_Iic (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b c : α} (H : b ∈ Set.Ioo a c)
Goal: Set.Ioo a c ∈ 𝓝[≤] b
-- Ico_mem_nhdsWithin_Iic
theorem Ico_mem_nhdsWithin_Iic (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b c : α} (H : b ∈ Set.Ioo a c)
Goal: Set.Ico a c ∈ 𝓝[≤] b
-- Ioc_mem_nhdsWithin_Iic
theorem Ioc_mem_nhdsWithin_Iic (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b c : α} (H : b ∈ Set.Ioc a c)
Goal: Set.Ioc a c ∈ 𝓝[≤] b
-- Icc_mem_nhdsWithin_Iic
theorem Icc_mem_nhdsWithin_Iic (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b c : α} (H : b ∈ Set.Ioc a c)
Goal: Set.Icc a c ∈ 𝓝[≤] b
-- Icc_mem_nhdsWithin_Iic'
theorem Icc_mem_nhdsWithin_Iic' (H : a < b) : Icc a b ∈ 𝓝[≤] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (H : a < b)
Goal: Set.Icc a b ∈ 𝓝[≤] b
-- nhdsWithin_Icc_eq_nhdsWithin_Iic
@[simp]
theorem nhdsWithin_Icc_eq_nhdsWithin_Iic (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (h : a < b)
Goal: 𝓝[Set.Icc a b] b = 𝓝[≤] b
-- nhdsWithin_Ioc_eq_nhdsWithin_Iic
@[simp]
theorem nhdsWithin_Ioc_eq_nhdsWithin_Iic (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {a b : α} (h : a < b)
Goal: 𝓝[Set.Ioc a b] b = 𝓝[≤] b
-- continuousWithinAt_Icc_iff_Iic
@[simp]
theorem continuousWithinAt_Icc_iff_Iic (h : a < b) :
    ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b)
Goal: ContinuousWithinAt f (Set.Icc a b) b ↔ ContinuousWithinAt f (Set.Iic b) b
-- continuousWithinAt_Ioc_iff_Iic
@[simp]
theorem continuousWithinAt_Ioc_iff_Iic (h : a < b) :
    ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b)
Goal: ContinuousWithinAt f (Set.Ioc a b) b ↔ ContinuousWithinAt f (Set.Iic b) b
-- isClosed_Ici
theorem isClosed_Ici {a : α} : IsClosed (Ici a) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] {a : α}
Goal: IsClosed (Set.Ici a)
-- ClosedIciTopology.isClosed_ge'
@[deprecated (since := "2024-02-15")]
lemma ClosedIciTopology.isClosed_ge' (a : α) : IsClosed {x | a ≤ x} 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] (a : α)
Goal: IsClosed {x | a ≤ x}
-- closure_Ici
@[simp]
theorem closure_Ici (a : α) : closure (Ici a) = Ici a 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] (a : α)
Goal: closure (Set.Ici a) = Set.Ici a
-- ge_of_tendsto_of_frequently
lemma ge_of_tendsto_of_frequently {x : Filter β} (lim : Tendsto f x (𝓝 a))
    (h : ∃ᶠ c in x, b ≤ f c) : b ≤ a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] {f : β → α} {a b : α} {x : Filter β} (lim : Filter.Tendsto f x (𝓝 a)) (h : ∃ᶠ (c : β) in x, b ≤ f c)
Goal: b ≤ a
-- ge_of_tendsto
theorem ge_of_tendsto {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))
    (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] {f : β → α} {a b : α} {x : Filter β} [x.NeBot] (lim : Filter.Tendsto f x (𝓝 a)) (h : ∀ᶠ (c : β) in x, b ≤ f c)
Goal: b ≤ a
-- ge_of_tendsto'
theorem ge_of_tendsto' {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))
    (h : ∀ c, b ≤ f c) : b ≤ a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] {f : β → α} {a b : α} {x : Filter β} [x.NeBot] (lim : Filter.Tendsto f x (𝓝 a)) (h : ∀ (c : β), b ≤ f c)
Goal: b ≤ a
-- lowerBounds_closure
@[simp] lemma lowerBounds_closure (s : Set α) : lowerBounds (closure s : Set α) = lowerBounds s 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] (s : Set α)
Goal: lowerBounds (closure s) = lowerBounds s
-- bddBelow_closure
@[simp] lemma bddBelow_closure : BddBelow (closure s) ↔ BddBelow s 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] {s : Set α}
Goal: BddBelow (closure s) ↔ BddBelow s
-- disjoint_nhds_atTop_iff
@[simp]
theorem disjoint_nhds_atTop_iff : Disjoint (𝓝 a) atTop ↔ ¬IsTop a 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] {a : α}
Goal: Disjoint (𝓝 a) Filter.atTop ↔ ¬IsTop a
-- disjoint_nhds_atTop
theorem disjoint_nhds_atTop (a : α) : Disjoint (𝓝 a) atTop 
Arguments: {α : Type u} [Preorder α] [NoTopOrder α] [TopologicalSpace α] [ClosedIciTopology α] (a : α)
Goal: Disjoint (𝓝 a) Filter.atTop
-- inf_nhds_atTop
@[simp]
theorem inf_nhds_atTop (a : α) : 𝓝 a ⊓ atTop = ⊥ 
Arguments: {α : Type u} [Preorder α] [NoTopOrder α] [TopologicalSpace α] [ClosedIciTopology α] (a : α)
Goal: 𝓝 a ⊓ Filter.atTop = ⊥
-- not_tendsto_nhds_of_tendsto_atTop
theorem not_tendsto_nhds_of_tendsto_atTop (hf : Tendsto f l atTop) (a : α) : ¬Tendsto f l (𝓝 a) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [NoTopOrder α] [TopologicalSpace α] [ClosedIciTopology α] {l : Filter β} [l.NeBot] {f : β → α} (hf : Filter.Tendsto f l Filter.atTop) (a : α)
Goal: ¬Filter.Tendsto f l (𝓝 a)
-- not_tendsto_atTop_of_tendsto_nhds
theorem not_tendsto_atTop_of_tendsto_nhds (hf : Tendsto f l (𝓝 a)) : ¬Tendsto f l atTop 
Arguments: {α : Type u} {β : Type v} [Preorder α] [NoTopOrder α] [TopologicalSpace α] [ClosedIciTopology α] {a : α} {l : Filter β} [l.NeBot] {f : β → α} (hf : Filter.Tendsto f l (𝓝 a))
Goal: ¬Filter.Tendsto f l Filter.atTop
-- isOpen_Iio
theorem isOpen_Iio : IsOpen (Iio a) 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a : α}
Goal: IsOpen (Set.Iio a)
-- interior_Iio
@[simp] theorem interior_Iio : interior (Iio a) = Iio a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a : α}
Goal: interior (Set.Iio a) = Set.Iio a
-- Iio_mem_nhds
theorem Iio_mem_nhds (h : a < b) : Iio b ∈ 𝓝 a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (h : a < b)
Goal: Set.Iio b ∈ 𝓝 a