ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CompleteLatticeHom.apply_liminf_iterate
theorem _root_.CompleteLatticeHom.apply_liminf_iterate (f : CompleteLatticeHom α α) (a : α) :
    f (liminf (fun n => f^[n] a) atTop) = liminf (fun n => f^[n] a) atTop 
Arguments: {α : Type u_1} [CompleteLattice α] (f : CompleteLatticeHom α α) (a : α)
Goal: f (Filter.liminf (fun n => (⇑f)^[n] a) Filter.atTop) = Filter.liminf (fun n => (⇑f)^[n] a) Filter.atTop
-- Filter.blimsup_mono
theorem blimsup_mono (h : ∀ 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 : ∀ (x : β), p x → q x)
Goal: Filter.blimsup u f p ≤ Filter.blimsup u f q
-- Filter.bliminf_antitone
theorem bliminf_antitone (h : ∀ x, p x → q x) : bliminf u f q ≤ bliminf u f p 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α} (h : ∀ (x : β), p x → q x)
Goal: Filter.bliminf u f q ≤ Filter.bliminf u f p
-- Filter.mono_blimsup'
theorem mono_blimsup' (h : ∀ᶠ x in f, p x → u x ≤ v x) : blimsup u f p ≤ blimsup v f p 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p : β → Prop} {u v : β → α} (h : ∀ᶠ (x : β) in f, p x → u x ≤ v x)
Goal: Filter.blimsup u f p ≤ Filter.blimsup v f p
-- Filter.mono_blimsup
theorem mono_blimsup (h : ∀ x, p x → u x ≤ v x) : blimsup u f p ≤ blimsup v f p 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p : β → Prop} {u v : β → α} (h : ∀ (x : β), p x → u x ≤ v x)
Goal: Filter.blimsup u f p ≤ Filter.blimsup v f p
-- Filter.mono_bliminf'
theorem mono_bliminf' (h : ∀ᶠ x in f, p x → u x ≤ v x) : bliminf u f p ≤ bliminf v f p 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p : β → Prop} {u v : β → α} (h : ∀ᶠ (x : β) in f, p x → u x ≤ v x)
Goal: Filter.bliminf u f p ≤ Filter.bliminf v f p
-- Filter.mono_bliminf
theorem mono_bliminf (h : ∀ x, p x → u x ≤ v x) : bliminf u f p ≤ bliminf v f p 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p : β → Prop} {u v : β → α} (h : ∀ (x : β), p x → u x ≤ v x)
Goal: Filter.bliminf u f p ≤ Filter.bliminf v f p
-- Filter.bliminf_antitone_filter
theorem bliminf_antitone_filter (h : f ≤ g) : bliminf u g p ≤ bliminf u f p 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f g : Filter β} {p : β → Prop} {u : β → α} (h : f ≤ g)
Goal: Filter.bliminf u g p ≤ Filter.bliminf u f p
-- Filter.blimsup_monotone_filter
theorem blimsup_monotone_filter (h : f ≤ g) : blimsup u f p ≤ blimsup u g p 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f g : Filter β} {p : β → Prop} {u : β → α} (h : f ≤ g)
Goal: Filter.blimsup u f p ≤ Filter.blimsup u g p
-- Filter.blimsup_and_le_inf
theorem blimsup_and_le_inf : (blimsup u f fun 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}
Goal: (Filter.blimsup u f fun x => p x ∧ q x) ≤ Filter.blimsup u f p ⊓ Filter.blimsup u f q
-- Filter.bliminf_sup_le_inf_aux_left
@[simp]
theorem bliminf_sup_le_inf_aux_left :
    (blimsup u f fun x => p x ∧ q x) ≤ blimsup u f p 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α}
Goal: (Filter.blimsup u f fun x => p x ∧ q x) ≤ Filter.blimsup u f p
-- Filter.bliminf_sup_le_inf_aux_right
@[simp]
theorem bliminf_sup_le_inf_aux_right :
    (blimsup u f fun x => p x ∧ q x) ≤ 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}
