ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.HasBasis.liminf_eq_ciSup_ciInf
theorem HasBasis.liminf_eq_ciSup_ciInf {v : Filter ι}
    {p : ι' → Prop} {s : ι' → Set ι} [Countable (Subtype p)] [Nonempty (Subtype p)]
    (hv : v.HasBasis p s) {f : ι → α} (hs : ∀ (j : Subtype p), (s j).Nonempty)
    (H : ∃ (j : Subtype p), BddBelow (range (fun (i : s j) ↦ f i))) :
    liminf f v = ⨆ (j : Subtype p), ⨅ (i : s (liminf_reparam f s p j)), f i 
Arguments: {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [ConditionallyCompleteLinearOrder α] {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι} [Countable (Subtype p)] [Nonempty (Subtype p)] (hv : v.HasBasis p s) {f : ι → α} (hs : ∀ (j : Subtype p), (s ↑j).Nonempty) (H : ∃ j, BddBelow (Set.range fun i => f ↑i))
Goal: Filter.liminf f v = ⨆ j, ⨅ i, f ↑i
-- Filter.HasBasis.liminf_eq_ite
theorem HasBasis.liminf_eq_ite {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι}
    [Countable (Subtype p)] [Nonempty (Subtype p)] (hv : v.HasBasis p s) (f : ι → α) :
    liminf f v = if ∃ (j : Subtype p), s j = ∅ then sSup univ else
      if ∀ (j : Subtype p), ¬BddBelow (range (fun (i : s j) ↦ f i)) then sSup ∅
      else ⨆ (j : Subtype p), ⨅ (i : s (liminf_reparam f s p j)), f i 
Arguments: {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [ConditionallyCompleteLinearOrder α] {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι} [Countable (Subtype p)] [Nonempty (Subtype p)] (hv : v.HasBasis p s) (f : ι → α)
Goal: Filter.liminf f v =
  if ∃ j, s ↑j = ∅ then sSup Set.univ
  else if ∀ (j : Subtype p), ¬BddBelow (Set.range fun i => f ↑i) then sSup ∅ else ⨆ j, ⨅ i, f ↑i
-- Filter.HasBasis.limsup_eq_ciInf_ciSup
theorem HasBasis.limsup_eq_ciInf_ciSup {v : Filter ι}
    {p : ι' → Prop} {s : ι' → Set ι} [Countable (Subtype p)] [Nonempty (Subtype p)]
    (hv : v.HasBasis p s) {f : ι → α} (hs : ∀ (j : Subtype p), (s j).Nonempty)
    (H : ∃ (j : Subtype p), BddAbove (range (fun (i : s j) ↦ f i))) :
    limsup f v = ⨅ (j : Subtype p), ⨆ (i : s (limsup_reparam f s p j)), f i 
Arguments: {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [ConditionallyCompleteLinearOrder α] {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι} [Countable (Subtype p)] [Nonempty (Subtype p)] (hv : v.HasBasis p s) {f : ι → α} (hs : ∀ (j : Subtype p), (s ↑j).Nonempty) (H : ∃ j, BddAbove (Set.range fun i => f ↑i))
Goal: Filter.limsup f v = ⨅ j, ⨆ i, f ↑i
-- Filter.HasBasis.limsup_eq_ite
theorem HasBasis.limsup_eq_ite {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι}
    [Countable (Subtype p)] [Nonempty (Subtype p)] (hv : v.HasBasis p s) (f : ι → α) :
    limsup f v = if ∃ (j : Subtype p), s j = ∅ then sInf univ else
      if ∀ (j : Subtype p), ¬BddAbove (range (fun (i : s j) ↦ f i)) then sInf ∅
      else ⨅ (j : Subtype p), ⨆ (i : s (limsup_reparam f s p j)), f i 
Arguments: {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [ConditionallyCompleteLinearOrder α] {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι} [Countable (Subtype p)] [Nonempty (Subtype p)] (hv : v.HasBasis p s) (f : ι → α)
