ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.limsSup_le_of_le
theorem limsSup_le_of_le {f : Filter α} {a}
    (hf : f.IsCobounded (· ≤ ·) := by isBoundedDefault)
    (h : ∀ᶠ n in f, n ≤ a) : limsSup f ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {f : Filter α} {a : α} (hf : Filter.IsCobounded (fun x x_1 => x ≤ x_1) f := by isBoundedDefault) (h : ∀ᶠ (n : α) in f, n ≤ a)
Goal: f.limsSup ≤ a
-- Filter.le_limsInf_of_le
theorem le_limsInf_of_le {f : Filter α} {a}
    (hf : f.IsCobounded (· ≥ ·) := by isBoundedDefault)
    (h : ∀ᶠ n in f, a ≤ n) : a ≤ limsInf f 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {f : Filter α} {a : α} (hf : Filter.IsCobounded (fun x x_1 => x ≥ x_1) f := by isBoundedDefault) (h : ∀ᶠ (n : α) in f, a ≤ n)
Goal: a ≤ f.limsInf
-- Filter.limsup_le_of_le
theorem limsup_le_of_le {f : Filter β} {u : β → α} {a}
    (hf : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h : ∀ᶠ n in f, u n ≤ a) : limsup u f ≤ a 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : Filter β} {u : β → α} {a : α} (hf : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (h : ∀ᶠ (n : β) in f, u n ≤ a)
Goal: Filter.limsup u f ≤ a
-- Filter.le_liminf_of_le
theorem le_liminf_of_le {f : Filter β} {u : β → α} {a}
    (hf : f.IsCoboundedUnder (· ≥ ·) u := by isBoundedDefault)
    (h : ∀ᶠ n in f, a ≤ u n) : a ≤ liminf u f 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : Filter β} {u : β → α} {a : α} (hf : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (h : ∀ᶠ (n : β) in f, a ≤ u n)
Goal: a ≤ Filter.liminf u f
-- Filter.le_limsSup_of_le
theorem le_limsSup_of_le {f : Filter α} {a}
    (hf : f.IsBounded (· ≤ ·) := by isBoundedDefault)
    (h : ∀ b, (∀ᶠ n in f, n ≤ b) → a ≤ b) : a ≤ limsSup f 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {f : Filter α} {a : α} (hf : Filter.IsBounded (fun x x_1 => x ≤ x_1) f := by isBoundedDefault) (h : ∀ (b : α), (∀ᶠ (n : α) in f, n ≤ b) → a ≤ b)
Goal: a ≤ f.limsSup
-- Filter.limsInf_le_of_le
theorem limsInf_le_of_le {f : Filter α} {a}
    (hf : f.IsBounded (· ≥ ·) := by isBoundedDefault)
    (h : ∀ b, (∀ᶠ n in f, b ≤ n) → b ≤ a) : limsInf f ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {f : Filter α} {a : α} (hf : Filter.IsBounded (fun x x_1 => x ≥ x_1) f := by isBoundedDefault) (h : ∀ (b : α), (∀ᶠ (n : α) in f, b ≤ n) → b ≤ a)
Goal: f.limsInf ≤ a
-- Filter.le_limsup_of_le
theorem le_limsup_of_le {f : Filter β} {u : β → α} {a}
    (hf : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h : ∀ b, (∀ᶠ n in f, u n ≤ b) → a ≤ b) : a ≤ limsup u f 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : Filter β} {u : β → α} {a : α} (hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (h : ∀ (b : α), (∀ᶠ (n : β) in f, u n ≤ b) → a ≤ b)
Goal: a ≤ Filter.limsup u f
-- Filter.liminf_le_of_le
theorem liminf_le_of_le {f : Filter β} {u : β → α} {a}
    (hf : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault)