Goal: (Filter.blimsup u f fun x => p x ∧ q x) ≤ Filter.blimsup u f q
-- Filter.bliminf_sup_le_and
theorem bliminf_sup_le_and : bliminf u f p ⊔ bliminf u f q ≤ bliminf u f fun x => p x ∧ q x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α}
Goal: Filter.bliminf u f p ⊔ Filter.bliminf u f q ≤ Filter.bliminf u f fun x => p x ∧ q x
-- Filter.bliminf_sup_le_and_aux_left
@[simp]
theorem bliminf_sup_le_and_aux_left : bliminf u f p ≤ bliminf u f fun x => p x ∧ q x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α}
Goal: Filter.bliminf u f p ≤ Filter.bliminf u f fun x => p x ∧ q x
-- Filter.bliminf_sup_le_and_aux_right
@[simp]
theorem bliminf_sup_le_and_aux_right : bliminf u f q ≤ bliminf u f fun x => p x ∧ q x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α}
Goal: Filter.bliminf u f q ≤ Filter.bliminf u f fun x => p x ∧ q x
-- Filter.blimsup_sup_le_or
-- @[simp] -- Porting note: simp_nf linter, lhs simplifies, added _aux simp versions below
theorem blimsup_sup_le_or : blimsup u f p ⊔ blimsup u f q ≤ blimsup u f fun x => p x ∨ q x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α}
Goal: Filter.blimsup u f p ⊔ Filter.blimsup u f q ≤ Filter.blimsup u f fun x => p x ∨ q x
-- Filter.bliminf_sup_le_or_aux_left
@[simp]
theorem bliminf_sup_le_or_aux_left : blimsup u f p ≤ blimsup u f fun x => p x ∨ q x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α}
Goal: Filter.blimsup u f p ≤ Filter.blimsup u f fun x => p x ∨ q x
-- Filter.bliminf_sup_le_or_aux_right
@[simp]
theorem bliminf_sup_le_or_aux_right : blimsup u f q ≤ blimsup u f fun x => p x ∨ q x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α}
Goal: Filter.blimsup u f q ≤ Filter.blimsup u f fun x => p x ∨ q x
-- Filter.bliminf_or_le_inf
--@[simp] -- Porting note: simp_nf linter, lhs simplifies, added _aux simp versions below
theorem bliminf_or_le_inf : (bliminf u f fun 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}
Goal: (Filter.bliminf u f fun x => p x ∨ q x) ≤ Filter.bliminf u f p ⊓ Filter.bliminf u f q
-- Filter.bliminf_or_le_inf_aux_left
@[simp]
theorem bliminf_or_le_inf_aux_left : (bliminf u f fun x => p x ∨ q x) ≤ bliminf u f p 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : Filter β} {p q : β → Prop} {u : β → α}
Goal: (Filter.bliminf u f fun x => p x ∨ q x) ≤ Filter.bliminf u f p
-- Filter.bliminf_or_le_inf_aux_right
@[simp]
theorem bliminf_or_le_inf_aux_right : (bliminf u f fun x => p x ∨ q x) ≤ 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}
Goal: (Filter.bliminf u f fun x => p x ∨ q x) ≤ Filter.bliminf u f q
-- OrderIso.apply_blimsup
theorem _root_.OrderIso.apply_blimsup [CompleteLattice γ] (e : α ≃o γ) :
    e (blimsup u f p) = blimsup (e ∘ u) f p 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CompleteLattice α] {f : Filter β} {p : β → Prop} {u : β → α} [CompleteLattice γ] (e : α ≃o γ)
Goal: e (Filter.blimsup u f p) = Filter.blimsup (⇑e ∘ u) f p
-- OrderIso.apply_bliminf
theorem _root_.OrderIso.apply_bliminf [CompleteLattice γ] (e : α ≃o γ) :
    e (bliminf u f p) = bliminf (e ∘ u) f p 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CompleteLattice α] {f : Filter β} {p : β → Prop} {u : β → α} [CompleteLattice γ] (e : α ≃o γ)