Goal: Filter.limsup f v =
  if ∃ j, s ↑j = ∅ then sInf Set.univ
  else if ∀ (j : Subtype p), ¬BddAbove (Set.range fun i => f ↑i) then sInf ∅ else ⨅ j, ⨆ i, f ↑i
-- Monotone.isBoundedUnder_le_comp_iff
theorem Monotone.isBoundedUnder_le_comp_iff [Nonempty β] [LinearOrder β] [Preorder γ] [NoMaxOrder γ]
    {g : β → γ} {f : α → β} {l : Filter α} (hg : Monotone g) (hg' : Tendsto g atTop atTop) :
    IsBoundedUnder (· ≤ ·) l (g ∘ f) ↔ IsBoundedUnder (· ≤ ·) l f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Nonempty β] [LinearOrder β] [Preorder γ] [NoMaxOrder γ] {g : β → γ} {f : α → β} {l : Filter α} (hg : Monotone g) (hg' : Filter.Tendsto g Filter.atTop Filter.atTop)
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (g ∘ f) ↔ Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l f
-- Monotone.isBoundedUnder_ge_comp_iff
theorem Monotone.isBoundedUnder_ge_comp_iff [Nonempty β] [LinearOrder β] [Preorder γ] [NoMinOrder γ]
    {g : β → γ} {f : α → β} {l : Filter α} (hg : Monotone g) (hg' : Tendsto g atBot atBot) :
    IsBoundedUnder (· ≥ ·) l (g ∘ f) ↔ IsBoundedUnder (· ≥ ·) l f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Nonempty β] [LinearOrder β] [Preorder γ] [NoMinOrder γ] {g : β → γ} {f : α → β} {l : Filter α} (hg : Monotone g) (hg' : Filter.Tendsto g Filter.atBot Filter.atBot)
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) l (g ∘ f) ↔ Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) l f
-- Antitone.isBoundedUnder_le_comp_iff
theorem Antitone.isBoundedUnder_le_comp_iff [Nonempty β] [LinearOrder β] [Preorder γ] [NoMaxOrder γ]
    {g : β → γ} {f : α → β} {l : Filter α} (hg : Antitone g) (hg' : Tendsto g atBot atTop) :
    IsBoundedUnder (· ≤ ·) l (g ∘ f) ↔ IsBoundedUnder (· ≥ ·) l f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Nonempty β] [LinearOrder β] [Preorder γ] [NoMaxOrder γ] {g : β → γ} {f : α → β} {l : Filter α} (hg : Antitone g) (hg' : Filter.Tendsto g Filter.atBot Filter.atTop)
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (g ∘ f) ↔ Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) l f
-- Antitone.isBoundedUnder_ge_comp_iff
theorem Antitone.isBoundedUnder_ge_comp_iff [Nonempty β] [LinearOrder β] [Preorder γ] [NoMinOrder γ]
    {g : β → γ} {f : α → β} {l : Filter α} (hg : Antitone g) (hg' : Tendsto g atTop atBot) :
    IsBoundedUnder (· ≥ ·) l (g ∘ f) ↔ IsBoundedUnder (· ≤ ·) l f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Nonempty β] [LinearOrder β] [Preorder γ] [NoMinOrder γ] {g : β → γ} {f : α → β} {l : Filter α} (hg : Antitone g) (hg' : Filter.Tendsto g Filter.atTop Filter.atBot)
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) l (g ∘ f) ↔ Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l f
-- GaloisConnection.l_limsup_le
theorem GaloisConnection.l_limsup_le [ConditionallyCompleteLattice β]
    [ConditionallyCompleteLattice γ] {f : Filter α} {v : α → β} {l : β → γ} {u : γ → β}
    (gc : GaloisConnection l u)
    (hlv : f.IsBoundedUnder (· ≤ ·) fun x => l (v x) := by isBoundedDefault)
