ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finpartition.biUnion_filter_atomise
theorem biUnion_filter_atomise (ht : t ∈ F) (hts : t ⊆ s) :
    ((atomise s F).parts.filter fun u ↦ u ⊆ t ∧ u.Nonempty).biUnion id = t 
Arguments: {α : Type u_1} [DecidableEq α] {s t : Finset α} [Fintype α] {F : Finset (Finset α)} (ht : t ∈ F) (hts : t ⊆ s)
Goal: (Finset.filter (fun u => u ⊆ t ∧ u.Nonempty) (Finpartition.atomise s F).parts).biUnion id = t
-- Finpartition.card_filter_atomise_le_two_pow
theorem card_filter_atomise_le_two_pow (ht : t ∈ F) :
    ((atomise s F).parts.filter fun u ↦ u ⊆ t ∧ u.Nonempty).card ≤ 2 ^ (F.card - 1) 
Arguments: {α : Type u_1} [DecidableEq α] {s t : Finset α} [Fintype α] {F : Finset (Finset α)} (ht : t ∈ F)
Goal: (Finset.filter (fun u => u ⊆ t ∧ u.Nonempty) (Finpartition.atomise s F).parts).card ≤ 2 ^ (F.card - 1)
-- ENNReal.eventually_le_limsup
theorem eventually_le_limsup [CountableInterFilter f] (u : α → ℝ≥0∞) :
    ∀ᶠ y in f, u y ≤ f.limsup u 
Arguments: {α : Type u_1} {f : Filter α} [CountableInterFilter f] (u : α → ℝ≥0∞)
Goal: ∀ᶠ (y : α) in f, u y ≤ Filter.limsup u f
-- ENNReal.limsup_eq_zero_iff
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : α → ℝ≥0∞} :
    f.limsup u = 0 ↔ u =ᶠ[f] 0 
Arguments: {α : Type u_1} {f : Filter α} [CountableInterFilter f] {u : α → ℝ≥0∞}
Goal: Filter.limsup u f = 0 ↔ u =ᶠ[f] 0
-- ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
    (f.limsup fun x : α => a * u x) = a * f.limsup u 
Arguments: {α : Type u_1} {f : Filter α} {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤)
Goal: Filter.limsup (fun x => a * u x) f = a * Filter.limsup u f
-- ENNReal.limsup_const_mul
theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
    f.limsup (a * u ·) = a * f.limsup u 
Arguments: {α : Type u_1} {f : Filter α} [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞}
Goal: Filter.limsup (fun x => a * u x) f = a * Filter.limsup u f
-- ENNReal.limsup_mul_le
theorem limsup_mul_le [CountableInterFilter f] (u v : α → ℝ≥0∞) :
    f.limsup (u * v) ≤ f.limsup u * f.limsup v 
Arguments: {α : Type u_1} {f : Filter α} [CountableInterFilter f] (u v : α → ℝ≥0∞)
Goal: Filter.limsup (u * v) f ≤ Filter.limsup u f * Filter.limsup v f
-- ENNReal.limsup_add_le
theorem limsup_add_le [CountableInterFilter f] (u v : α → ℝ≥0∞) :
    f.limsup (u + v) ≤ f.limsup u + f.limsup v 
Arguments: {α : Type u_1} {f : Filter α} [CountableInterFilter f] (u v : α → ℝ≥0∞)
Goal: Filter.limsup (u + v) f ≤ Filter.limsup u f + Filter.limsup v f
-- ENNReal.limsup_liminf_le_liminf_limsup
theorem limsup_liminf_le_liminf_limsup {β} [Countable β] {f : Filter α} [CountableInterFilter f]
    {g : Filter β} (u : α → β → ℝ≥0∞) :
    (f.limsup fun a : α => g.liminf fun b : β => u a b) ≤
      g.liminf fun b => f.limsup fun a => u a b 
Arguments: {α : Type u_1} {β : Type u_2} [Countable β] {f : Filter α} [CountableInterFilter f] {g : Filter β} (u : α → β → ℝ≥0∞)
Goal: Filter.limsup (fun a => Filter.liminf (fun b => u a b) g) f ≤
  Filter.liminf (fun b => Filter.limsup (fun a => u a b) f) g
-- Ultrafilter.neBot'