Goal: e (Filter.bliminf u f p) = Filter.bliminf (⇑e ∘ u) f p
-- sSupHom.apply_blimsup_le
theorem _root_.sSupHom.apply_blimsup_le [CompleteLattice γ] (g : sSupHom α γ) :
    g (blimsup u f p) ≤ blimsup (g ∘ u) f p 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CompleteLattice α] {f : Filter β} {p : β → Prop} {u : β → α} [CompleteLattice γ] (g : sSupHom α γ)
Goal: g (Filter.blimsup u f p) ≤ Filter.blimsup (⇑g ∘ u) f p
-- sInfHom.le_apply_bliminf
theorem _root_.sInfHom.le_apply_bliminf [CompleteLattice γ] (g : sInfHom α γ) :
    bliminf (g ∘ u) f p ≤ g (bliminf u f p) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CompleteLattice α] {f : Filter β} {p : β → Prop} {u : β → α} [CompleteLattice γ] (g : sInfHom α γ)
Goal: Filter.bliminf (⇑g ∘ u) f p ≤ g (Filter.bliminf u f p)
-- Filter.limsup_sup_filter
lemma limsup_sup_filter {g} : limsup u (f ⊔ g) = limsup u f ⊔ limsup u g 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {u : β → α} {g : Filter β}
Goal: Filter.limsup u (f ⊔ g) = Filter.limsup u f ⊔ Filter.limsup u g
-- Filter.liminf_sup_filter
lemma liminf_sup_filter {g} : liminf u (f ⊔ g) = liminf u f ⊓ liminf u g 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {u : β → α} {g : Filter β}
Goal: Filter.liminf u (f ⊔ g) = Filter.liminf u f ⊓ Filter.liminf u g
-- Filter.blimsup_or_eq_sup
@[simp]
theorem blimsup_or_eq_sup : (blimsup u f fun x => p x ∨ q x) = blimsup u f p ⊔ blimsup u f q 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {p q : β → Prop} {u : β → α}
Goal: (Filter.blimsup u f fun x => p x ∨ q x) = Filter.blimsup u f p ⊔ Filter.blimsup u f q
-- Filter.bliminf_or_eq_inf
@[simp]
theorem bliminf_or_eq_inf : (bliminf u f fun x => p x ∨ q x) = bliminf u f p ⊓ bliminf u f q 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {p q : β → Prop} {u : β → α}
Goal: (Filter.bliminf u f fun x => p x ∨ q x) = Filter.bliminf u f p ⊓ Filter.bliminf u f q
-- Filter.blimsup_sup_not
@[simp]
lemma blimsup_sup_not : blimsup u f p ⊔ blimsup u f (¬p ·) = limsup u f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {p : β → Prop} {u : β → α}
Goal: (Filter.blimsup u f p ⊔ Filter.blimsup u f fun x => ¬p x) = Filter.limsup u f
-- Filter.bliminf_inf_not
@[simp]
lemma bliminf_inf_not : bliminf u f p ⊓ bliminf u f (¬p ·) = liminf u f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {p : β → Prop} {u : β → α}
Goal: (Filter.bliminf u f p ⊓ Filter.bliminf u f fun x => ¬p x) = Filter.liminf u f
-- Filter.blimsup_not_sup
@[simp]
lemma blimsup_not_sup : blimsup u f (¬p ·) ⊔ blimsup u f p = limsup u f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {p : β → Prop} {u : β → α}
Goal: (Filter.blimsup u f fun x => ¬p x) ⊔ Filter.blimsup u f p = Filter.limsup u f
-- Filter.bliminf_not_inf
@[simp]
lemma bliminf_not_inf : bliminf u f (¬p ·) ⊓ bliminf u f p = liminf u f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {p : β → Prop} {u : β → α}
Goal: (Filter.bliminf u f fun x => ¬p x) ⊓ Filter.bliminf u f p = Filter.liminf u f
-- Filter.limsup_piecewise
lemma limsup_piecewise {s : Set β} [DecidablePred (· ∈ s)] {v} :