    (h : ∀ b, (∀ᶠ n in f, b ≤ u n) → b ≤ a) : liminf u f ≤ a 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : Filter β} {u : β → α} {a : α} (hf : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (h : ∀ (b : α), (∀ᶠ (n : β) in f, b ≤ u n) → b ≤ a)
Goal: Filter.liminf u f ≤ a
-- Filter.limsInf_le_limsSup
theorem limsInf_le_limsSup {f : Filter α} [NeBot f]
    (h₁ : f.IsBounded (· ≤ ·) := by isBoundedDefault)
    (h₂ : f.IsBounded (· ≥ ·) := by isBoundedDefault) :
    limsInf f ≤ limsSup f 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {f : Filter α} [f.NeBot] (h₁ : Filter.IsBounded (fun x x_1 => x ≤ x_1) f := by isBoundedDefault) (h₂ : Filter.IsBounded (fun x x_1 => x ≥ x_1) f := by isBoundedDefault)
Goal: f.limsInf ≤ f.limsSup
-- Filter.liminf_le_limsup
theorem liminf_le_limsup {f : Filter β} [NeBot f] {u : β → α}
    (h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) :
    liminf u f ≤ limsup u f 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : Filter β} [f.NeBot] {u : β → α} (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.liminf u f ≤ Filter.limsup u f
-- Filter.limsSup_le_limsSup
theorem limsSup_le_limsSup {f g : Filter α}
    (hf : f.IsCobounded (· ≤ ·) := by isBoundedDefault)
    (hg : g.IsBounded (· ≤ ·) := by isBoundedDefault)
    (h : ∀ a, (∀ᶠ n in g, n ≤ a) → ∀ᶠ n in f, n ≤ a) : limsSup f ≤ limsSup g 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {f g : Filter α} (hf : Filter.IsCobounded (fun x x_1 => x ≤ x_1) f := by isBoundedDefault) (hg : Filter.IsBounded (fun x x_1 => x ≤ x_1) g := by isBoundedDefault) (h : ∀ (a : α), (∀ᶠ (n : α) in g, n ≤ a) → ∀ᶠ (n : α) in f, n ≤ a)
Goal: f.limsSup ≤ g.limsSup
-- Filter.limsInf_le_limsInf
theorem limsInf_le_limsInf {f g : Filter α}
    (hf : f.IsBounded (· ≥ ·) := by isBoundedDefault)
    (hg : g.IsCobounded (· ≥ ·) := by isBoundedDefault)
    (h : ∀ a, (∀ᶠ n in f, a ≤ n) → ∀ᶠ n in g, a ≤ n) : limsInf f ≤ limsInf g 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {f g : Filter α} (hf : Filter.IsBounded (fun x x_1 => x ≥ x_1) f := by isBoundedDefault) (hg : Filter.IsCobounded (fun x x_1 => x ≥ x_1) g := by isBoundedDefault) (h : ∀ (a : α), (∀ᶠ (n : α) in f, a ≤ n) → ∀ᶠ (n : α) in g, a ≤ n)
Goal: f.limsInf ≤ g.limsInf
-- Filter.limsup_le_limsup
theorem limsup_le_limsup {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β}
    (h : u ≤ᶠ[f] v)
    (hu : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
    (hv : f.IsBoundedUnder (· ≤ ·) v := by isBoundedDefault) :
    limsup u f ≤ limsup v f 
Arguments: {β : Type u_2} {α : Type u_6} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β} (h : u ≤ᶠ[f] v) (hu : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (hv : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f v := by isBoundedDefault)
Goal: Filter.limsup u f ≤ Filter.limsup v f
-- Filter.liminf_le_liminf
theorem liminf_le_liminf {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β}
