ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.tendsto_nhds_min_left
theorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :
    Tendsto (fun i => min (f i) a) l (𝓝[<] a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f : β → α} {l : Filter β} {a : α} (h : Filter.Tendsto f l (𝓝[<] a))
Goal: Filter.Tendsto (fun i => min (f i) a) l (𝓝[<] a)
-- Dense.exists_between
theorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :
    ∃ z ∈ s, z ∈ Ioo x y 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y)
Goal: ∃ z ∈ s, z ∈ Set.Ioo x y
-- Dense.Ioi_eq_biUnion
theorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :
    Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α)
Goal: Set.Ioi x = ⋃ y ∈ s ∩ Set.Ioi x, Set.Ioi y
-- Dense.Iio_eq_biUnion
theorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :
    Iio x = ⋃ y ∈ s ∩ Iio x, Iio y 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α)
Goal: Set.Iio x = ⋃ y ∈ s ∩ Set.Iio x, Set.Iio y
-- Monotone.map_sSup_of_continuousAt'
theorem Monotone.map_sSup_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A))
    (Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
    f (sSup A) = sSup (f '' A) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A)) (Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault)
Goal: f (sSup A) = sSup (f '' A)
-- Monotone.map_iSup_of_continuousAt'
theorem Monotone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iSup g)) (Mf : Monotone f)
    (bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨆ i, f (g i) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {ι : Sort u_4} [Nonempty ι] {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (bdd : BddAbove (Set.range g) := by bddDefault)
Goal: f (⨆ i, g i) = ⨆ i, f (g i)
-- Monotone.map_sInf_of_continuousAt'
theorem Monotone.map_sInf_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
    (Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
    f (sInf A) = sInf (f '' A) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A)) (Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault)
Goal: f (sInf A) = sInf (f '' A)
-- Monotone.map_iInf_of_continuousAt'
theorem Monotone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
    (bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨅ i, f (g i) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {ι : Sort u_4} [Nonempty ι] {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iInf g)) (Mf : Monotone f) (bdd : BddBelow (Set.range g) := by bddDefault)
Goal: f (⨅ i, g i) = ⨅ i, f (g i)
-- Antitone.map_sInf_of_continuousAt'
theorem Antitone.map_sInf_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
    (Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
    f (sInf A) = sSup (f '' A) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A)) (Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault)
Goal: f (sInf A) = sSup (f '' A)
-- Antitone.map_iInf_of_continuousAt'
theorem Antitone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iInf g)) (Af : Antitone f)
    (bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨆ i, f (g i) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {ι : Sort u_4} [Nonempty ι] {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iInf g)) (Af : Antitone f) (bdd : BddBelow (Set.range g) := by bddDefault)
Goal: f (⨅ i, g i) = ⨆ i, f (g i)
-- Antitone.map_sSup_of_continuousAt'
theorem Antitone.map_sSup_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A))
    (Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
    f (sSup A) = sInf (f '' A) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A)) (Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault)
Goal: f (sSup A) = sInf (f '' A)
-- Antitone.map_iSup_of_continuousAt'
theorem Antitone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iSup g)) (Af : Antitone f)
    (bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨅ i, f (g i) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {ι : Sort u_4} [Nonempty ι] {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iSup g)) (Af : Antitone f) (bdd : BddAbove (Set.range g) := by bddDefault)
Goal: f (⨆ i, g i) = ⨅ i, f (g i)
-- sSup_mem_closure
theorem sSup_mem_closure {s : Set α} (hs : s.Nonempty) : sSup s ∈ closure s 
Arguments: {α : Type u_1} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hs : s.Nonempty)
Goal: sSup s ∈ closure s
-- sInf_mem_closure
theorem sInf_mem_closure {s : Set α} (hs : s.Nonempty) : sInf s ∈ closure s 
Arguments: {α : Type u_1} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hs : s.Nonempty)