    limsup (s.piecewise u v) f = blimsup u f (· ∈ s) ⊔ blimsup v f (· ∉ s) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {u : β → α} {s : Set β} [DecidablePred fun x => x ∈ s] {v : β → α}
Goal: Filter.limsup (s.piecewise u v) f = (Filter.blimsup u f fun x => x ∈ s) ⊔ Filter.blimsup v f fun x => x ∉ s
-- Filter.liminf_piecewise
lemma liminf_piecewise {s : Set β} [DecidablePred (· ∈ s)] {v} :
    liminf (s.piecewise u v) f = bliminf u f (· ∈ s) ⊓ bliminf v f (· ∉ s) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {u : β → α} {s : Set β} [DecidablePred fun x => x ∈ s] {v : β → α}
Goal: Filter.liminf (s.piecewise u v) f = (Filter.bliminf u f fun x => x ∈ s) ⊓ Filter.bliminf v f fun x => x ∉ s
-- Filter.sup_limsup
theorem sup_limsup [NeBot f] (a : α) : a ⊔ limsup u f = limsup (fun x => a ⊔ u x) f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {u : β → α} [f.NeBot] (a : α)
Goal: a ⊔ Filter.limsup u f = Filter.limsup (fun x => a ⊔ u x) f
-- Filter.inf_liminf
theorem inf_liminf [NeBot f] (a : α) : a ⊓ liminf u f = liminf (fun x => a ⊓ u x) f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {u : β → α} [f.NeBot] (a : α)
Goal: a ⊓ Filter.liminf u f = Filter.liminf (fun x => a ⊓ u x) f
-- Filter.sup_liminf
theorem sup_liminf (a : α) : a ⊔ liminf u f = liminf (fun x => a ⊔ u x) f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {u : β → α} (a : α)
Goal: a ⊔ Filter.liminf u f = Filter.liminf (fun x => a ⊔ u x) f
-- Filter.inf_limsup
theorem inf_limsup (a : α) : a ⊓ limsup u f = limsup (fun x => a ⊓ u x) f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteDistribLattice α] {f : Filter β} {u : β → α} (a : α)
Goal: a ⊓ Filter.limsup u f = Filter.limsup (fun x => a ⊓ u x) f
-- Filter.limsup_compl
theorem limsup_compl : (limsup u f)ᶜ = liminf (compl ∘ u) f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteBooleanAlgebra α] (f : Filter β) (u : β → α)
Goal: (Filter.limsup u f)ᶜ = Filter.liminf (compl ∘ u) f
-- Filter.liminf_compl
theorem liminf_compl : (liminf u f)ᶜ = limsup (compl ∘ u) f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteBooleanAlgebra α] (f : Filter β) (u : β → α)
Goal: (Filter.liminf u f)ᶜ = Filter.limsup (compl ∘ u) f
-- Filter.limsup_sdiff
theorem limsup_sdiff (a : α) : limsup u f \ a = limsup (fun b => u b \ a) f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteBooleanAlgebra α] (f : Filter β) (u : β → α) (a : α)
Goal: Filter.limsup u f \ a = Filter.limsup (fun b => u b \ a) f
-- Filter.liminf_sdiff
theorem liminf_sdiff [NeBot f] (a : α) : liminf u f \ a = liminf (fun b => u b \ a) f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteBooleanAlgebra α] (f : Filter β) (u : β → α) [f.NeBot] (a : α)
Goal: Filter.liminf u f \ a = Filter.liminf (fun b => u b \ a) f
-- Filter.sdiff_limsup
theorem sdiff_limsup [NeBot f] (a : α) : a \ limsup u f = liminf (fun b => a \ u b) f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteBooleanAlgebra α] (f : Filter β) (u : β → α) [f.NeBot] (a : α)
Goal: a \ Filter.limsup u f = Filter.liminf (fun b => a \ u b) f
-- Filter.sdiff_liminf
theorem sdiff_liminf (a : α) : a \ liminf u f = limsup (fun b => a \ u b) f 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteBooleanAlgebra α] (f : Filter β) (u : β → α) (a : α)