    (h : ∀ᶠ a in f, u a ≤ v a)
    (hu : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault)
    (hv : f.IsCoboundedUnder (· ≥ ·) v := by isBoundedDefault) :
    liminf u f ≤ liminf v f 
Arguments: {β : Type u_2} {α : Type u_6} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β} (h : ∀ᶠ (a : α) in f, u a ≤ v a) (hu : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (hv : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f v := by isBoundedDefault)
Goal: Filter.liminf u f ≤ Filter.liminf v f
-- Filter.limsSup_le_limsSup_of_le
theorem limsSup_le_limsSup_of_le {f g : Filter α} (h : f ≤ g)
    (hf : f.IsCobounded (· ≤ ·) := by isBoundedDefault)
    (hg : g.IsBounded (· ≤ ·) := by isBoundedDefault) :
    limsSup f ≤ limsSup g 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {f g : Filter α} (h : f ≤ g) (hf : Filter.IsCobounded (fun x x_1 => x ≤ x_1) f := by isBoundedDefault) (hg : Filter.IsBounded (fun x x_1 => x ≤ x_1) g := by isBoundedDefault)
Goal: f.limsSup ≤ g.limsSup
-- Filter.limsInf_le_limsInf_of_le
theorem limsInf_le_limsInf_of_le {f g : Filter α} (h : g ≤ f)
    (hf : f.IsBounded (· ≥ ·) := by isBoundedDefault)
    (hg : g.IsCobounded (· ≥ ·) := by isBoundedDefault) :
    limsInf f ≤ limsInf g 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {f g : Filter α} (h : g ≤ f) (hf : Filter.IsBounded (fun x x_1 => x ≥ x_1) f := by isBoundedDefault) (hg : Filter.IsCobounded (fun x x_1 => x ≥ x_1) g := by isBoundedDefault)
Goal: f.limsInf ≤ g.limsInf
-- Filter.limsup_le_limsup_of_le
theorem limsup_le_limsup_of_le {α β} [ConditionallyCompleteLattice β] {f g : Filter α} (h : f ≤ g)
    {u : α → β}
    (hf : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
    (hg : g.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault) :
    limsup u f ≤ limsup u g 
Arguments: {α : Type u_6} {β : Type u_7} [ConditionallyCompleteLattice β] {f g : Filter α} (h : f ≤ g) {u : α → β} (hf : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (hg : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) g u := by isBoundedDefault)
Goal: Filter.limsup u f ≤ Filter.limsup u g
-- Filter.liminf_le_liminf_of_le
theorem liminf_le_liminf_of_le {α β} [ConditionallyCompleteLattice β] {f g : Filter α} (h : g ≤ f)
    {u : α → β}
    (hf : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault)
    (hg : g.IsCoboundedUnder (· ≥ ·) u := by isBoundedDefault) :
    liminf u f ≤ liminf u g 
Arguments: {α : Type u_6} {β : Type u_7} [ConditionallyCompleteLattice β] {f g : Filter α} (h : g ≤ f) {u : α → β} (hf : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (hg : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) g u := by isBoundedDefault)
Goal: Filter.liminf u f ≤ Filter.liminf u g
-- Filter.limsSup_principal
theorem limsSup_principal {s : Set α} (h : BddAbove s) (hs : s.Nonempty) :
    limsSup (𝓟 s) = sSup s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} (h : BddAbove s) (hs : s.Nonempty)