    (hv_co : f.IsCoboundedUnder (· ≤ ·) v := by isBoundedDefault) :
    l (limsup v f) ≤ limsup (fun x => l (v x)) f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [ConditionallyCompleteLattice β] [ConditionallyCompleteLattice γ] {f : Filter α} {v : α → β} {l : β → γ} {u : γ → β} (gc : GaloisConnection l u) (hlv : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => l (v x) := by isBoundedDefault) (hv_co : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f v := by isBoundedDefault)
Goal: l (Filter.limsup v f) ≤ Filter.limsup (fun x => l (v x)) f
-- OrderIso.limsup_apply
theorem OrderIso.limsup_apply {γ} [ConditionallyCompleteLattice β] [ConditionallyCompleteLattice γ]
    {f : Filter α} {u : α → β} (g : β ≃o γ)
    (hu : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
    (hu_co : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
    (hgu : f.IsBoundedUnder (· ≤ ·) fun x => g (u x) := by isBoundedDefault)
    (hgu_co : f.IsCoboundedUnder (· ≤ ·) fun x => g (u x) := by isBoundedDefault) :
    g (limsup u f) = limsup (fun x => g (u x)) f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_6} [ConditionallyCompleteLattice β] [ConditionallyCompleteLattice γ] {f : Filter α} {u : α → β} (g : β ≃o γ) (hu : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (hu_co : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (hgu : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x) := by isBoundedDefault) (hgu_co : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x) := by isBoundedDefault)
Goal: g (Filter.limsup u f) = Filter.limsup (fun x => g (u x)) f
-- OrderIso.liminf_apply
theorem OrderIso.liminf_apply {γ} [ConditionallyCompleteLattice β] [ConditionallyCompleteLattice γ]
    {f : Filter α} {u : α → β} (g : β ≃o γ)
    (hu : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault)
    (hu_co : f.IsCoboundedUnder (· ≥ ·) u := by isBoundedDefault)
    (hgu : f.IsBoundedUnder (· ≥ ·) fun x => g (u x) := by isBoundedDefault)
    (hgu_co : f.IsCoboundedUnder (· ≥ ·) fun x => g (u x) := by isBoundedDefault) :
    g (liminf u f) = liminf (fun x => g (u x)) f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_6} [ConditionallyCompleteLattice β] [ConditionallyCompleteLattice γ] {f : Filter α} {u : α → β} (g : β ≃o γ) (hu : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (hu_co : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (hgu : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f fun x => g (u x) := by isBoundedDefault) (hgu_co : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f fun x => g (u x) := by isBoundedDefault)
Goal: g (Filter.liminf u f) = Filter.liminf (fun x => g (u x)) f
-- isCoboundedUnder_le_max
theorem isCoboundedUnder_le_max [LinearOrder β] {f : Filter α} {u v : α → β}
    (h : f.IsCoboundedUnder (· ≤ ·) u ∨ f.IsCoboundedUnder (· ≤ ·) v) :
    f.IsCoboundedUnder (· ≤ ·) (fun a ↦ max (u a) (v a)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder β] {f : Filter α} {u v : α → β} (h : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u ∨ Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f v)