Goal: a \ Filter.liminf u f = Filter.limsup (fun b => a \ u b) f
-- Filter.mem_liminf_iff_eventually_mem
lemma mem_liminf_iff_eventually_mem : (a ∈ liminf s 𝓕) ↔ (∀ᶠ i in 𝓕, a ∈ s i) 
Arguments: {α : Type u_1} {ι : Type u_4} {s : ι → Set α} {𝓕 : Filter ι} {a : α}
Goal: a ∈ Filter.liminf s 𝓕 ↔ ∀ᶠ (i : ι) in 𝓕, a ∈ s i
-- Filter.mem_limsup_iff_frequently_mem
lemma mem_limsup_iff_frequently_mem : (a ∈ limsup s 𝓕) ↔ (∃ᶠ i in 𝓕, a ∈ s i) 
Arguments: {α : Type u_1} {ι : Type u_4} {s : ι → Set α} {𝓕 : Filter ι} {a : α}
Goal: a ∈ Filter.limsup s 𝓕 ↔ ∃ᶠ (i : ι) in 𝓕, a ∈ s i
-- Filter.cofinite.blimsup_set_eq
theorem cofinite.blimsup_set_eq :
    blimsup s cofinite p = { x | { n | p n ∧ x ∈ s n }.Infinite } 
Arguments: {α : Type u_1} {ι : Type u_4} {p : ι → Prop} {s : ι → Set α}
Goal: Filter.blimsup s Filter.cofinite p = {x | {n | p n ∧ x ∈ s n}.Infinite}
-- Filter.cofinite.bliminf_set_eq
theorem cofinite.bliminf_set_eq : bliminf s cofinite p = { x | { n | p n ∧ x ∉ s n }.Finite } 
Arguments: {α : Type u_1} {ι : Type u_4} {p : ι → Prop} {s : ι → Set α}
Goal: Filter.bliminf s Filter.cofinite p = {x | {n | p n ∧ x ∉ s n}.Finite}
-- Filter.cofinite.limsup_set_eq
theorem cofinite.limsup_set_eq : limsup s cofinite = { x | { n | x ∈ s n }.Infinite } 
Arguments: {α : Type u_1} {ι : Type u_4} {s : ι → Set α}
Goal: Filter.limsup s Filter.cofinite = {x | {n | x ∈ s n}.Infinite}
-- Filter.cofinite.liminf_set_eq
theorem cofinite.liminf_set_eq : liminf s cofinite = { x | { n | x ∉ s n }.Finite } 
Arguments: {α : Type u_1} {ι : Type u_4} {s : ι → Set α}
Goal: Filter.liminf s Filter.cofinite = {x | {n | x ∉ s n}.Finite}
-- Filter.exists_forall_mem_of_hasBasis_mem_blimsup
theorem exists_forall_mem_of_hasBasis_mem_blimsup {l : Filter β} {b : ι → Set β} {q : ι → Prop}
    (hl : l.HasBasis q b) {u : β → Set α} {p : β → Prop} {x : α} (hx : x ∈ blimsup u l p) :
    ∃ f : { i | q i } → β, ∀ i, x ∈ u (f i) ∧ p (f i) ∧ f i ∈ b i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} {l : Filter β} {b : ι → Set β} {q : ι → Prop} (hl : l.HasBasis q b) {u : β → Set α} {p : β → Prop} {x : α} (hx : x ∈ Filter.blimsup u l p)
Goal: ∃ f, ∀ (i : ↑{i | q i}), x ∈ u (f i) ∧ p (f i) ∧ f i ∈ b ↑i
-- Filter.exists_forall_mem_of_hasBasis_mem_blimsup'
theorem exists_forall_mem_of_hasBasis_mem_blimsup' {l : Filter β} {b : ι → Set β}
    (hl : l.HasBasis (fun _ => True) b) {u : β → Set α} {p : β → Prop} {x : α}
    (hx : x ∈ blimsup u l p) : ∃ f : ι → β, ∀ i, x ∈ u (f i) ∧ p (f i) ∧ f i ∈ b i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} {l : Filter β} {b : ι → Set β} (hl : l.HasBasis (fun x => True) b) {u : β → Set α} {p : β → Prop} {x : α} (hx : x ∈ Filter.blimsup u l p)
Goal: ∃ f, ∀ (i : ι), x ∈ u (f i) ∧ p (f i) ∧ f i ∈ b i
-- Filter.frequently_lt_of_lt_limsSup
theorem frequently_lt_of_lt_limsSup {f : Filter α} [ConditionallyCompleteLinearOrder α] {a : α}
    (hf : f.IsCobounded (· ≤ ·) := by isBoundedDefault)
    (h : a < limsSup f) : ∃ᶠ n in f, a < n 
Arguments: {α : Type u_1} {f : Filter α} [ConditionallyCompleteLinearOrder α] {a : α} (hf : Filter.IsCobounded (fun x x_1 => x ≤ x_1) f := by isBoundedDefault) (h : a < f.limsSup)