Goal: sInf s ∈ closure s
-- IsClosed.sSup_mem
theorem IsClosed.sSup_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sSup s ∈ s 
Arguments: {α : Type u_1} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hs : s.Nonempty) (hc : IsClosed s)
Goal: sSup s ∈ s
-- IsClosed.sInf_mem
theorem IsClosed.sInf_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sInf s ∈ s 
Arguments: {α : Type u_1} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hs : s.Nonempty) (hc : IsClosed s)
Goal: sInf s ∈ s
-- Monotone.map_sSup_of_continuousAt
theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
    (Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [CompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s)) (Mf : Monotone f) (fbot : f ⊥ = ⊥)
Goal: f (sSup s) = sSup (f '' s)
-- Monotone.map_iSup_of_continuousAt
theorem Monotone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fbot : f ⊥ = ⊥) :
    f (⨆ i, g i) = ⨆ i, f (g i) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [CompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {ι : Sort u_4} {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fbot : f ⊥ = ⊥)
Goal: f (⨆ i, g i) = ⨆ i, f (g i)
-- Monotone.map_sInf_of_continuousAt
theorem Monotone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
    (Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (sInf s) = sInf (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [CompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s)) (Mf : Monotone f) (ftop : f ⊤ = ⊤)
Goal: f (sInf s) = sInf (f '' s)
-- Monotone.map_iInf_of_continuousAt
theorem Monotone.map_iInf_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iInf g)) (Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (iInf g) = iInf (f ∘ g) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [CompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {ι : Sort u_4} {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iInf g)) (Mf : Monotone f) (ftop : f ⊤ = ⊤)
Goal: f (iInf g) = iInf (f ∘ g)
-- Antitone.map_sSup_of_continuousAt
theorem Antitone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
    (Af : Antitone f) (fbot : f ⊥ = ⊤) : f (sSup s) = sInf (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [CompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s)) (Af : Antitone f) (fbot : f ⊥ = ⊤)
Goal: f (sSup s) = sInf (f '' s)
-- Antitone.map_iSup_of_continuousAt
theorem Antitone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iSup g)) (Af : Antitone f) (fbot : f ⊥ = ⊤) :
    f (⨆ i, g i) = ⨅ i, f (g i) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [CompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {ι : Sort u_4} {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iSup g)) (Af : Antitone f) (fbot : f ⊥ = ⊤)
Goal: f (⨆ i, g i) = ⨅ i, f (g i)
-- Antitone.map_sInf_of_continuousAt
theorem Antitone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
    (Af : Antitone f) (ftop : f ⊤ = ⊥) : f (sInf s) = sSup (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [CompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s)) (Af : Antitone f) (ftop : f ⊤ = ⊥)
Goal: f (sInf s) = sSup (f '' s)
-- Antitone.map_iInf_of_continuousAt
theorem Antitone.map_iInf_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
    (Cf : ContinuousAt f (iInf g)) (Af : Antitone f) (ftop : f ⊤ = ⊥) : f (iInf g) = iSup (f ∘ g) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [CompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {ι : Sort u_4} {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iInf g)) (Af : Antitone f) (ftop : f ⊤ = ⊥)
Goal: f (iInf g) = iSup (f ∘ g)
-- csSup_mem_closure
theorem csSup_mem_closure {s : Set α} (hs : s.Nonempty) (B : BddAbove s) : sSup s ∈ closure s 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hs : s.Nonempty) (B : BddAbove s)
Goal: sSup s ∈ closure s
-- csInf_mem_closure
theorem csInf_mem_closure {s : Set α} (hs : s.Nonempty) (B : BddBelow s) : sInf s ∈ closure s 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hs : s.Nonempty) (B : BddBelow s)
Goal: sInf s ∈ closure s
-- IsClosed.csSup_mem
theorem IsClosed.csSup_mem {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddAbove s) :
    sSup s ∈ s 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddAbove s)
Goal: sSup s ∈ s
-- IsClosed.csInf_mem
theorem IsClosed.csInf_mem {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddBelow s) :
    sInf s ∈ s 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddBelow s)
Goal: sInf s ∈ s
-- IsClosed.isLeast_csInf
