ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- interior_Icc
@[simp]
theorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α] {a b : α}
Goal: interior (Set.Icc a b) = Set.Ioo a b
-- Icc_mem_nhds_iff
@[simp]
theorem Icc_mem_nhds_iff [NoMinOrder α] [NoMaxOrder α] {a b x : α} :
    Icc a b ∈ 𝓝 x ↔ x ∈ Ioo a b 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α] {a b x : α}
Goal: Set.Icc a b ∈ 𝓝 x ↔ x ∈ Set.Ioo a b
-- interior_Ico
@[simp]
theorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] {a b : α}
Goal: interior (Set.Ico a b) = Set.Ioo a b
-- Ico_mem_nhds_iff
@[simp]
theorem Ico_mem_nhds_iff [NoMinOrder α] {a b x : α} : Ico a b ∈ 𝓝 x ↔ x ∈ Ioo a b 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] {a b x : α}
Goal: Set.Ico a b ∈ 𝓝 x ↔ x ∈ Set.Ioo a b
-- interior_Ioc
@[simp]
theorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMaxOrder α] {a b : α}
Goal: interior (Set.Ioc a b) = Set.Ioo a b
-- Ioc_mem_nhds_iff
@[simp]
theorem Ioc_mem_nhds_iff [NoMaxOrder α] {a b x : α} : Ioc a b ∈ 𝓝 x ↔ x ∈ Ioo a b 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMaxOrder α] {a b x : α}
Goal: Set.Ioc a b ∈ 𝓝 x ↔ x ∈ Set.Ioo a b
-- closure_interior_Icc
theorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} (h : a ≠ b)
Goal: closure (interior (Set.Icc a b)) = Set.Icc a b
-- Ioc_subset_closure_interior
theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] (a b : α)
Goal: Set.Ioc a b ⊆ closure (interior (Set.Ioc a b))
-- Ico_subset_closure_interior
theorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] (a b : α)
Goal: Set.Ico a b ⊆ closure (interior (Set.Ico a b))
-- frontier_Ici'
@[simp]
theorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} (ha : (Set.Iio a).Nonempty)
Goal: frontier (Set.Ici a) = {a}
-- frontier_Ici
theorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] {a : α}
Goal: frontier (Set.Ici a) = {a}
-- frontier_Iic'
@[simp]
theorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} (ha : (Set.Ioi a).Nonempty)
Goal: frontier (Set.Iic a) = {a}
-- frontier_Iic
theorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMaxOrder α] {a : α}
Goal: frontier (Set.Iic a) = {a}
-- frontier_Ioi'
@[simp]
theorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} (ha : (Set.Ioi a).Nonempty)
Goal: frontier (Set.Ioi a) = {a}
-- frontier_Ioi
theorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMaxOrder α] {a : α}
Goal: frontier (Set.Ioi a) = {a}
-- frontier_Iio'
@[simp]
theorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} (ha : (Set.Iio a).Nonempty)
Goal: frontier (Set.Iio a) = {a}
-- frontier_Iio
theorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] {a : α}
Goal: frontier (Set.Iio a) = {a}
-- frontier_Icc
@[simp]
theorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :
    frontier (Icc a b) = {a, b} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b)
Goal: frontier (Set.Icc a b) = {a, b}
-- frontier_Ioo
@[simp]
theorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: frontier (Set.Ioo a b) = {a, b}