Goal: ∃ᶠ (n : α) in f, a < n
-- Filter.frequently_lt_of_limsInf_lt
theorem frequently_lt_of_limsInf_lt {f : Filter α} [ConditionallyCompleteLinearOrder α] {a : α}
    (hf : f.IsCobounded (· ≥ ·) := by isBoundedDefault)
    (h : limsInf f < a) : ∃ᶠ n in f, n < a 
Arguments: {α : Type u_1} {f : Filter α} [ConditionallyCompleteLinearOrder α] {a : α} (hf : Filter.IsCobounded (fun x x_1 => x ≥ x_1) f := by isBoundedDefault) (h : f.limsInf < a)
Goal: ∃ᶠ (n : α) in f, n < a
-- Filter.eventually_lt_of_lt_liminf
theorem eventually_lt_of_lt_liminf {f : Filter α} [ConditionallyCompleteLinearOrder β] {u : α → β}
    {b : β} (h : b < liminf u f)
    (hu : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) :
    ∀ᶠ a in f, b < u a 
Arguments: {α : Type u_1} {β : Type u_2} {f : Filter α} [ConditionallyCompleteLinearOrder β] {u : α → β} {b : β} (h : b < Filter.liminf u f) (hu : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault)
Goal: ∀ᶠ (a : α) in f, b < u a
-- Filter.eventually_lt_of_limsup_lt
theorem eventually_lt_of_limsup_lt {f : Filter α} [ConditionallyCompleteLinearOrder β] {u : α → β}
    {b : β} (h : limsup u f < b)
    (hu : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault) :
    ∀ᶠ a in f, u a < b 
Arguments: {α : Type u_1} {β : Type u_2} {f : Filter α} [ConditionallyCompleteLinearOrder β] {u : α → β} {b : β} (h : Filter.limsup u f < b) (hu : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault)
Goal: ∀ᶠ (a : α) in f, u a < b
-- Filter.le_limsup_of_frequently_le
theorem le_limsup_of_frequently_le {α β} [ConditionallyCompleteLinearOrder β] {f : Filter α}
    {u : α → β} {b : β} (hu_le : ∃ᶠ x in f, b ≤ u x)
    (hu : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault) :
    b ≤ limsup u f 
Arguments: {α : Type u_6} {β : Type u_7} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {b : β} (hu_le : ∃ᶠ (x : α) in f, b ≤ u x) (hu : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault)
Goal: b ≤ Filter.limsup u f
-- Filter.liminf_le_of_frequently_le
theorem liminf_le_of_frequently_le {α β} [ConditionallyCompleteLinearOrder β] {f : Filter α}
    {u : α → β} {b : β} (hu_le : ∃ᶠ x in f, u x ≤ b)
    (hu : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) :
    liminf u f ≤ b 
Arguments: {α : Type u_6} {β : Type u_7} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {b : β} (hu_le : ∃ᶠ (x : α) in f, u x ≤ b) (hu : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault)
Goal: Filter.liminf u f ≤ b
-- Filter.frequently_lt_of_lt_limsup
theorem frequently_lt_of_lt_limsup {α β} [ConditionallyCompleteLinearOrder β] {f : Filter α}
    {u : α → β} {b : β}
    (hu : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h : b < limsup u f) : ∃ᶠ x in f, b < u x 
Arguments: {α : Type u_6} {β : Type u_7} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {b : β} (hu : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (h : b < Filter.limsup u f)
Goal: ∃ᶠ (x : α) in f, b < u x
-- Filter.frequently_lt_of_liminf_lt
theorem frequently_lt_of_liminf_lt {α β} [ConditionallyCompleteLinearOrder β] {f : Filter α}
    {u : α → β} {b : β}
    (hu : f.IsCoboundedUnder (· ≥ ·) u := by isBoundedDefault)