Goal: (𝓟 s).limsSup = sSup s
-- Filter.limsInf_principal
theorem limsInf_principal {s : Set α} (h : BddBelow s) (hs : s.Nonempty) : limsInf (𝓟 s) = sInf s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} (h : BddBelow s) (hs : s.Nonempty)
Goal: (𝓟 s).limsInf = sInf s
-- Filter.limsup_congr
theorem limsup_congr {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β}
    (h : ∀ᶠ a in f, u a = v a) : limsup u f = limsup v f 
Arguments: {β : Type u_2} {α : Type u_6} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β} (h : ∀ᶠ (a : α) in f, u a = v a)
Goal: Filter.limsup u f = Filter.limsup v f
-- Filter.blimsup_congr
theorem blimsup_congr {f : Filter β} {u v : β → α} {p : β → Prop} (h : ∀ᶠ a in f, p a → u a = v a) :
    blimsup u f p = blimsup v f p 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : Filter β} {u v : β → α} {p : β → Prop} (h : ∀ᶠ (a : β) in f, p a → u a = v a)
Goal: Filter.blimsup u f p = Filter.blimsup v f p
-- Filter.bliminf_congr
theorem bliminf_congr {f : Filter β} {u v : β → α} {p : β → Prop} (h : ∀ᶠ a in f, p a → u a = v a) :
    bliminf u f p = bliminf v f p 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : Filter β} {u v : β → α} {p : β → Prop} (h : ∀ᶠ (a : β) in f, p a → u a = v a)
Goal: Filter.bliminf u f p = Filter.bliminf v f p
-- Filter.liminf_congr
theorem liminf_congr {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β}
    (h : ∀ᶠ a in f, u a = v a) : liminf u f = liminf v f 
Arguments: {β : Type u_2} {α : Type u_6} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β} (h : ∀ᶠ (a : α) in f, u a = v a)
Goal: Filter.liminf u f = Filter.liminf v f
-- Filter.limsup_const
@[simp]
theorem limsup_const {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} [NeBot f]
    (b : β) : limsup (fun _ => b) f = b 
Arguments: {β : Type u_2} {α : Type u_6} [ConditionallyCompleteLattice β] {f : Filter α} [f.NeBot] (b : β)
Goal: Filter.limsup (fun x => b) f = b
-- Filter.liminf_const
@[simp]
theorem liminf_const {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} [NeBot f]
    (b : β) : liminf (fun _ => b) f = b 
Arguments: {β : Type u_2} {α : Type u_6} [ConditionallyCompleteLattice β] {f : Filter α} [f.NeBot] (b : β)
Goal: Filter.liminf (fun x => b) f = b
-- Filter.HasBasis.liminf_eq_sSup_iUnion_iInter
theorem HasBasis.liminf_eq_sSup_iUnion_iInter {ι ι' : Type*} {f : ι → α} {v : Filter ι}
    {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) :
    liminf f v = sSup (⋃ (j : Subtype p), ⋂ (i : s j), Iic (f i)) 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {ι : Type u_6} {ι' : Type u_7} {f : ι → α} {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s)
Goal: Filter.liminf f v = sSup (⋃ j, ⋂ i, Set.Iic (f ↑i))
-- Filter.HasBasis.liminf_eq_sSup_univ_of_empty
theorem HasBasis.liminf_eq_sSup_univ_of_empty {f : ι → α} {v : Filter ι}