Goal: Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun a => max (u a) (v a)
-- limsup_max
theorem limsup_max [ConditionallyCompleteLinearOrder β] {f : Filter α} {u v : α → β}
    (h₁ : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h₂ : f.IsCoboundedUnder (· ≤ ·) v := by isBoundedDefault)
    (h₃ : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
    (h₄ : f.IsBoundedUnder (· ≤ ·) v := by isBoundedDefault) :
    limsup (fun a ↦ max (u a) (v a)) f = max (limsup u f) (limsup v f) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u v : α → β} (h₁ : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (h₂ : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f v := by isBoundedDefault) (h₃ : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u := by isBoundedDefault) (h₄ : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f v := by isBoundedDefault)
Goal: Filter.limsup (fun a => max (u a) (v a)) f = max (Filter.limsup u f) (Filter.limsup v f)
-- liminf_min
theorem liminf_min [ConditionallyCompleteLinearOrder β] {f : Filter α} {u v : α → β}
    (h₁ : f.IsCoboundedUnder (· ≥ ·) u := by isBoundedDefault)
    (h₂ : f.IsCoboundedUnder (· ≥ ·) v := by isBoundedDefault)
    (h₃ : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault)
    (h₄ : f.IsBoundedUnder (· ≥ ·) v := by isBoundedDefault) :
    liminf (fun a ↦ min (u a) (v a)) f = min (liminf u f) (liminf v f) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u v : α → β} (h₁ : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (h₂ : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f v := by isBoundedDefault) (h₃ : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault) (h₄ : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f v := by isBoundedDefault)
Goal: Filter.liminf (fun a => min (u a) (v a)) f = min (Filter.liminf u f) (Filter.liminf v f)
-- isBoundedUnder_le_finset_sup'
theorem isBoundedUnder_le_finset_sup' [LinearOrder β] [Nonempty β] {f : Filter α} {F : ι → α → β}
    {s : Finset ι} (hs : s.Nonempty) (h : ∀ i ∈ s, f.IsBoundedUnder (· ≤ ·) (F i)) :
    f.IsBoundedUnder (· ≤ ·) (fun a ↦ sup' s hs (fun i ↦ F i a)) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [LinearOrder β] [Nonempty β] {f : Filter α} {F : ι → α → β} {s : Finset ι} (hs : s.Nonempty) (h : ∀ i ∈ s, Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f (F i))
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun a => s.sup' hs fun i => F i a
-- isCoboundedUnder_le_finset_sup'
theorem isCoboundedUnder_le_finset_sup' [LinearOrder β] {f : Filter α} {F : ι → α → β}
    {s : Finset ι} (hs : s.Nonempty) (h : ∃ i ∈ s, f.IsCoboundedUnder (· ≤ ·) (F i)) :
    f.IsCoboundedUnder (· ≤ ·) (fun a ↦ sup' s hs (fun i ↦ F i a)) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [LinearOrder β] {f : Filter α} {F : ι → α → β} {s : Finset ι} (hs : s.Nonempty) (h : ∃ i ∈ s, Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f (F i))
Goal: Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun a => s.sup' hs fun i => F i a