-- frontier_Ico
@[simp]
theorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] {a b : α} (h : a < b)
Goal: frontier (Set.Ico a b) = {a, b}
-- frontier_Ioc
@[simp]
theorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMaxOrder α] {a b : α} (h : a < b)
Goal: frontier (Set.Ioc a b) = {a, b}
-- nhdsWithin_Ioi_neBot'
theorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :
    NeBot (𝓝[Ioi a] b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} (H₁ : (Set.Ioi a).Nonempty) (H₂ : a ≤ b)
Goal: (𝓝[Set.Ioi a] b).NeBot
-- nhdsWithin_Ioi_neBot
theorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMaxOrder α] {a b : α} (H : a ≤ b)
Goal: (𝓝[Set.Ioi a] b).NeBot
-- nhdsWithin_Ioi_self_neBot'
theorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} (H : (Set.Ioi a).Nonempty)
Goal: (𝓝[>] a).NeBot
-- nhdsWithin_Iio_neBot'
theorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :
    NeBot (𝓝[Iio c] b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {b c : α} (H₁ : (Set.Iio c).Nonempty) (H₂ : b ≤ c)
Goal: (𝓝[Set.Iio c] b).NeBot
-- nhdsWithin_Iio_neBot
theorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] {a b : α} (H : a ≤ b)
Goal: (𝓝[Set.Iio b] a).NeBot
-- nhdsWithin_Iio_self_neBot'
theorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {b : α} (H : (Set.Iio b).Nonempty)
Goal: (𝓝[<] b).NeBot
-- right_nhdsWithin_Ico_neBot
theorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} (H : a < b)
Goal: (𝓝[Set.Ico a b] b).NeBot
-- left_nhdsWithin_Ioc_neBot
theorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} (H : a < b)
Goal: (𝓝[Set.Ioc a b] a).NeBot
-- left_nhdsWithin_Ioo_neBot
theorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} (H : a < b)
Goal: (𝓝[Set.Ioo a b] a).NeBot
-- right_nhdsWithin_Ioo_neBot
theorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} (H : a < b)
Goal: (𝓝[Set.Ioo a b] b).NeBot
-- comap_coe_nhdsWithin_Iio_of_Ioo_subset
theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
    (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {b : α} {s : Set α} (hb : s ⊆ Set.Iio b) (hs : s.Nonempty → ∃ a < b, Set.Ioo a b ⊆ s)
Goal: Filter.comap Subtype.val (𝓝[<] b) = Filter.atTop
-- comap_coe_nhdsWithin_Ioi_of_Ioo_subset
theorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)
    (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} {s : Set α} (ha : s ⊆ Set.Ioi a) (hs : s.Nonempty → ∃ b > a, Set.Ioo a b ⊆ s)
Goal: Filter.comap Subtype.val (𝓝[>] a) = Filter.atBot
-- map_coe_atTop_of_Ioo_subset
theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
    map ((↑) : s → α) atTop = 𝓝[<] b 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {b : α} {s : Set α} (hb : s ⊆ Set.Iio b) (hs : ∀ a' < b, ∃ a < b, Set.Ioo a b ⊆ s)
Goal: Filter.map Subtype.val Filter.atTop = 𝓝[<] b
-- map_coe_atBot_of_Ioo_subset
theorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :
    map ((↑) : s → α) atBot = 𝓝[>] a 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} {s : Set α} (ha : s ⊆ Set.Ioi a) (hs : ∀ b' > a, ∃ b > a, Set.Ioo a b ⊆ s)