    {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) (i : ι') (hi : p i) (h'i : s i = ∅) :
    liminf f v = sSup univ 
Arguments: {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [ConditionallyCompleteLattice α] {f : ι → α} {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) (i : ι') (hi : p i) (h'i : s i = ∅)
Goal: Filter.liminf f v = sSup Set.univ
-- Filter.HasBasis.limsup_eq_sInf_iUnion_iInter
theorem HasBasis.limsup_eq_sInf_iUnion_iInter {ι ι' : Type*} {f : ι → α} {v : Filter ι}
    {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) :
    limsup f v = sInf (⋃ (j : Subtype p), ⋂ (i : s j), Ici (f i)) 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {ι : Type u_6} {ι' : Type u_7} {f : ι → α} {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s)
Goal: Filter.limsup f v = sInf (⋃ j, ⋂ i, Set.Ici (f ↑i))
-- Filter.HasBasis.limsup_eq_sInf_univ_of_empty
theorem HasBasis.limsup_eq_sInf_univ_of_empty {f : ι → α} {v : Filter ι}
    {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) (i : ι') (hi : p i) (h'i : s i = ∅) :
    limsup f v = sInf univ 
Arguments: {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [ConditionallyCompleteLattice α] {f : ι → α} {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) (i : ι') (hi : p i) (h'i : s i = ∅)
Goal: Filter.limsup f v = sInf Set.univ
-- Filter.liminf_nat_add
@[simp, nolint simpNF]
theorem liminf_nat_add (f : ℕ → α) (k : ℕ) :
    liminf (fun i => f (i + k)) atTop = liminf f atTop 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] (f : ℕ → α) (k : ℕ)
Goal: Filter.liminf (fun i => f (i + k)) Filter.atTop = Filter.liminf f Filter.atTop
-- Filter.limsup_nat_add
@[simp, nolint simpNF]
theorem limsup_nat_add (f : ℕ → α) (k : ℕ) : limsup (fun i => f (i + k)) atTop = limsup f atTop 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] (f : ℕ → α) (k : ℕ)
Goal: Filter.limsup (fun i => f (i + k)) Filter.atTop = Filter.limsup f Filter.atTop
-- Filter.limsSup_bot
@[simp]
theorem limsSup_bot : limsSup (⊥ : Filter α) = ⊥ 
Arguments: {α : Type u_1} [CompleteLattice α]
Goal: ⊥.limsSup = ⊥
-- Filter.limsup_bot
@[simp] theorem limsup_bot (f : β → α) : limsup f ⊥ = ⊥ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] (f : β → α)
Goal: Filter.limsup f ⊥ = ⊥
-- Filter.limsInf_bot
@[simp]
theorem limsInf_bot : limsInf (⊥ : Filter α) = ⊤ 
Arguments: {α : Type u_1} [CompleteLattice α]
Goal: ⊥.limsInf = ⊤
-- Filter.liminf_bot
@[simp] theorem liminf_bot (f : β → α) : liminf f ⊥ = ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] (f : β → α)
Goal: Filter.liminf f ⊥ = ⊤
-- Filter.limsSup_top
@[simp]
theorem limsSup_top : limsSup (⊤ : Filter α) = ⊤ 
Arguments: {α : Type u_1} [CompleteLattice α]
Goal: ⊤.limsSup = ⊤
-- Filter.limsInf_top
@[simp]
theorem limsInf_top : limsInf (⊤ : Filter α) = ⊥ 
Arguments: {α : Type u_1} [CompleteLattice α]
Goal: ⊤.limsInf = ⊥
-- Filter.blimsup_false
@[simp]