theorem IsClosed.isLeast_csInf {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddBelow s) :
    IsLeast s (sInf s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddBelow s)
Goal: IsLeast s (sInf s)
-- IsClosed.isGreatest_csSup
theorem IsClosed.isGreatest_csSup {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddAbove s) :
    IsGreatest s (sSup s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddAbove s)
Goal: IsGreatest s (sSup s)
-- Monotone.map_csSup_of_continuousAt
theorem Monotone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
    (Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s) : f (sSup s) = sSup (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s)) (Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s)
Goal: f (sSup s) = sSup (f '' s)
-- Monotone.map_ciSup_of_continuousAt
theorem Monotone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
    (Mf : Monotone f) (H : BddAbove (range g)) : f (⨆ i, g i) = ⨆ i, f (g i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] [Nonempty γ] {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i)) (Mf : Monotone f) (H : BddAbove (Set.range g))
Goal: f (⨆ i, g i) = ⨆ i, f (g i)
-- Monotone.map_csInf_of_continuousAt
theorem Monotone.map_csInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
    (Mf : Monotone f) (ne : s.Nonempty) (H : BddBelow s) : f (sInf s) = sInf (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s)) (Mf : Monotone f) (ne : s.Nonempty) (H : BddBelow s)
Goal: f (sInf s) = sInf (f '' s)
-- Monotone.map_ciInf_of_continuousAt
theorem Monotone.map_ciInf_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨅ i, g i))
    (Mf : Monotone f) (H : BddBelow (range g)) : f (⨅ i, g i) = ⨅ i, f (g i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] [Nonempty γ] {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨅ i, g i)) (Mf : Monotone f) (H : BddBelow (Set.range g))
Goal: f (⨅ i, g i) = ⨅ i, f (g i)
-- Antitone.map_csSup_of_continuousAt
theorem Antitone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
    (Af : Antitone f) (ne : s.Nonempty) (H : BddAbove s) : f (sSup s) = sInf (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s)) (Af : Antitone f) (ne : s.Nonempty) (H : BddAbove s)
Goal: f (sSup s) = sInf (f '' s)
-- Antitone.map_ciSup_of_continuousAt
theorem Antitone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
    (Af : Antitone f) (H : BddAbove (range g)) : f (⨆ i, g i) = ⨅ i, f (g i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] [Nonempty γ] {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i)) (Af : Antitone f) (H : BddAbove (Set.range g))
Goal: f (⨆ i, g i) = ⨅ i, f (g i)
-- Antitone.map_csInf_of_continuousAt
theorem Antitone.map_csInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
    (Af : Antitone f) (ne : s.Nonempty) (H : BddBelow s) : f (sInf s) = sSup (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s)) (Af : Antitone f) (ne : s.Nonempty) (H : BddBelow s)
Goal: f (sInf s) = sSup (f '' s)
-- Antitone.map_ciInf_of_continuousAt
theorem Antitone.map_ciInf_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨅ i, g i))
    (Af : Antitone f) (H : BddBelow (range g)) : f (⨅ i, g i) = ⨆ i, f (g i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] [Nonempty γ] {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨅ i, g i)) (Af : Antitone f) (H : BddBelow (Set.range g))
Goal: f (⨅ i, g i) = ⨆ i, f (g i)
-- Monotone.tendsto_nhdsWithin_Iio
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
    [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
    {f : α → β} (Mf : Monotone f) (x : α) : Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x))) 
Arguments: {α : Type u_4} {β : Type u_5} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (Mf : Monotone f) (x : α)
Goal: Filter.Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Set.Iio x)))
-- Monotone.tendsto_nhdsWithin_Ioi
theorem Monotone.tendsto_nhdsWithin_Ioi {α β : Type*} [LinearOrder α] [TopologicalSpace α]
    [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
    {f : α → β} (Mf : Monotone f) (x : α) : Tendsto f (𝓝[>] x) (𝓝 (sInf (f '' Ioi x))) 
Arguments: {α : Type u_4} {β : Type u_5} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (Mf : Monotone f) (x : α)
Goal: Filter.Tendsto f (𝓝[>] x) (𝓝 (sInf (f '' Set.Ioi x)))
-- SupConvergenceClass.tendsto_coe_atTop_isLUB
class SupConvergenceClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
    tendsto_coe_atTop_isLUB :
    ∀ (a : α) (s : Set α), IsLUB s a → Tendsto (CoeTC.coe : s → α) atTop (𝓝 a)
