ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- le_tsum'
@[to_additive]
theorem le_tprod' (hf : Multipliable f) (i : ι) : f i ≤ ∏' i, f i 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (hf : Summable f) (i : ι)
Goal: f i ≤ ∑' (i : ι), f i
-- le_tprod'
@[to_additive]
theorem le_tprod' (hf : Multipliable f) (i : ι) : f i ≤ ∏' i, f i 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (hf : Multipliable f) (i : ι)
Goal: f i ≤ ∏' (i : ι), f i
-- hasSum_zero_iff
@[to_additive]
theorem hasProd_one_iff : HasProd f 1 ↔ ∀ x, f x = 1 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α}
Goal: HasSum f 0 ↔ ∀ (x : ι), f x = 0
-- hasProd_one_iff
@[to_additive]
theorem hasProd_one_iff : HasProd f 1 ↔ ∀ x, f x = 1 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α}
Goal: HasProd f 1 ↔ ∀ (x : ι), f x = 1
-- tsum_eq_zero_iff
@[to_additive]
theorem tprod_eq_one_iff (hf : Multipliable f) : ∏' i, f i = 1 ↔ ∀ x, f x = 1 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (hf : Summable f)
Goal: ∑' (i : ι), f i = 0 ↔ ∀ (x : ι), f x = 0
-- tprod_eq_one_iff
@[to_additive]
theorem tprod_eq_one_iff (hf : Multipliable f) : ∏' i, f i = 1 ↔ ∀ x, f x = 1 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (hf : Multipliable f)
Goal: ∏' (i : ι), f i = 1 ↔ ∀ (x : ι), f x = 1
-- tsum_ne_zero_iff
@[to_additive]
theorem tprod_ne_one_iff (hf : Multipliable f) : ∏' i, f i ≠ 1 ↔ ∃ x, f x ≠ 1 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (hf : Summable f)
Goal: ∑' (i : ι), f i ≠ 0 ↔ ∃ x, f x ≠ 0
-- tprod_ne_one_iff
@[to_additive]
theorem tprod_ne_one_iff (hf : Multipliable f) : ∏' i, f i ≠ 1 ↔ ∃ x, f x ≠ 1 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (hf : Multipliable f)
Goal: ∏' (i : ι), f i ≠ 1 ↔ ∃ x, f x ≠ 1
-- isLUB_hasSum'
@[to_additive]
theorem isLUB_hasProd' (hf : HasProd f a) : IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) a 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a : α} (hf : HasSum f a)
Goal: IsLUB (Set.range fun s => ∑ i ∈ s, f i) a
-- isLUB_hasProd'
@[to_additive]
theorem isLUB_hasProd' (hf : HasProd f a) : IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) a 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a : α} (hf : HasProd f a)
Goal: IsLUB (Set.range fun s => ∏ i ∈ s, f i) a
-- hasSum_of_isLUB_of_nonneg
@[to_additive]
theorem hasProd_of_isLUB_of_one_le [LinearOrderedCommMonoid α] [TopologicalSpace α]
    [OrderTopology α] {f : ι → α} (i : α) (h : ∀ i, 1 ≤ f i)
    (hf : IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) i) : HasProd f i 
Arguments: {ι : Type u_1} {α : Type u_3} [LinearOrderedAddCommMonoid α] [TopologicalSpace α] [OrderTopology α] {f : ι → α} (i : α) (h : ∀ (i : ι), 0 ≤ f i) (hf : IsLUB (Set.range fun s => ∑ i ∈ s, f i) i)
Goal: HasSum f i
-- hasProd_of_isLUB_of_one_le
@[to_additive]
theorem hasProd_of_isLUB_of_one_le [LinearOrderedCommMonoid α] [TopologicalSpace α]
    [OrderTopology α] {f : ι → α} (i : α) (h : ∀ i, 1 ≤ f i)
    (hf : IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) i) : HasProd f i 
Arguments: {ι : Type u_1} {α : Type u_3} [LinearOrderedCommMonoid α] [TopologicalSpace α] [OrderTopology α] {f : ι → α} (i : α) (h : ∀ (i : ι), 1 ≤ f i) (hf : IsLUB (Set.range fun s => ∏ i ∈ s, f i) i)