Goal: Filter.map Subtype.val Filter.atBot = 𝓝[>] a
-- comap_coe_Ioo_nhdsWithin_Iio
theorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] (a b : α)
Goal: Filter.comap Subtype.val (𝓝[<] b) = Filter.atTop
-- comap_coe_Ioo_nhdsWithin_Ioi
theorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] (a b : α)
Goal: Filter.comap Subtype.val (𝓝[>] a) = Filter.atBot
-- comap_coe_Ioi_nhdsWithin_Ioi
theorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] (a : α)
Goal: Filter.comap Subtype.val (𝓝[>] a) = Filter.atBot
-- comap_coe_Iio_nhdsWithin_Iio
theorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] (a : α)
Goal: Filter.comap Subtype.val (𝓝[<] a) = Filter.atTop
-- map_coe_Ioo_atTop
@[simp]
theorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: Filter.map Subtype.val Filter.atTop = 𝓝[<] b
-- map_coe_Ioo_atBot
@[simp]
theorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: Filter.map Subtype.val Filter.atBot = 𝓝[>] a
-- map_coe_Ioi_atBot
@[simp]
theorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] (a : α)
Goal: Filter.map Subtype.val Filter.atBot = 𝓝[>] a
-- map_coe_Iio_atTop
@[simp]
theorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] (a : α)
Goal: Filter.map Subtype.val Filter.atTop = 𝓝[<] a
-- tendsto_comp_coe_Ioo_atTop
@[simp]
theorem tendsto_comp_coe_Ioo_atTop (h : a < b) :
    Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} {l : Filter β} {f : α → β} (h : a < b)
Goal: Filter.Tendsto (fun x => f ↑x) Filter.atTop l ↔ Filter.Tendsto f (𝓝[<] b) l
-- tendsto_comp_coe_Ioo_atBot
@[simp]
theorem tendsto_comp_coe_Ioo_atBot (h : a < b) :
    Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} {l : Filter β} {f : α → β} (h : a < b)
Goal: Filter.Tendsto (fun x => f ↑x) Filter.atBot l ↔ Filter.Tendsto f (𝓝[>] a) l
-- tendsto_comp_coe_Ioi_atBot
@[simp, nolint simpNF]
theorem tendsto_comp_coe_Ioi_atBot :
    Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} {l : Filter β} {f : α → β}
Goal: Filter.Tendsto (fun x => f ↑x) Filter.atBot l ↔ Filter.Tendsto f (𝓝[>] a) l
-- tendsto_comp_coe_Iio_atTop
@[simp, nolint simpNF]
theorem tendsto_comp_coe_Iio_atTop :
    Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} {l : Filter β} {f : α → β}
Goal: Filter.Tendsto (fun x => f ↑x) Filter.atTop l ↔ Filter.Tendsto f (𝓝[<] a) l
-- tendsto_Ioo_atTop
@[simp]
theorem tendsto_Ioo_atTop {f : β → Ioo a b} :
    Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} {l : Filter β} {f : β → ↑(Set.Ioo a b)}
Goal: Filter.Tendsto f l Filter.atTop ↔ Filter.Tendsto (fun x => ↑(f x)) l (𝓝[<] b)
-- tendsto_Ioo_atBot
@[simp]
theorem tendsto_Ioo_atBot {f : β → Ioo a b} :
    Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} {l : Filter β} {f : β → ↑(Set.Ioo a b)}
Goal: Filter.Tendsto f l Filter.atBot ↔ Filter.Tendsto (fun x => ↑(f x)) l (𝓝[>] a)
-- tendsto_Ioi_atBot
@[simp]
theorem tendsto_Ioi_atBot {f : β → Ioi a} :
    Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} {l : Filter β} {f : β → ↑(Set.Ioi a)}