theorem blimsup_false {f : Filter β} {u : β → α} : (blimsup u f fun _ => False) = ⊥ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {u : β → α}
Goal: (Filter.blimsup u f fun x => False) = ⊥
-- Filter.bliminf_false
@[simp]
theorem bliminf_false {f : Filter β} {u : β → α} : (bliminf u f fun _ => False) = ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {u : β → α}
Goal: (Filter.bliminf u f fun x => False) = ⊤
-- Filter.limsup_const_bot
@[simp]
theorem limsup_const_bot {f : Filter β} : limsup (fun _ : β => (⊥ : α)) f = (⊥ : α) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β}
Goal: Filter.limsup (fun x => ⊥) f = ⊥
-- Filter.liminf_const_top
@[simp]
theorem liminf_const_top {f : Filter β} : liminf (fun _ : β => (⊤ : α)) f = (⊤ : α) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β}
Goal: Filter.liminf (fun x => ⊤) f = ⊤
-- Filter.HasBasis.limsSup_eq_iInf_sSup
theorem HasBasis.limsSup_eq_iInf_sSup {ι} {p : ι → Prop} {s} {f : Filter α} (h : f.HasBasis p s) :
    limsSup f = ⨅ (i) (_ : p i), sSup (s i) 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Sort u_6} {p : ι → Prop} {s : ι → Set α} {f : Filter α} (h : f.HasBasis p s)
Goal: f.limsSup = ⨅ i, ⨅ (_ : p i), sSup (s i)
-- Filter.HasBasis.limsInf_eq_iSup_sInf
theorem HasBasis.limsInf_eq_iSup_sInf {p : ι → Prop} {s : ι → Set α} {f : Filter α}
    (h : f.HasBasis p s) : limsInf f = ⨆ (i) (_ : p i), sInf (s i) 
Arguments: {α : Type u_1} {ι : Type u_4} [CompleteLattice α] {p : ι → Prop} {s : ι → Set α} {f : Filter α} (h : f.HasBasis p s)
Goal: f.limsInf = ⨆ i, ⨆ (_ : p i), sInf (s i)
-- Filter.limsSup_eq_iInf_sSup
theorem limsSup_eq_iInf_sSup {f : Filter α} : limsSup f = ⨅ s ∈ f, sSup s 
Arguments: {α : Type u_1} [CompleteLattice α] {f : Filter α}
Goal: f.limsSup = ⨅ s ∈ f, sSup s
-- Filter.limsInf_eq_iSup_sInf
theorem limsInf_eq_iSup_sInf {f : Filter α} : limsInf f = ⨆ s ∈ f, sInf s 
Arguments: {α : Type u_1} [CompleteLattice α] {f : Filter α}
Goal: f.limsInf = ⨆ s ∈ f, sInf s
-- Filter.limsup_le_iSup
theorem limsup_le_iSup {f : Filter β} {u : β → α} : limsup u f ≤ ⨆ n, u n 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {u : β → α}
Goal: Filter.limsup u f ≤ ⨆ n, u n
-- Filter.iInf_le_liminf
theorem iInf_le_liminf {f : Filter β} {u : β → α} : ⨅ n, u n ≤ liminf u f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {u : β → α}
Goal: ⨅ n, u n ≤ Filter.liminf u f
-- Filter.limsup_eq_iInf_iSup
theorem limsup_eq_iInf_iSup {f : Filter β} {u : β → α} : limsup u f = ⨅ s ∈ f, ⨆ a ∈ s, u a 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {u : β → α}
Goal: Filter.limsup u f = ⨅ s ∈ f, ⨆ a ∈ s, u a
-- Filter.limsup_eq_iInf_iSup_of_nat
theorem limsup_eq_iInf_iSup_of_nat {u : ℕ → α} : limsup u atTop = ⨅ n : ℕ, ⨆ i ≥ n, u i 
Arguments: {α : Type u_1} [CompleteLattice α] {u : ℕ → α}
Goal: Filter.limsup u Filter.atTop = ⨅ n, ⨆ i, ⨆ (_ : i ≥ n), u i
-- Filter.limsup_eq_iInf_iSup_of_nat'
theorem limsup_eq_iInf_iSup_of_nat' {u : ℕ → α} : limsup u atTop = ⨅ n : ℕ, ⨆ i : ℕ, u (i + n) 