Goal: HasProd f i
-- hasProd_of_isLUB
@[to_additive]
theorem hasProd_of_isLUB [CanonicallyLinearOrderedCommMonoid α] [TopologicalSpace α]
    [OrderTopology α] {f : ι → α} (b : α) (hf : IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) b) :
    HasProd f b 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyLinearOrderedCommMonoid α] [TopologicalSpace α] [OrderTopology α] {f : ι → α} (b : α) (hf : IsLUB (Set.range fun s => ∏ i ∈ s, f i) b)
Goal: HasProd f b
-- hasSum_of_isLUB
@[to_additive]
theorem hasProd_of_isLUB [CanonicallyLinearOrderedCommMonoid α] [TopologicalSpace α]
    [OrderTopology α] {f : ι → α} (b : α) (hf : IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) b) :
    HasProd f b 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyLinearOrderedAddCommMonoid α] [TopologicalSpace α] [OrderTopology α] {f : ι → α} (b : α) (hf : IsLUB (Set.range fun s => ∑ i ∈ s, f i) b)
Goal: HasSum f b
-- summable_abs_iff
@[to_additive]
theorem multipliable_mabs_iff [LinearOrderedCommGroup α] [UniformSpace α] [UniformGroup α]
    [CompleteSpace α] {f : ι → α} : (Multipliable fun x ↦ mabs (f x)) ↔ Multipliable f 
Arguments: {ι : Type u_1} {α : Type u_3} [LinearOrderedAddCommGroup α] [UniformSpace α] [UniformAddGroup α] [CompleteSpace α] {f : ι → α}
Goal: (Summable fun x => |f x|) ↔ Summable f
-- multipliable_mabs_iff
@[to_additive]
theorem multipliable_mabs_iff [LinearOrderedCommGroup α] [UniformSpace α] [UniformGroup α]
    [CompleteSpace α] {f : ι → α} : (Multipliable fun x ↦ mabs (f x)) ↔ Multipliable f 
Arguments: {ι : Type u_1} {α : Type u_3} [LinearOrderedCommGroup α] [UniformSpace α] [UniformGroup α] [CompleteSpace α] {f : ι → α}
Goal: (Multipliable fun x => mabs (f x)) ↔ Multipliable f
-- Finite.of_summable_const
theorem Finite.of_summable_const [LinearOrderedAddCommGroup α] [TopologicalSpace α] [Archimedean α]
    [OrderClosedTopology α] {b : α} (hb : 0 < b) (hf : Summable fun _ : ι ↦ b) :
    Finite ι 
Arguments: {ι : Type u_1} {α : Type u_3} [LinearOrderedAddCommGroup α] [TopologicalSpace α] [Archimedean α] [OrderClosedTopology α] {b : α} (hb : 0 < b) (hf : Summable fun x => b)
Goal: Finite ι
-- Set.Finite.of_summable_const
theorem Set.Finite.of_summable_const [LinearOrderedAddCommGroup α] [TopologicalSpace α]
    [Archimedean α] [OrderClosedTopology α] {b : α} (hb : 0 < b) (hf : Summable fun _ : ι ↦ b) :
    (Set.univ : Set ι).Finite 
Arguments: {ι : Type u_1} {α : Type u_3} [LinearOrderedAddCommGroup α] [TopologicalSpace α] [Archimedean α] [OrderClosedTopology α] {b : α} (hb : 0 < b) (hf : Summable fun x => b)
Goal: Set.univ.Finite
-- Summable.tendsto_atTop_of_pos
theorem Summable.tendsto_atTop_of_pos [LinearOrderedField α] [TopologicalSpace α] [OrderTopology α]
    {f : ℕ → α} (hf : Summable f⁻¹) (hf' : ∀ n, 0 < f n) : Tendsto f atTop atTop 
Arguments: {α : Type u_3} [LinearOrderedField α] [TopologicalSpace α] [OrderTopology α] {f : ℕ → α} (hf : Summable f⁻¹) (hf' : ∀ (n : ℕ), 0 < f n)
Goal: Filter.Tendsto f Filter.atTop Filter.atTop
-- BoundedLENhdsClass.isBounded_le_nhds
class BoundedLENhdsClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
  isBounded_le_nhds (a : α) : (𝓝 a).IsBounded (· ≤ ·)