Goal: Filter.Tendsto f l Filter.atBot ↔ Filter.Tendsto (fun x => ↑(f x)) l (𝓝[>] a)
-- tendsto_Iio_atTop
@[simp]
theorem tendsto_Iio_atTop {f : β → Iio a} :
    Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a : α} {l : Filter β} {f : β → ↑(Set.Iio a)}
Goal: Filter.Tendsto f l Filter.atTop ↔ Filter.Tendsto (fun x => ↑(f x)) l (𝓝[<] a)
-- Dense.exists_countable_dense_subset_no_bot_top
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s]
    (hs : Dense s) :
    ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [Nontrivial α] {s : Set α} [TopologicalSpace.SeparableSpace ↑s] (hs : Dense s)
Goal: ∃ t ⊆ s, t.Countable ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∉ t) ∧ ∀ (x : α), IsTop x → x ∉ t
-- exists_countable_dense_no_bot_top
theorem exists_countable_dense_no_bot_top [SeparableSpace α] [Nontrivial α] :
    ∃ s : Set α, s.Countable ∧ Dense s ∧ (∀ x, IsBot x → x ∉ s) ∧ ∀ x, IsTop x → x ∉ s 
Arguments: (α : Type u_1) [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [TopologicalSpace.SeparableSpace α] [Nontrivial α]
Goal: ∃ s, s.Countable ∧ Dense s ∧ (∀ (x : α), IsBot x → x ∉ s) ∧ ∀ (x : α), IsTop x → x ∉ s
-- PNat.dist_eq
theorem dist_eq (x y : ℕ+) : dist x y = |(↑x : ℝ) - ↑y| 
Arguments: (x y : ℕ+)
Goal: dist x y = |↑↑x - ↑↑y|
-- PNat.dist_coe
@[simp, norm_cast]
theorem dist_coe (x y : ℕ+) : dist (↑x : ℕ) (↑y : ℕ) = dist x y 
Arguments: (x y : ℕ+)
Goal: dist ↑x ↑y = dist x y
-- PNat.uniformEmbedding_coe
theorem uniformEmbedding_coe : UniformEmbedding ((↑) : ℕ+ → ℕ) 
Goal: UniformEmbedding PNat.val
-- continuousAt_sign_of_pos
theorem continuousAt_sign_of_pos {a : α} (h : 0 < a) : ContinuousAt SignType.sign a 
Arguments: {α : Type u_1} [Zero α] [TopologicalSpace α] [PartialOrder α] [DecidableRel fun x x_1 => x < x_1] [OrderTopology α] {a : α} (h : 0 < a)
Goal: ContinuousAt (⇑SignType.sign) a
-- continuousAt_sign_of_neg
theorem continuousAt_sign_of_neg {a : α} (h : a < 0) : ContinuousAt SignType.sign a 
Arguments: {α : Type u_1} [Zero α] [TopologicalSpace α] [PartialOrder α] [DecidableRel fun x x_1 => x < x_1] [OrderTopology α] {a : α} (h : a < 0)
Goal: ContinuousAt (⇑SignType.sign) a
-- continuousAt_sign_of_ne_zero
theorem continuousAt_sign_of_ne_zero {a : α} (h : a ≠ 0) : ContinuousAt SignType.sign a 
Arguments: {α : Type u_1} [Zero α] [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} (h : a ≠ 0)
Goal: ContinuousAt (⇑SignType.sign) a
-- continuous_real_toNNReal
theorem _root_.continuous_real_toNNReal : Continuous Real.toNNReal 
Goal: Continuous Real.toNNReal
-- ContinuousMap.realToNNReal_apply
@[simps (config := .asFn)]
noncomputable def _root_.ContinuousMap.realToNNReal : C(ℝ, ℝ≥0) 
Goal: ⇑ContinuousMap.realToNNReal = Real.toNNReal
-- NNReal.continuous_coe
theorem continuous_coe : Continuous ((↑) : ℝ≥0 → ℝ) 
Goal: Continuous NNReal.toReal
-- ContinuousMap.coeNNRealReal_apply
@[simps (config := .asFn)]
def _root_.ContinuousMap.coeNNRealReal : C(ℝ≥0, ℝ) 