-- isBoundedUnder_le_finset_sup
theorem isBoundedUnder_le_finset_sup [LinearOrder β] [OrderBot β] {f : Filter α} {F : ι → α → β}
    {s : Finset ι} (h : ∀ i ∈ s, f.IsBoundedUnder (· ≤ ·) (F i)) :
    f.IsBoundedUnder (· ≤ ·) (fun a ↦ sup s (fun i ↦ F i a)) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [LinearOrder β] [OrderBot β] {f : Filter α} {F : ι → α → β} {s : Finset ι} (h : ∀ i ∈ s, Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f (F i))
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun a => s.sup fun i => F i a
-- isBoundedUnder_ge_finset_inf'
theorem isBoundedUnder_ge_finset_inf' [LinearOrder β] [Nonempty β] {f : Filter α} {F : ι → α → β}
    {s : Finset ι} (hs : s.Nonempty) (h : ∀ i ∈ s, f.IsBoundedUnder (· ≥ ·) (F i)) :
    f.IsBoundedUnder (· ≥ ·) (fun a ↦ inf' s hs (fun i ↦ F i a)) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [LinearOrder β] [Nonempty β] {f : Filter α} {F : ι → α → β} {s : Finset ι} (hs : s.Nonempty) (h : ∀ i ∈ s, Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f (F i))
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f fun a => s.inf' hs fun i => F i a
-- isCoboundedUnder_ge_finset_inf'
theorem isCoboundedUnder_ge_finset_inf' [LinearOrder β] {f : Filter α} {F : ι → α → β}
    {s : Finset ι} (hs : s.Nonempty) (h : ∃ i ∈ s, f.IsCoboundedUnder (· ≥ ·) (F i)) :
    f.IsCoboundedUnder (· ≥ ·) (fun a ↦ inf' s hs (fun i ↦ F i a)) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [LinearOrder β] {f : Filter α} {F : ι → α → β} {s : Finset ι} (hs : s.Nonempty) (h : ∃ i ∈ s, Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f (F i))
Goal: Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f fun a => s.inf' hs fun i => F i a
-- isBoundedUnder_ge_finset_inf
theorem isBoundedUnder_ge_finset_inf [LinearOrder β] [OrderTop β] {f : Filter α} {F : ι → α → β}
    {s : Finset ι} (h : ∀ i ∈ s, f.IsBoundedUnder (· ≥ ·) (F i)) :
    f.IsBoundedUnder (· ≥ ·) (fun a ↦ inf s (fun i ↦ F i a)) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [LinearOrder β] [OrderTop β] {f : Filter α} {F : ι → α → β} {s : Finset ι} (h : ∀ i ∈ s, Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f (F i))
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f fun a => s.inf fun i => F i a
-- limsup_finset_sup'
theorem limsup_finset_sup' [ConditionallyCompleteLinearOrder β] {f : Filter α}
    {F : ι → α → β} {s : Finset ι} (hs : s.Nonempty)
    (h₁ : ∀ i ∈ s, f.IsCoboundedUnder (· ≤ ·) (F i) := by exact fun _ _ ↦ by isBoundedDefault)
    (h₂ : ∀ i ∈ s, f.IsBoundedUnder (· ≤ ·) (F i) := by exact fun _ _ ↦ by isBoundedDefault) :
    limsup (fun a ↦ sup' s hs (fun i ↦ F i a)) f = sup' s hs (fun i ↦ limsup (F i) f) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [ConditionallyCompleteLinearOrder β] {f : Filter α} {F : ι → α → β} {s : Finset ι} (hs : s.Nonempty) (h₁ : ∀ i ∈ s, Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f (F i) := by exact fun _ _ ↦ by isBoundedDefault) (h₂ : ∀ i ∈ s, Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f (F i) := by exact fun _ _ ↦ by isBoundedDefault)