Arguments: {α : Type u_7} [Preorder α] [TopologicalSpace α] [self : BoundedLENhdsClass α] (a : α)
Goal: Filter.IsBounded (fun x x_1 => x ≤ x_1) (𝓝 a)
-- BoundedGENhdsClass.isBounded_ge_nhds
class BoundedGENhdsClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
  isBounded_ge_nhds (a : α) : (𝓝 a).IsBounded (· ≥ ·)
Arguments: {α : Type u_7} [Preorder α] [TopologicalSpace α] [self : BoundedGENhdsClass α] (a : α)
Goal: Filter.IsBounded (fun x x_1 => x ≥ x_1) (𝓝 a)
-- isBounded_le_nhds
theorem isBounded_le_nhds (a : α) : (𝓝 a).IsBounded (· ≤ ·) 
Arguments: {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedLENhdsClass α] (a : α)
Goal: Filter.IsBounded (fun x x_1 => x ≤ x_1) (𝓝 a)
-- Filter.Tendsto.isBoundedUnder_le
theorem Filter.Tendsto.isBoundedUnder_le (h : Tendsto u f (𝓝 a)) : f.IsBoundedUnder (· ≤ ·) u 
Arguments: {ι : Type u_1} {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedLENhdsClass α] {f : Filter ι} {u : ι → α} {a : α} (h : Filter.Tendsto u f (𝓝 a))
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u
-- Filter.Tendsto.bddAbove_range_of_cofinite
theorem Filter.Tendsto.bddAbove_range_of_cofinite [IsDirected α (· ≤ ·)]
    (h : Tendsto u cofinite (𝓝 a)) : BddAbove (Set.range u) 
Arguments: {ι : Type u_1} {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedLENhdsClass α] {u : ι → α} {a : α} [IsDirected α fun x x_1 => x ≤ x_1] (h : Filter.Tendsto u Filter.cofinite (𝓝 a))
Goal: BddAbove (Set.range u)
-- Filter.Tendsto.bddAbove_range
theorem Filter.Tendsto.bddAbove_range [IsDirected α (· ≤ ·)] {u : ℕ → α}
    (h : Tendsto u atTop (𝓝 a)) : BddAbove (Set.range u) 
Arguments: {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedLENhdsClass α] {a : α} [IsDirected α fun x x_1 => x ≤ x_1] {u : ℕ → α} (h : Filter.Tendsto u Filter.atTop (𝓝 a))
Goal: BddAbove (Set.range u)
-- isCobounded_ge_nhds
theorem isCobounded_ge_nhds (a : α) : (𝓝 a).IsCobounded (· ≥ ·) 
Arguments: {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedLENhdsClass α] (a : α)
Goal: Filter.IsCobounded (fun x x_1 => x ≥ x_1) (𝓝 a)
-- Filter.Tendsto.isCoboundedUnder_ge
theorem Filter.Tendsto.isCoboundedUnder_ge [NeBot f] (h : Tendsto u f (𝓝 a)) :
    f.IsCoboundedUnder (· ≥ ·) u 
Arguments: {ι : Type u_1} {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedLENhdsClass α] {f : Filter ι} {u : ι → α} {a : α} [f.NeBot] (h : Filter.Tendsto u f (𝓝 a))
Goal: Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f u
-- isBounded_ge_nhds
theorem isBounded_ge_nhds (a : α) : (𝓝 a).IsBounded (· ≥ ·) 
Arguments: {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedGENhdsClass α] (a : α)
Goal: Filter.IsBounded (fun x x_1 => x ≥ x_1) (𝓝 a)
-- Filter.Tendsto.isBoundedUnder_ge
theorem Filter.Tendsto.isBoundedUnder_ge (h : Tendsto u f (𝓝 a)) : f.IsBoundedUnder (· ≥ ·) u 
Arguments: {ι : Type u_1} {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedGENhdsClass α] {f : Filter ι} {u : ι → α} {a : α} (h : Filter.Tendsto u f (𝓝 a))
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u
-- Filter.Tendsto.bddBelow_range_of_cofinite
theorem Filter.Tendsto.bddBelow_range_of_cofinite [IsDirected α (· ≥ ·)]
    (h : Tendsto u cofinite (𝓝 a)) : BddBelow (Set.range u) 