Arguments: {α : Type u_1} [CompleteLattice α] {u : ℕ → α}
Goal: Filter.limsup u Filter.atTop = ⨅ n, ⨆ i, u (i + n)
-- Filter.HasBasis.limsup_eq_iInf_iSup
theorem HasBasis.limsup_eq_iInf_iSup {p : ι → Prop} {s : ι → Set β} {f : Filter β} {u : β → α}
    (h : f.HasBasis p s) : limsup u f = ⨅ (i) (_ : p i), ⨆ a ∈ s i, u a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [CompleteLattice α] {p : ι → Prop} {s : ι → Set β} {f : Filter β} {u : β → α} (h : f.HasBasis p s)
Goal: Filter.limsup u f = ⨅ i, ⨅ (_ : p i), ⨆ a ∈ s i, u a
-- Filter.blimsup_congr'
theorem blimsup_congr' {f : Filter β} {p q : β → Prop} {u : β → α}
    (h : ∀ᶠ x in f, u x ≠ ⊥ → (p x ↔ q x)) : blimsup u f p = blimsup u f q 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α} (h : ∀ᶠ (x : β) in f, u x ≠ ⊥ → (p x ↔ q x))
Goal: Filter.blimsup u f p = Filter.blimsup u f q
-- Filter.bliminf_congr'
theorem bliminf_congr' {f : Filter β} {p q : β → Prop} {u : β → α}
    (h : ∀ᶠ x in f, u x ≠ ⊤ → (p x ↔ q x)) : bliminf u f p = bliminf u f q 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α} (h : ∀ᶠ (x : β) in f, u x ≠ ⊤ → (p x ↔ q x))
Goal: Filter.bliminf u f p = Filter.bliminf u f q
-- Filter.HasBasis.blimsup_eq_iInf_iSup
lemma HasBasis.blimsup_eq_iInf_iSup {p : ι → Prop} {s : ι → Set β} {f : Filter β} {u : β → α}
    (hf : f.HasBasis p s) {q : β → Prop} :
    blimsup u f q = ⨅ (i) (_ : p i), ⨆ a ∈ s i, ⨆ (_ : q a), u a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [CompleteLattice α] {p : ι → Prop} {s : ι → Set β} {f : Filter β} {u : β → α} (hf : f.HasBasis p s) {q : β → Prop}
Goal: Filter.blimsup u f q = ⨅ i, ⨅ (_ : p i), ⨆ a ∈ s i, ⨆ (_ : q a), u a
-- Filter.blimsup_eq_iInf_biSup
theorem blimsup_eq_iInf_biSup {f : Filter β} {p : β → Prop} {u : β → α} :
    blimsup u f p = ⨅ s ∈ f, ⨆ (b) (_ : p b ∧ b ∈ s), u b 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p : β → Prop} {u : β → α}
Goal: Filter.blimsup u f p = ⨅ s ∈ f, ⨆ b, ⨆ (_ : p b ∧ b ∈ s), u b
-- Filter.blimsup_eq_iInf_biSup_of_nat
theorem blimsup_eq_iInf_biSup_of_nat {p : ℕ → Prop} {u : ℕ → α} :
    blimsup u atTop p = ⨅ i, ⨆ (j) (_ : p j ∧ i ≤ j), u j 
Arguments: {α : Type u_1} [CompleteLattice α] {p : ℕ → Prop} {u : ℕ → α}
Goal: Filter.blimsup u Filter.atTop p = ⨅ i, ⨆ j, ⨆ (_ : p j ∧ i ≤ j), u j
-- Filter.liminf_eq_iSup_iInf
theorem liminf_eq_iSup_iInf {f : Filter β} {u : β → α} : liminf u f = ⨆ s ∈ f, ⨅ a ∈ s, u a 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {u : β → α}
Goal: Filter.liminf u f = ⨆ s ∈ f, ⨅ a ∈ s, u a
-- Filter.liminf_eq_iSup_iInf_of_nat
theorem liminf_eq_iSup_iInf_of_nat {u : ℕ → α} : liminf u atTop = ⨆ n : ℕ, ⨅ i ≥ n, u i 
Arguments: {α : Type u_1} [CompleteLattice α] {u : ℕ → α}
Goal: Filter.liminf u Filter.atTop = ⨆ n, ⨅ i, ⨅ (_ : i ≥ n), u i
-- Filter.liminf_eq_iSup_iInf_of_nat'
theorem liminf_eq_iSup_iInf_of_nat' {u : ℕ → α} : liminf u atTop = ⨆ n : ℕ, ⨅ i : ℕ, u (i + n) 