Goal: Filter.limsup (fun a => s.sup' hs fun i => F i a) f = s.sup' hs fun i => Filter.limsup (F i) f
-- limsup_finset_sup
theorem limsup_finset_sup [ConditionallyCompleteLinearOrder β] [OrderBot β] {f : Filter α}
    {F : ι → α → β} {s : Finset ι}
    (h₁ : ∀ i ∈ s, f.IsCoboundedUnder (· ≤ ·) (F i) := by exact fun _ _ ↦ by isBoundedDefault)
    (h₂ : ∀ i ∈ s, f.IsBoundedUnder (· ≤ ·) (F i) := by exact fun _ _ ↦ by isBoundedDefault) :
    limsup (fun a ↦ sup s (fun i ↦ F i a)) f = sup s (fun i ↦ limsup (F i) f) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [ConditionallyCompleteLinearOrder β] [OrderBot β] {f : Filter α} {F : ι → α → β} {s : Finset ι} (h₁ : ∀ i ∈ s, Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) f (F i) := by exact fun _ _ ↦ by isBoundedDefault) (h₂ : ∀ i ∈ s, Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f (F i) := by exact fun _ _ ↦ by isBoundedDefault)
Goal: Filter.limsup (fun a => s.sup fun i => F i a) f = s.sup fun i => Filter.limsup (F i) f
-- liminf_finset_inf'
theorem liminf_finset_inf' [ConditionallyCompleteLinearOrder β] {f : Filter α}
    {F : ι → α → β} {s : Finset ι} (hs : s.Nonempty)
    (h₁ : ∀ i ∈ s, f.IsCoboundedUnder (· ≥ ·) (F i) := by exact fun _ _ ↦ by isBoundedDefault)
    (h₂ : ∀ i ∈ s, f.IsBoundedUnder (· ≥ ·) (F i) := by exact fun _ _ ↦ by isBoundedDefault) :
    liminf (fun a ↦ inf' s hs (fun i ↦ F i a)) f = inf' s hs (fun i ↦ liminf (F i) f) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [ConditionallyCompleteLinearOrder β] {f : Filter α} {F : ι → α → β} {s : Finset ι} (hs : s.Nonempty) (h₁ : ∀ i ∈ s, Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f (F i) := by exact fun _ _ ↦ by isBoundedDefault) (h₂ : ∀ i ∈ s, Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f (F i) := by exact fun _ _ ↦ by isBoundedDefault)
Goal: Filter.liminf (fun a => s.inf' hs fun i => F i a) f = s.inf' hs fun i => Filter.liminf (F i) f
-- liminf_finset_inf
theorem liminf_finset_inf [ConditionallyCompleteLinearOrder β] [OrderTop β] {f : Filter α}
    {F : ι → α → β} {s : Finset ι}
    (h₁ : ∀ i ∈ s, f.IsCoboundedUnder (· ≥ ·) (F i) := by exact fun _ _ ↦ by isBoundedDefault)
    (h₂ : ∀ i ∈ s, f.IsBoundedUnder (· ≥ ·) (F i) := by exact fun _ _ ↦ by isBoundedDefault) :
    liminf (fun a ↦ inf s (fun i ↦ F i a)) f = inf s (fun i ↦ liminf (F i) f) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [ConditionallyCompleteLinearOrder β] [OrderTop β] {f : Filter α} {F : ι → α → β} {s : Finset ι} (h₁ : ∀ i ∈ s, Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f (F i) := by exact fun _ _ ↦ by isBoundedDefault) (h₂ : ∀ i ∈ s, Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f (F i) := by exact fun _ _ ↦ by isBoundedDefault)
Goal: Filter.liminf (fun a => s.inf fun i => F i a) f = s.inf fun i => Filter.liminf (F i) f
-- Filter.IsCobounded.frequently_ge
lemma IsCobounded.frequently_ge [NeBot F] (cobdd : IsCobounded (· ≤ ·) F) :
    ∃ l, ∃ᶠ x in F, l ≤ x 
Arguments: {R : Type u_6} {F : Filter R} [LinearOrder R] [F.NeBot] (cobdd : Filter.IsCobounded (fun x x_1 => x ≤ x_1) F)