Goal: ⇑ContinuousMap.coeNNRealReal = NNReal.toReal
-- NNReal.tendsto_coe
@[simp, norm_cast]
theorem tendsto_coe {f : Filter α} {m : α → ℝ≥0} {x : ℝ≥0} :
    Tendsto (fun a => (m a : ℝ)) f (𝓝 (x : ℝ)) ↔ Tendsto m f (𝓝 x) 
Arguments: {α : Type u_1} {f : Filter α} {m : α → NNReal} {x : NNReal}
Goal: Filter.Tendsto (fun a => ↑(m a)) f (𝓝 ↑x) ↔ Filter.Tendsto m f (𝓝 x)
-- NNReal.tendsto_coe'
theorem tendsto_coe' {f : Filter α} [NeBot f] {m : α → ℝ≥0} {x : ℝ} :
    Tendsto (fun a => m a : α → ℝ) f (𝓝 x) ↔ ∃ hx : 0 ≤ x, Tendsto m f (𝓝 ⟨x, hx⟩) 
Arguments: {α : Type u_1} {f : Filter α} [f.NeBot] {m : α → NNReal} {x : ℝ}
Goal: Filter.Tendsto (fun a => ↑(m a)) f (𝓝 x) ↔ ∃ (hx : 0 ≤ x), Filter.Tendsto m f (𝓝 ⟨x, hx⟩)
-- NNReal.map_coe_atTop
@[simp] theorem map_coe_atTop : map toReal atTop = atTop 
Goal: Filter.map NNReal.toReal Filter.atTop = Filter.atTop
-- NNReal.comap_coe_atTop
@[simp]
theorem comap_coe_atTop : comap toReal atTop = atTop 
Goal: Filter.comap NNReal.toReal Filter.atTop = Filter.atTop
-- NNReal.tendsto_coe_atTop
@[simp, norm_cast]
theorem tendsto_coe_atTop {f : Filter α} {m : α → ℝ≥0} :
    Tendsto (fun a => (m a : ℝ)) f atTop ↔ Tendsto m f atTop 
Arguments: {α : Type u_1} {f : Filter α} {m : α → NNReal}
Goal: Filter.Tendsto (fun a => ↑(m a)) f Filter.atTop ↔ Filter.Tendsto m f Filter.atTop
-- tendsto_real_toNNReal
theorem _root_.tendsto_real_toNNReal {f : Filter α} {m : α → ℝ} {x : ℝ} (h : Tendsto m f (𝓝 x)) :
    Tendsto (fun a => Real.toNNReal (m a)) f (𝓝 (Real.toNNReal x)) 
Arguments: {α : Type u_1} {f : Filter α} {m : α → ℝ} {x : ℝ} (h : Filter.Tendsto m f (𝓝 x))
Goal: Filter.Tendsto (fun a => (m a).toNNReal) f (𝓝 x.toNNReal)
-- Real.map_toNNReal_atTop
@[simp]
theorem _root_.Real.map_toNNReal_atTop : map Real.toNNReal atTop = atTop 
Goal: Filter.map Real.toNNReal Filter.atTop = Filter.atTop
-- tendsto_real_toNNReal_atTop
theorem _root_.tendsto_real_toNNReal_atTop : Tendsto Real.toNNReal atTop atTop 
Goal: Filter.Tendsto Real.toNNReal Filter.atTop Filter.atTop
-- Real.comap_toNNReal_atTop
@[simp]
theorem _root_.Real.comap_toNNReal_atTop : comap Real.toNNReal atTop = atTop 
Goal: Filter.comap Real.toNNReal Filter.atTop = Filter.atTop
-- Real.tendsto_toNNReal_atTop_iff
@[simp]
theorem _root_.Real.tendsto_toNNReal_atTop_iff {l : Filter α} {f : α → ℝ} :
    Tendsto (fun x ↦ (f x).toNNReal) l atTop ↔ Tendsto f l atTop 
Arguments: {α : Type u_1} {l : Filter α} {f : α → ℝ}
Goal: Filter.Tendsto (fun x => (f x).toNNReal) l Filter.atTop ↔ Filter.Tendsto f l Filter.atTop
-- Real.tendsto_toNNReal_atTop
theorem _root_.Real.tendsto_toNNReal_atTop : Tendsto Real.toNNReal atTop atTop 
Goal: Filter.Tendsto Real.toNNReal Filter.atTop Filter.atTop
-- NNReal.nhds_zero
theorem nhds_zero : 𝓝 (0 : ℝ≥0) = ⨅ (a : ℝ≥0) (_ : a ≠ 0), 𝓟 (Iio a) 
Goal: 𝓝 0 = ⨅ a, ⨅ (_ : a ≠ 0), 𝓟 (Set.Iio a)
-- NNReal.nhds_zero_basis
theorem nhds_zero_basis : (𝓝 (0 : ℝ≥0)).HasBasis (fun a : ℝ≥0 => 0 < a) fun a => Iio a 
Goal: (𝓝 0).HasBasis (fun a => 0 < a) fun a => Set.Iio a
-- NNReal.hasSum_coe