Arguments: {ι : Type u_1} {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedGENhdsClass α] {u : ι → α} {a : α} [IsDirected α fun x x_1 => x ≥ x_1] (h : Filter.Tendsto u Filter.cofinite (𝓝 a))
Goal: BddBelow (Set.range u)
-- Filter.Tendsto.bddBelow_range
theorem Filter.Tendsto.bddBelow_range [IsDirected α (· ≥ ·)] {u : ℕ → α}
    (h : Tendsto u atTop (𝓝 a)) : BddBelow (Set.range u) 
Arguments: {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedGENhdsClass α] {a : α} [IsDirected α fun x x_1 => x ≥ x_1] {u : ℕ → α} (h : Filter.Tendsto u Filter.atTop (𝓝 a))
Goal: BddBelow (Set.range u)
-- isCobounded_le_nhds
theorem isCobounded_le_nhds (a : α) : (𝓝 a).IsCobounded (· ≤ ·) 
Arguments: {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedGENhdsClass α] (a : α)
Goal: Filter.IsCobounded (fun x x_1 => x ≤ x_1) (𝓝 a)
-- Filter.Tendsto.isCoboundedUnder_le
theorem Filter.Tendsto.isCoboundedUnder_le [NeBot f] (h : Tendsto u f (𝓝 a)) :
    f.IsCoboundedUnder (· ≤ ·) u 
Arguments: {ι : Type u_1} {α : Type u_2} [Preorder α] [TopologicalSpace α] [BoundedGENhdsClass α] {f : Filter ι} {u : ι → α} {a : α} [f.NeBot] (h : Filter.Tendsto u f (𝓝 a))
Goal: Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u
-- le_nhds_of_limsSup_eq_limsInf
theorem le_nhds_of_limsSup_eq_limsInf {f : Filter α} {a : α} (hl : f.IsBounded (· ≤ ·))
    (hg : f.IsBounded (· ≥ ·)) (hs : f.limsSup = a) (hi : f.limsInf = a) : f ≤ 𝓝 a 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {f : Filter α} {a : α} (hl : Filter.IsBounded (fun x x_1 => x ≤ x_1) f) (hg : Filter.IsBounded (fun x x_1 => x ≥ x_1) f) (hs : f.limsSup = a) (hi : f.limsInf = a)
Goal: f ≤ 𝓝 a
-- limsSup_nhds
theorem limsSup_nhds (a : α) : limsSup (𝓝 a) = a 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] (a : α)
Goal: (𝓝 a).limsSup = a
-- limsInf_nhds
theorem limsInf_nhds : ∀ a : α, limsInf (𝓝 a) = a 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] (a : α)
Goal: (𝓝 a).limsInf = a
-- limsInf_eq_of_le_nhds
theorem limsInf_eq_of_le_nhds {f : Filter α} {a : α} [NeBot f] (h : f ≤ 𝓝 a) : f.limsInf = a 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {f : Filter α} {a : α} [f.NeBot] (h : f ≤ 𝓝 a)
Goal: f.limsInf = a
-- limsSup_eq_of_le_nhds
theorem limsSup_eq_of_le_nhds : ∀ {f : Filter α} {a : α} [NeBot f], f ≤ 𝓝 a → f.limsSup = a 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {f : Filter α} {a : α} [f.NeBot]
Goal: f ≤ 𝓝 a → f.limsSup = a
-- Filter.Tendsto.limsup_eq
theorem Filter.Tendsto.limsup_eq {f : Filter β} {u : β → α} {a : α} [NeBot f]
    (h : Tendsto u f (𝓝 a)) : limsup u f = a 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {f : Filter β} {u : β → α} {a : α} [f.NeBot] (h : Filter.Tendsto u f (𝓝 a))
Goal: Filter.limsup u f = a
-- Filter.Tendsto.liminf_eq
theorem Filter.Tendsto.liminf_eq {f : Filter β} {u : β → α} {a : α} [NeBot f]
    (h : Tendsto u f (𝓝 a)) : liminf u f = a 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {f : Filter β} {u : β → α} {a : α} [f.NeBot] (h : Filter.Tendsto u f (𝓝 a))
Goal: Filter.liminf u f = a
-- tendsto_of_liminf_eq_limsup
theorem tendsto_of_liminf_eq_limsup {f : Filter β} {u : β → α} {a : α} (hinf : liminf u f = a)