Arguments: {α : Type u_3} [Preorder α] [TopologicalSpace α] [self : SupConvergenceClass α] (a : α) (s : Set α)
Goal: IsLUB s a → Filter.Tendsto CoeTC.coe Filter.atTop (𝓝 a)
-- InfConvergenceClass.tendsto_coe_atBot_isGLB
class InfConvergenceClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
    tendsto_coe_atBot_isGLB :
    ∀ (a : α) (s : Set α), IsGLB s a → Tendsto (CoeTC.coe : s → α) atBot (𝓝 a)
Arguments: {α : Type u_3} [Preorder α] [TopologicalSpace α] [self : InfConvergenceClass α] (a : α) (s : Set α)
Goal: IsGLB s a → Filter.Tendsto CoeTC.coe Filter.atBot (𝓝 a)
-- tendsto_atTop_isLUB
theorem tendsto_atTop_isLUB (h_mono : Monotone f) (ha : IsLUB (Set.range f) a) :
    Tendsto f atTop (𝓝 a) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [Preorder α] [SupConvergenceClass α] {f : ι → α} {a : α} (h_mono : Monotone f) (ha : IsLUB (Set.range f) a)
Goal: Filter.Tendsto f Filter.atTop (𝓝 a)
-- tendsto_atBot_isLUB
theorem tendsto_atBot_isLUB (h_anti : Antitone f) (ha : IsLUB (Set.range f) a) :
    Tendsto f atBot (𝓝 a) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [Preorder α] [SupConvergenceClass α] {f : ι → α} {a : α} (h_anti : Antitone f) (ha : IsLUB (Set.range f) a)
Goal: Filter.Tendsto f Filter.atBot (𝓝 a)
-- tendsto_atBot_isGLB
theorem tendsto_atBot_isGLB (h_mono : Monotone f) (ha : IsGLB (Set.range f) a) :
    Tendsto f atBot (𝓝 a) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [Preorder α] [InfConvergenceClass α] {f : ι → α} {a : α} (h_mono : Monotone f) (ha : IsGLB (Set.range f) a)
Goal: Filter.Tendsto f Filter.atBot (𝓝 a)
-- tendsto_atTop_isGLB
theorem tendsto_atTop_isGLB (h_anti : Antitone f) (ha : IsGLB (Set.range f) a) :
    Tendsto f atTop (𝓝 a) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [Preorder α] [InfConvergenceClass α] {f : ι → α} {a : α} (h_anti : Antitone f) (ha : IsGLB (Set.range f) a)
Goal: Filter.Tendsto f Filter.atTop (𝓝 a)
-- tendsto_atTop_ciSup
theorem tendsto_atTop_ciSup (h_mono : Monotone f) (hbdd : BddAbove <| range f) :
    Tendsto f atTop (𝓝 (⨆ i, f i)) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [ConditionallyCompleteLattice α] [SupConvergenceClass α] {f : ι → α} (h_mono : Monotone f) (hbdd : BddAbove (Set.range f))
Goal: Filter.Tendsto f Filter.atTop (𝓝 (⨆ i, f i))
-- tendsto_atBot_ciSup
theorem tendsto_atBot_ciSup (h_anti : Antitone f) (hbdd : BddAbove <| range f) :
    Tendsto f atBot (𝓝 (⨆ i, f i)) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [ConditionallyCompleteLattice α] [SupConvergenceClass α] {f : ι → α} (h_anti : Antitone f) (hbdd : BddAbove (Set.range f))
Goal: Filter.Tendsto f Filter.atBot (𝓝 (⨆ i, f i))
-- tendsto_atBot_ciInf
theorem tendsto_atBot_ciInf (h_mono : Monotone f) (hbdd : BddBelow <| range f) :
    Tendsto f atBot (𝓝 (⨅ i, f i)) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [ConditionallyCompleteLattice α] [InfConvergenceClass α] {f : ι → α} (h_mono : Monotone f) (hbdd : BddBelow (Set.range f))
Goal: Filter.Tendsto f Filter.atBot (𝓝 (⨅ i, f i))
-- tendsto_atTop_ciInf
theorem tendsto_atTop_ciInf (h_anti : Antitone f) (hbdd : BddBelow <| range f) :
    Tendsto f atTop (𝓝 (⨅ i, f i)) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [ConditionallyCompleteLattice α] [InfConvergenceClass α] {f : ι → α} (h_anti : Antitone f) (hbdd : BddBelow (Set.range f))
Goal: Filter.Tendsto f Filter.atTop (𝓝 (⨅ i, f i))
-- tendsto_atTop_iSup
theorem tendsto_atTop_iSup (h_mono : Monotone f) : Tendsto f atTop (𝓝 (⨆ i, f i)) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [CompleteLattice α] [SupConvergenceClass α] {f : ι → α} (h_mono : Monotone f)
Goal: Filter.Tendsto f Filter.atTop (𝓝 (⨆ i, f i))
-- tendsto_atBot_iSup
theorem tendsto_atBot_iSup (h_anti : Antitone f) : Tendsto f atBot (𝓝 (⨆ i, f i)) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [CompleteLattice α] [SupConvergenceClass α] {f : ι → α} (h_anti : Antitone f)
Goal: Filter.Tendsto f Filter.atBot (𝓝 (⨆ i, f i))
-- tendsto_atBot_iInf
theorem tendsto_atBot_iInf (h_mono : Monotone f) : Tendsto f atBot (𝓝 (⨅ i, f i)) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [CompleteLattice α] [InfConvergenceClass α] {f : ι → α} (h_mono : Monotone f)
Goal: Filter.Tendsto f Filter.atBot (𝓝 (⨅ i, f i))
-- tendsto_atTop_iInf
theorem tendsto_atTop_iInf (h_anti : Antitone f) : Tendsto f atTop (𝓝 (⨅ i, f i)) 
Arguments: {α : Type u_1} {ι : Type u_3} [Preorder ι] [TopologicalSpace α] [CompleteLattice α] [InfConvergenceClass α] {f : ι → α} (h_anti : Antitone f)
Goal: Filter.Tendsto f Filter.atTop (𝓝 (⨅ i, f i))