Goal: ∃ l, ∃ᶠ (x : R) in F, l ≤ x
-- Filter.IsCobounded.frequently_le
lemma IsCobounded.frequently_le [NeBot F] (cobdd : IsCobounded (· ≥ ·) F) :
    ∃ u, ∃ᶠ x in F, x ≤ u 
Arguments: {R : Type u_6} {F : Filter R} [LinearOrder R] [F.NeBot] (cobdd : Filter.IsCobounded (fun x x_1 => x ≥ x_1) F)
Goal: ∃ u, ∃ᶠ (x : R) in F, x ≤ u
-- Filter.IsCobounded.of_frequently_ge
lemma IsCobounded.of_frequently_ge {l : R} (freq_ge : ∃ᶠ x in F, l ≤ x) :
    IsCobounded (· ≤ ·) F 
Arguments: {R : Type u_6} {F : Filter R} [LinearOrder R] {l : R} (freq_ge : ∃ᶠ (x : R) in F, l ≤ x)
Goal: Filter.IsCobounded (fun x x_1 => x ≤ x_1) F
-- Filter.IsCobounded.of_frequently_le
lemma IsCobounded.of_frequently_le {u : R} (freq_le : ∃ᶠ r in F, r ≤ u) :
    IsCobounded (· ≥ ·) F 
Arguments: {R : Type u_6} {F : Filter R} [LinearOrder R] {u : R} (freq_le : ∃ᶠ (r : R) in F, r ≤ u)
Goal: Filter.IsCobounded (fun x x_1 => x ≥ x_1) F
-- Monotone.frequently_ge_map_of_frequently_ge
lemma Monotone.frequently_ge_map_of_frequently_ge {f : R → S} (f_incr : Monotone f)
    {l : R} (freq_ge : ∃ᶠ x in F, l ≤ x) :
    ∃ᶠ x' in F.map f, f l ≤ x' 
Arguments: {R : Type u_6} {S : Type u_7} {F : Filter R} [LinearOrder R] [LinearOrder S] {f : R → S} (f_incr : Monotone f) {l : R} (freq_ge : ∃ᶠ (x : R) in F, l ≤ x)
Goal: ∃ᶠ (x' : S) in Filter.map f F, f l ≤ x'
-- Monotone.frequently_le_map_of_frequently_le
lemma Monotone.frequently_le_map_of_frequently_le {f : R → S} (f_incr : Monotone f)
    {u : R} (freq_le : ∃ᶠ x in F, x ≤ u) :
    ∃ᶠ y in F.map f, y ≤ f u 
Arguments: {R : Type u_6} {S : Type u_7} {F : Filter R} [LinearOrder R] [LinearOrder S] {f : R → S} (f_incr : Monotone f) {u : R} (freq_le : ∃ᶠ (x : R) in F, x ≤ u)
Goal: ∃ᶠ (y : S) in Filter.map f F, y ≤ f u
-- Antitone.frequently_le_map_of_frequently_ge
lemma Antitone.frequently_le_map_of_frequently_ge {f : R → S} (f_decr : Antitone f)
    {l : R} (frbdd : ∃ᶠ x in F, l ≤ x) :
    ∃ᶠ y in F.map f, y ≤ f l 
Arguments: {R : Type u_6} {S : Type u_7} {F : Filter R} [LinearOrder R] [LinearOrder S] {f : R → S} (f_decr : Antitone f) {l : R} (frbdd : ∃ᶠ (x : R) in F, l ≤ x)
Goal: ∃ᶠ (y : S) in Filter.map f F, y ≤ f l
-- Antitone.frequently_ge_map_of_frequently_le
lemma Antitone.frequently_ge_map_of_frequently_le {f : R → S} (f_decr : Antitone f)
    {u : R} (frbdd : ∃ᶠ x in F, x ≤ u) :
    ∃ᶠ y in F.map f, f u ≤ y 
Arguments: {R : Type u_6} {S : Type u_7} {F : Filter R} [LinearOrder R] [LinearOrder S] {f : R → S} (f_decr : Antitone f) {u : R} (frbdd : ∃ᶠ (x : R) in F, x ≤ u)
Goal: ∃ᶠ (y : S) in Filter.map f F, f u ≤ y
-- Monotone.isCoboundedUnder_le_of_isCobounded
lemma Monotone.isCoboundedUnder_le_of_isCobounded {f : R → S} (f_incr : Monotone f)
    [NeBot F] (cobdd : IsCobounded (· ≤ ·) F) :
    F.IsCoboundedUnder (· ≤ ·) f 
Arguments: {R : Type u_6} {S : Type u_7} {F : Filter R} [LinearOrder R] [LinearOrder S] {f : R → S} (f_incr : Monotone f) [F.NeBot] (cobdd : Filter.IsCobounded (fun x x_1 => x ≤ x_1) F)
Goal: Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) F f