    (h : liminf u f < b) : ∃ᶠ x in f, u x < b 
Arguments: {α : Type u_6} {β : Type u_7} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {b : β} (hu : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (h : Filter.liminf u f < b)
Goal: ∃ᶠ (x : α) in f, u x < b
-- Filter.limsup_le_iff
theorem limsup_le_iff {α β} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {x : β}
    (h₁ : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h₂ : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault) :
    limsup u f ≤ x ↔ ∀ y > x, ∀ᶠ a in f, u a < y 
Arguments: {α : Type u_6} {β : Type u_7} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {x : β} (h₁ : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (h₂ : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault)
Goal: Filter.limsup u f ≤ x ↔ ∀ y > x, ∀ᶠ (a : α) in f, u a < y
-- Filter.le_limsup_iff
theorem le_limsup_iff {α β} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {x : β}
    (h₁ : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h₂ : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault) :
    x ≤ limsup u f ↔ ∀ y < x, ∃ᶠ a in f, y < u a 
Arguments: {α : Type u_6} {β : Type u_7} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {x : β} (h₁ : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (h₂ : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault)
Goal: x ≤ Filter.limsup u f ↔ ∀ y < x, ∃ᶠ (a : α) in f, y < u a
-- Filter.le_liminf_iff
theorem le_liminf_iff {α β} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {x : β}
    (h₁ : f.IsCoboundedUnder (· ≥ ·) u := by isBoundedDefault)
    (h₂ : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) :
    x ≤ liminf u f ↔ ∀ y < x, ∀ᶠ a in f, y < u a 
Arguments: {α : Type u_6} {β : Type u_7} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {x : β} (h₁ : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (h₂ : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault)
Goal: x ≤ Filter.liminf u f ↔ ∀ y < x, ∀ᶠ (a : α) in f, y < u a
-- Filter.liminf_le_iff
theorem liminf_le_iff {α β} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {x : β}
    (h₁ : f.IsCoboundedUnder (· ≥ ·) u := by isBoundedDefault)
    (h₂ : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) :
    liminf u f ≤ x ↔ ∀ y > x, ∃ᶠ a in f, u a < y 
Arguments: {α : Type u_6} {β : Type u_7} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {x : β} (h₁ : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (h₂ : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault)
Goal: Filter.liminf u f ≤ x ↔ ∀ y > x, ∃ᶠ (a : α) in f, u a < y
-- Filter.lt_mem_sets_of_limsSup_lt
theorem lt_mem_sets_of_limsSup_lt (h : f.IsBounded (· ≤ ·)) (l : f.limsSup < b) :
    ∀ᶠ a in f, a < b 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {f : Filter α} {b : α} (h : Filter.IsBounded (fun x x_1 => x ≤ x_1) f) (l : f.limsSup < b)
Goal: ∀ᶠ (a : α) in f, a < b
-- Filter.gt_mem_sets_of_limsInf_gt
theorem gt_mem_sets_of_limsInf_gt : f.IsBounded (· ≥ ·) → b < f.limsInf → ∀ᶠ a in f, b < a 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {f : Filter α} {b : α}
Goal: Filter.IsBounded (fun x x_1 => x ≥ x_1) f → b < f.limsInf → ∀ᶠ (a : α) in f, b < a