    (hsup : limsup u f = a) (h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) : Tendsto u f (𝓝 a) 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {f : Filter β} {u : β → α} {a : α} (hinf : Filter.liminf u f = a) (hsup : Filter.limsup u f = a) (h : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (h' : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault)
Goal: Filter.Tendsto u f (𝓝 a)
-- tendsto_of_le_liminf_of_limsup_le
theorem tendsto_of_le_liminf_of_limsup_le {f : Filter β} {u : β → α} {a : α} (hinf : a ≤ liminf u f)
    (hsup : limsup u f ≤ a) (h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) : Tendsto u f (𝓝 a) 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {f : Filter β} {u : β → α} {a : α} (hinf : a ≤ Filter.liminf u f) (hsup : Filter.limsup u f ≤ a) (h : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (h' : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault)
Goal: Filter.Tendsto u f (𝓝 a)
-- tendsto_of_no_upcrossings
theorem tendsto_of_no_upcrossings [DenselyOrdered α] {f : Filter β} {u : β → α} {s : Set α}
    (hs : Dense s) (H : ∀ a ∈ s, ∀ b ∈ s, a < b → ¬((∃ᶠ n in f, u n < a) ∧ ∃ᶠ n in f, b < u n))
    (h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) :
    ∃ c : α, Tendsto u f (𝓝 c) 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {f : Filter β} {u : β → α} {s : Set α} (hs : Dense s) (H : ∀ a ∈ s, ∀ b ∈ s, a < b → ¬((∃ᶠ (n : β) in f, u n < a) ∧ ∃ᶠ (n : β) in f, b < u n)) (h : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (h' : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault)
Goal: ∃ c, Filter.Tendsto u f (𝓝 c)
-- eventually_le_limsup
theorem eventually_le_limsup (hf : IsBoundedUnder (· ≤ ·) f u := by isBoundedDefault) :
    ∀ᶠ b in f, u b ≤ f.limsup u 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {f : Filter β} [CountableInterFilter f] {u : β → α} (hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault)
Goal: ∀ᶠ (b : β) in f, u b ≤ Filter.limsup u f
-- eventually_liminf_le
theorem eventually_liminf_le (hf : IsBoundedUnder (· ≥ ·) f u := by isBoundedDefault) :
    ∀ᶠ b in f, f.liminf u ≤ u b 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {f : Filter β} [CountableInterFilter f] {u : β → α} (hf : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault)
Goal: ∀ᶠ (b : β) in f, Filter.liminf u f ≤ u b
-- limsup_eq_bot
@[simp]
theorem limsup_eq_bot : f.limsup u = ⊥ ↔ u =ᶠ[f] ⊥ 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLinearOrder α] [TopologicalSpace α] [FirstCountableTopology α] [OrderTopology α] {f : Filter β} [CountableInterFilter f] {u : β → α}
Goal: Filter.limsup u f = ⊥ ↔ u =ᶠ[f] ⊥
-- liminf_eq_top
@[simp]
theorem liminf_eq_top : f.liminf u = ⊤ ↔ u =ᶠ[f] ⊤ 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLinearOrder α] [TopologicalSpace α] [FirstCountableTopology α] [OrderTopology α] {f : Filter β} [CountableInterFilter f] {u : β → α}
Goal: Filter.liminf u f = ⊤ ↔ u =ᶠ[f] ⊤
-- Antitone.map_limsSup_of_continuousAt
theorem Antitone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
    (f_decr : Antitone f) (f_cont : ContinuousAt f F.limsSup)
    (bdd_above : F.IsBounded (· ≤ ·) := by isBoundedDefault)
    (cobdd : F.IsCobounded (· ≤ ·) := by isBoundedDefault) :