Arguments: {α : Type u_1} [CompleteLattice α] {u : ℕ → α}
Goal: Filter.liminf u Filter.atTop = ⨆ n, ⨅ i, u (i + n)
-- Filter.HasBasis.liminf_eq_iSup_iInf
theorem HasBasis.liminf_eq_iSup_iInf {p : ι → Prop} {s : ι → Set β} {f : Filter β} {u : β → α}
    (h : f.HasBasis p s) : liminf u f = ⨆ (i) (_ : p i), ⨅ a ∈ s i, u a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [CompleteLattice α] {p : ι → Prop} {s : ι → Set β} {f : Filter β} {u : β → α} (h : f.HasBasis p s)
Goal: Filter.liminf u f = ⨆ i, ⨆ (_ : p i), ⨅ a ∈ s i, u a
-- Filter.bliminf_eq_iSup_biInf
theorem bliminf_eq_iSup_biInf {f : Filter β} {p : β → Prop} {u : β → α} :
    bliminf u f p = ⨆ s ∈ f, ⨅ (b) (_ : p b ∧ b ∈ s), u b 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p : β → Prop} {u : β → α}
Goal: Filter.bliminf u f p = ⨆ s ∈ f, ⨅ b, ⨅ (_ : p b ∧ b ∈ s), u b
-- Filter.bliminf_eq_iSup_biInf_of_nat
theorem bliminf_eq_iSup_biInf_of_nat {p : ℕ → Prop} {u : ℕ → α} :
    bliminf u atTop p = ⨆ i, ⨅ (j) (_ : p j ∧ i ≤ j), u j 
Arguments: {α : Type u_1} [CompleteLattice α] {p : ℕ → Prop} {u : ℕ → α}
Goal: Filter.bliminf u Filter.atTop p = ⨆ i, ⨅ j, ⨅ (_ : p j ∧ i ≤ j), u j
-- Filter.limsup_eq_sInf_sSup
theorem limsup_eq_sInf_sSup {ι R : Type*} (F : Filter ι) [CompleteLattice R] (a : ι → R) :
    limsup a F = sInf ((fun I => sSup (a '' I)) '' F.sets) 
Arguments: {ι : Type u_6} {R : Type u_7} (F : Filter ι) [CompleteLattice R] (a : ι → R)
Goal: Filter.limsup a F = sInf ((fun I => sSup (a '' I)) '' F.sets)
-- Filter.liminf_eq_sSup_sInf
theorem liminf_eq_sSup_sInf {ι R : Type*} (F : Filter ι) [CompleteLattice R] (a : ι → R) :
    liminf a F = sSup ((fun I => sInf (a '' I)) '' F.sets) 
Arguments: {ι : Type u_6} {R : Type u_7} (F : Filter ι) [CompleteLattice R] (a : ι → R)
Goal: Filter.liminf a F = sSup ((fun I => sInf (a '' I)) '' F.sets)
-- Filter.liminf_le_of_frequently_le'
theorem liminf_le_of_frequently_le' {α β} [CompleteLattice β] {f : Filter α} {u : α → β} {x : β}
    (h : ∃ᶠ a in f, u a ≤ x) : liminf u f ≤ x 
Arguments: {α : Type u_6} {β : Type u_7} [CompleteLattice β] {f : Filter α} {u : α → β} {x : β} (h : ∃ᶠ (a : α) in f, u a ≤ x)
Goal: Filter.liminf u f ≤ x
-- Filter.le_limsup_of_frequently_le'
theorem le_limsup_of_frequently_le' {α β} [CompleteLattice β] {f : Filter α} {u : α → β} {x : β}
    (h : ∃ᶠ a in f, x ≤ u a) : x ≤ limsup u f 
Arguments: {α : Type u_6} {β : Type u_7} [CompleteLattice β] {f : Filter α} {u : α → β} {x : β} (h : ∃ᶠ (a : α) in f, x ≤ u a)
Goal: x ≤ Filter.limsup u f
-- CompleteLatticeHom.apply_limsup_iterate
@[simp]
theorem _root_.CompleteLatticeHom.apply_limsup_iterate (f : CompleteLatticeHom α α) (a : α) :
    f (limsup (fun n => f^[n] a) atTop) = limsup (fun n => f^[n] a) atTop 
Arguments: {α : Type u_1} [CompleteLattice α] (f : CompleteLatticeHom α α) (a : α)
Goal: f (Filter.limsup (fun n => (⇑f)^[n] a) Filter.atTop) = Filter.limsup (fun n => (⇑f)^[n] a) Filter.atTop