@[norm_cast]
theorem hasSum_coe {f : α → ℝ≥0} {r : ℝ≥0} : HasSum (fun a => (f a : ℝ)) (r : ℝ) ↔ HasSum f r 
Arguments: {α : Type u_1} {f : α → NNReal} {r : NNReal}
Goal: HasSum (fun a => ↑(f a)) ↑r ↔ HasSum f r
-- HasSum.toNNReal
protected theorem _root_.HasSum.toNNReal {f : α → ℝ} {y : ℝ} (hf₀ : ∀ n, 0 ≤ f n)
    (hy : HasSum f y) : HasSum (fun x => Real.toNNReal (f x)) y.toNNReal 
Arguments: {α : Type u_1} {f : α → ℝ} {y : ℝ} (hf₀ : ∀ (n : α), 0 ≤ f n) (hy : HasSum f y)
Goal: HasSum (fun x => (f x).toNNReal) y.toNNReal
-- NNReal.hasSum_real_toNNReal_of_nonneg
theorem hasSum_real_toNNReal_of_nonneg {f : α → ℝ} (hf_nonneg : ∀ n, 0 ≤ f n) (hf : Summable f) :
    HasSum (fun n => Real.toNNReal (f n)) (Real.toNNReal (∑' n, f n)) 
Arguments: {α : Type u_1} {f : α → ℝ} (hf_nonneg : ∀ (n : α), 0 ≤ f n) (hf : Summable f)
Goal: HasSum (fun n => (f n).toNNReal) (∑' (n : α), f n).toNNReal
-- NNReal.summable_coe
@[norm_cast]
theorem summable_coe {f : α → ℝ≥0} : (Summable fun a => (f a : ℝ)) ↔ Summable f 
Arguments: {α : Type u_1} {f : α → NNReal}
Goal: (Summable fun a => ↑(f a)) ↔ Summable f
-- NNReal.summable_mk
theorem summable_mk {f : α → ℝ} (hf : ∀ n, 0 ≤ f n) :
    (@Summable ℝ≥0 _ _ _ fun n => ⟨f n, hf n⟩) ↔ Summable f 
Arguments: {α : Type u_1} {f : α → ℝ} (hf : ∀ (n : α), 0 ≤ f n)
Goal: (Summable fun n => ⟨f n, ⋯⟩) ↔ Summable f
-- NNReal.coe_tsum
@[norm_cast]
theorem coe_tsum {f : α → ℝ≥0} : ↑(∑' a, f a) = ∑' a, (f a : ℝ) 
Arguments: {α : Type u_1} {f : α → NNReal}
Goal: ↑(∑' (a : α), f a) = ∑' (a : α), ↑(f a)
-- NNReal.coe_tsum_of_nonneg
theorem coe_tsum_of_nonneg {f : α → ℝ} (hf₁ : ∀ n, 0 ≤ f n) :
    (⟨∑' n, f n, tsum_nonneg hf₁⟩ : ℝ≥0) = (∑' n, ⟨f n, hf₁ n⟩ : ℝ≥0) 
Arguments: {α : Type u_1} {f : α → ℝ} (hf₁ : ∀ (n : α), 0 ≤ f n)
Goal: ⟨∑' (n : α), f n, ⋯⟩ = ∑' (n : α), ⟨f n, ⋯⟩
-- NNReal.tsum_mul_left
nonrec theorem tsum_mul_left (a : ℝ≥0) (f : α → ℝ≥0) : ∑' x, a * f x = a * ∑' x, f x 
Arguments: {α : Type u_1} (a : NNReal) (f : α → NNReal)
Goal: ∑' (x : α), a * f x = a * ∑' (x : α), f x
-- NNReal.tsum_mul_right
nonrec theorem tsum_mul_right (f : α → ℝ≥0) (a : ℝ≥0) : ∑' x, f x * a = (∑' x, f x) * a 
Arguments: {α : Type u_1} (f : α → NNReal) (a : NNReal)
Goal: ∑' (x : α), f x * a = (∑' (x : α), f x) * a
-- NNReal.summable_comp_injective
theorem summable_comp_injective {β : Type*} {f : α → ℝ≥0} (hf : Summable f) {i : β → α}
    (hi : Function.Injective i) : Summable (f ∘ i) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → NNReal} (hf : Summable f) {i : β → α} (hi : Function.Injective i)
Goal: Summable (f ∘ i)
-- NNReal.summable_nat_add
theorem summable_nat_add (f : ℕ → ℝ≥0) (hf : Summable f) (k : ℕ) : Summable fun i => f (i + k) 
Arguments: (f : ℕ → NNReal) (hf : Summable f) (k : ℕ)
Goal: Summable fun i => f (i + k)
-- NNReal.summable_nat_add_iff
nonrec theorem summable_nat_add_iff {f : ℕ → ℝ≥0} (k : ℕ) :
    (Summable fun i => f (i + k)) ↔ Summable f 
Arguments: {f : ℕ → NNReal} (k : ℕ)
Goal: (Summable fun i => f (i + k)) ↔ Summable f