    f F.limsSup = F.liminf f 
Arguments: {R : Type u_4} {S : Type u_5} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] [ConditionallyCompleteLinearOrder S] [TopologicalSpace S] [OrderTopology S] {F : Filter R} [F.NeBot] {f : R → S} (f_decr : Antitone f) (f_cont : ContinuousAt f F.limsSup) (bdd_above : Filter.IsBounded (fun x x_1 => x ≤ x_1) F := by isBoundedDefault) (cobdd : Filter.IsCobounded (fun x x_1 => x ≤ x_1) F := by isBoundedDefault)
Goal: f F.limsSup = Filter.liminf f F
-- Antitone.map_limsup_of_continuousAt
theorem Antitone.map_limsup_of_continuousAt {f : R → S} (f_decr : Antitone f) (a : ι → R)
    (f_cont : ContinuousAt f (F.limsup a))
    (bdd_above : F.IsBoundedUnder (· ≤ ·) a := by isBoundedDefault)
    (cobdd : F.IsCoboundedUnder (· ≤ ·) a := by isBoundedDefault) :
    f (F.limsup a) = F.liminf (f ∘ a) 
Arguments: {ι : Type u_1} {R : Type u_4} {S : Type u_5} {F : Filter ι} [F.NeBot] [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] [ConditionallyCompleteLinearOrder S] [TopologicalSpace S] [OrderTopology S] {f : R → S} (f_decr : Antitone f) (a : ι → R) (f_cont : ContinuousAt f (Filter.limsup a F)) (bdd_above : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) F a := by isBoundedDefault) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) F a := by isBoundedDefault)
Goal: f (Filter.limsup a F) = Filter.liminf (f ∘ a) F
-- Antitone.map_limsInf_of_continuousAt
theorem Antitone.map_limsInf_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
    (f_decr : Antitone f) (f_cont : ContinuousAt f F.limsInf)
    (cobdd : F.IsCobounded (· ≥ ·) := by isBoundedDefault)
    (bdd_below : F.IsBounded (· ≥ ·) := by isBoundedDefault) : f F.limsInf = F.limsup f 
Arguments: {R : Type u_4} {S : Type u_5} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] [ConditionallyCompleteLinearOrder S] [TopologicalSpace S] [OrderTopology S] {F : Filter R} [F.NeBot] {f : R → S} (f_decr : Antitone f) (f_cont : ContinuousAt f F.limsInf) (cobdd : Filter.IsCobounded (fun x x_1 => x ≥ x_1) F := by isBoundedDefault) (bdd_below : Filter.IsBounded (fun x x_1 => x ≥ x_1) F := by isBoundedDefault)
Goal: f F.limsInf = Filter.limsup f F
-- Antitone.map_liminf_of_continuousAt
theorem Antitone.map_liminf_of_continuousAt {f : R → S} (f_decr : Antitone f) (a : ι → R)
    (f_cont : ContinuousAt f (F.liminf a))
    (cobdd : F.IsCoboundedUnder (· ≥ ·) a := by isBoundedDefault)
    (bdd_below : F.IsBoundedUnder (· ≥ ·) a := by isBoundedDefault) :
    f (F.liminf a) = F.limsup (f ∘ a) 
Arguments: {ι : Type u_1} {R : Type u_4} {S : Type u_5} {F : Filter ι} [F.NeBot] [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] [ConditionallyCompleteLinearOrder S] [TopologicalSpace S] [OrderTopology S] {f : R → S} (f_decr : Antitone f) (a : ι → R) (f_cont : ContinuousAt f (Filter.liminf a F)) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) F a := by isBoundedDefault) (bdd_below : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) F a := by isBoundedDefault)
Goal: f (Filter.liminf a F) = Filter.limsup (f ∘ a) F
-- Monotone.map_limsSup_of_continuousAt
theorem Monotone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