-- Monotone.isCoboundedUnder_ge_of_isCobounded
lemma Monotone.isCoboundedUnder_ge_of_isCobounded {f : R → S} (f_incr : Monotone f)
    [NeBot F] (cobdd : IsCobounded (· ≥ ·) F) :
    F.IsCoboundedUnder (· ≥ ·) f 
Arguments: {R : Type u_6} {S : Type u_7} {F : Filter R} [LinearOrder R] [LinearOrder S] {f : R → S} (f_incr : Monotone f) [F.NeBot] (cobdd : Filter.IsCobounded (fun x x_1 => x ≥ x_1) F)
Goal: Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) F f
-- Antitone.isCoboundedUnder_le_of_isCobounded
lemma Antitone.isCoboundedUnder_le_of_isCobounded {f : R → S} (f_decr : Antitone f)
    [NeBot F] (cobdd : IsCobounded (· ≥ ·) F) :
    F.IsCoboundedUnder (· ≤ ·) f 
Arguments: {R : Type u_6} {S : Type u_7} {F : Filter R} [LinearOrder R] [LinearOrder S] {f : R → S} (f_decr : Antitone f) [F.NeBot] (cobdd : Filter.IsCobounded (fun x x_1 => x ≥ x_1) F)
Goal: Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) F f
-- Antitone.isCoboundedUnder_ge_of_isCobounded
lemma Antitone.isCoboundedUnder_ge_of_isCobounded {f : R → S} (f_decr : Antitone f)
    [NeBot F] (cobdd : IsCobounded (· ≤ ·) F) :
    F.IsCoboundedUnder (· ≥ ·) f 
Arguments: {R : Type u_6} {S : Type u_7} {F : Filter R} [LinearOrder R] [LinearOrder S] {f : R → S} (f_decr : Antitone f) [F.NeBot] (cobdd : Filter.IsCobounded (fun x x_1 => x ≤ x_1) F)
Goal: Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) F f
-- Order.Frame.MinimalAxioms.inf_sSup_le_iSup_inf
class Order.Frame.MinimalAxioms (α : Type u) extends CompleteLattice α where
  inf_sSup_le_iSup_inf (a : α) (s : Set α) : a ⊓ sSup s ≤ ⨆ b ∈ s, a ⊓ b
Arguments: {α : Type u} [self : Order.Frame.MinimalAxioms α] (a : α) (s : Set α)
Goal: a ⊓ sSup s ≤ ⨆ b ∈ s, a ⊓ b
-- Order.Coframe.MinimalAxioms.iInf_sup_le_sup_sInf
class Order.Coframe.MinimalAxioms (α : Type u) extends CompleteLattice α where
  iInf_sup_le_sup_sInf (a : α) (s : Set α) : ⨅ b ∈ s, a ⊔ b ≤ a ⊔ sInf s
Arguments: {α : Type u} [self : Order.Coframe.MinimalAxioms α] (a : α) (s : Set α)
Goal: ⨅ b ∈ s, a ⊔ b ≤ a ⊔ sInf s
-- Order.Frame.inf_sSup_le_iSup_inf
class Order.Frame (α : Type*) extends CompleteLattice α, HeytingAlgebra α where
    inf_sSup_le_iSup_inf (a : α) (s : Set α) : a ⊓ sSup s ≤ ⨆ b ∈ s, a ⊓ b
Arguments: {α : Type u_1} [self : Order.Frame α] (a : α) (s : Set α)
Goal: a ⊓ sSup s ≤ ⨆ b ∈ s, a ⊓ b
-- Order.Coframe.iInf_sup_le_sup_sInf
class Order.Coframe (α : Type*) extends CompleteLattice α, CoheytingAlgebra α where
    iInf_sup_le_sup_sInf (a : α) (s : Set α) : ⨅ b ∈ s, a ⊔ b ≤ a ⊔ sInf s
Arguments: {α : Type u_1} [self : Order.Coframe α] (a : α) (s : Set α)
Goal: ⨅ b ∈ s, a ⊔ b ≤ a ⊔ sInf s
-- CompletelyDistribLattice.MinimalAxioms.iInf_iSup_eq
structure CompletelyDistribLattice.MinimalAxioms (α : Type u) extends CompleteLattice α where
  protected iInf_iSup_eq {ι : Type u} {κ : ι → Type u} (f : ∀ a, κ a → α) :
    (⨅ a, ⨆ b, f a b) = ⨆ g : ∀ a, κ a, ⨅ a, f a (g a)
Arguments: {α : Type u} (self : CompletelyDistribLattice.MinimalAxioms α) {ι : Type u} {κ : ι → Type u} (f : (a : ι) → κ a → α)
Goal: ⨅ a, ⨆ b, f a b = ⨆ g, ⨅ a, f a (g a)