    (f_incr : Monotone f) (f_cont : ContinuousAt f F.limsSup)
    (bdd_above : F.IsBounded (· ≤ ·) := by isBoundedDefault)
    (cobdd : F.IsCobounded (· ≤ ·) := by isBoundedDefault) : f F.limsSup = F.limsup f 
Arguments: {R : Type u_4} {S : Type u_5} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] [ConditionallyCompleteLinearOrder S] [TopologicalSpace S] [OrderTopology S] {F : Filter R} [F.NeBot] {f : R → S} (f_incr : Monotone f) (f_cont : ContinuousAt f F.limsSup) (bdd_above : Filter.IsBounded (fun x x_1 => x ≤ x_1) F := by isBoundedDefault) (cobdd : Filter.IsCobounded (fun x x_1 => x ≤ x_1) F := by isBoundedDefault)
Goal: f F.limsSup = Filter.limsup f F
-- Monotone.map_limsup_of_continuousAt
theorem Monotone.map_limsup_of_continuousAt {f : R → S} (f_incr : Monotone f) (a : ι → R)
    (f_cont : ContinuousAt f (F.limsup a))
    (bdd_above : F.IsBoundedUnder (· ≤ ·) a := by isBoundedDefault)
    (cobdd : F.IsCoboundedUnder (· ≤ ·) a := by isBoundedDefault) :
    f (F.limsup a) = F.limsup (f ∘ a) 
Arguments: {ι : Type u_1} {R : Type u_4} {S : Type u_5} {F : Filter ι} [F.NeBot] [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] [ConditionallyCompleteLinearOrder S] [TopologicalSpace S] [OrderTopology S] {f : R → S} (f_incr : Monotone f) (a : ι → R) (f_cont : ContinuousAt f (Filter.limsup a F)) (bdd_above : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) F a := by isBoundedDefault) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) F a := by isBoundedDefault)
Goal: f (Filter.limsup a F) = Filter.limsup (f ∘ a) F
-- Monotone.map_limsInf_of_continuousAt
theorem Monotone.map_limsInf_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
    (f_incr : Monotone f) (f_cont : ContinuousAt f F.limsInf)
    (cobdd : F.IsCobounded (· ≥ ·) := by isBoundedDefault)
    (bdd_below : F.IsBounded (· ≥ ·) := by isBoundedDefault) : f F.limsInf = F.liminf f 
Arguments: {R : Type u_4} {S : Type u_5} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] [ConditionallyCompleteLinearOrder S] [TopologicalSpace S] [OrderTopology S] {F : Filter R} [F.NeBot] {f : R → S} (f_incr : Monotone f) (f_cont : ContinuousAt f F.limsInf) (cobdd : Filter.IsCobounded (fun x x_1 => x ≥ x_1) F := by isBoundedDefault) (bdd_below : Filter.IsBounded (fun x x_1 => x ≥ x_1) F := by isBoundedDefault)
Goal: f F.limsInf = Filter.liminf f F
-- Monotone.map_liminf_of_continuousAt
theorem Monotone.map_liminf_of_continuousAt {f : R → S} (f_incr : Monotone f) (a : ι → R)
    (f_cont : ContinuousAt f (F.liminf a))
    (cobdd : F.IsCoboundedUnder (· ≥ ·) a := by isBoundedDefault)
    (bdd_below : F.IsBoundedUnder (· ≥ ·) a := by isBoundedDefault) :
    f (F.liminf a) = F.liminf (f ∘ a) 
Arguments: {ι : Type u_1} {R : Type u_4} {S : Type u_5} {F : Filter ι} [F.NeBot] [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] [ConditionallyCompleteLinearOrder S] [TopologicalSpace S] [OrderTopology S] {f : R → S} (f_incr : Monotone f) (a : ι → R) (f_cont : ContinuousAt f (Filter.liminf a F)) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) F a := by isBoundedDefault) (bdd_below : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) F a := by isBoundedDefault)
Goal: f (Filter.liminf a F) = Filter.liminf (f ∘ a) F