structure Ultrafilter (α : Type*) extends Filter α where
    protected neBot' : NeBot toFilter
    protected le_of_le : ∀ g, Filter.NeBot g → g ≤ toFilter → toFilter ≤ g
Arguments: {α : Type u_2} (self : Ultrafilter α)
Goal: (↑self).NeBot
-- Ultrafilter.le_of_le
structure Ultrafilter (α : Type*) extends Filter α where
    protected neBot' : NeBot toFilter
    protected le_of_le : ∀ g, Filter.NeBot g → g ≤ toFilter → toFilter ≤ g
Arguments: {α : Type u_2} (self : Ultrafilter α) (g : Filter α)
Goal: g.NeBot → g ≤ ↑self → ↑self ≤ g
-- Ultrafilter.unique
theorem unique (f : Ultrafilter α) {g : Filter α} (h : g ≤ f) (hne : NeBot g := by infer_instance) :
    g = f 
Arguments: {α : Type u} (f : Ultrafilter α) {g : Filter α} (h : g ≤ ↑f) (hne : g.NeBot := by infer_instance)
Goal: g = ↑f
-- Ultrafilter.isAtom
protected theorem isAtom (f : Ultrafilter α) : IsAtom (f : Filter α) 
Arguments: {α : Type u} (f : Ultrafilter α)
Goal: IsAtom ↑f
-- Ultrafilter.mem_coe
@[simp, norm_cast]
theorem mem_coe : s ∈ (f : Filter α) ↔ s ∈ f 
Arguments: {α : Type u} {f : Ultrafilter α} {s : Set α}
Goal: s ∈ ↑f ↔ s ∈ f
-- Ultrafilter.coe_injective
theorem coe_injective : Injective ((↑) : Ultrafilter α → Filter α)
  
Arguments: {α : Type u}
Goal: Function.Injective Ultrafilter.toFilter
-- Ultrafilter.eq_of_le
theorem eq_of_le {f g : Ultrafilter α} (h : (f : Filter α) ≤ g) : f = g 
Arguments: {α : Type u} {f g : Ultrafilter α} (h : ↑f ≤ ↑g)
Goal: f = g
-- Ultrafilter.coe_le_coe
@[simp, norm_cast]
theorem coe_le_coe {f g : Ultrafilter α} : (f : Filter α) ≤ g ↔ f = g 
Arguments: {α : Type u} {f g : Ultrafilter α}
Goal: ↑f ≤ ↑g ↔ f = g
-- Ultrafilter.coe_inj
@[simp, norm_cast]
theorem coe_inj : (f : Filter α) = g ↔ f = g 
Arguments: {α : Type u} {f g : Ultrafilter α}
Goal: ↑f = ↑g ↔ f = g
-- Ultrafilter.ext
@[ext]
theorem ext ⦃f g : Ultrafilter α⦄ (h : ∀ s, s ∈ f ↔ s ∈ g) : f = g 
Arguments: {α : Type u} ⦃f g : Ultrafilter α⦄ (h : ∀ (s : Set α), s ∈ f ↔ s ∈ g)
Goal: f = g
-- Ultrafilter.le_of_inf_neBot
theorem le_of_inf_neBot (f : Ultrafilter α) {g : Filter α} (hg : NeBot (↑f ⊓ g)) : ↑f ≤ g 
Arguments: {α : Type u} (f : Ultrafilter α) {g : Filter α} (hg : (↑f ⊓ g).NeBot)
Goal: ↑f ≤ g
-- Ultrafilter.le_of_inf_neBot'
theorem le_of_inf_neBot' (f : Ultrafilter α) {g : Filter α} (hg : NeBot (g ⊓ f)) : ↑f ≤ g 
Arguments: {α : Type u} (f : Ultrafilter α) {g : Filter α} (hg : (g ⊓ ↑f).NeBot)
Goal: ↑f ≤ g
-- Ultrafilter.inf_neBot_iff
theorem inf_neBot_iff {f : Ultrafilter α} {g : Filter α} : NeBot (↑f ⊓ g) ↔ ↑f ≤ g 
Arguments: {α : Type u} {f : Ultrafilter α} {g : Filter α}
Goal: (↑f ⊓ g).NeBot ↔ ↑f ≤ g
-- Ultrafilter.disjoint_iff_not_le
theorem disjoint_iff_not_le {f : Ultrafilter α} {g : Filter α} : Disjoint (↑f) g ↔ ¬↑f ≤ g 
Arguments: {α : Type u} {f : Ultrafilter α} {g : Filter α}
Goal: Disjoint (↑f) g ↔ ¬↑f ≤ g
-- Ultrafilter.compl_not_mem_iff
@[simp]
theorem compl_not_mem_iff : sᶜ ∉ f ↔ s ∈ f 
Arguments: {α : Type u} {f : Ultrafilter α} {s : Set α}
Goal: sᶜ ∉ f ↔ s ∈ f
-- Ultrafilter.frequently_iff_eventually
@[simp]
theorem frequently_iff_eventually : (∃ᶠ x in f, p x) ↔ ∀ᶠ x in f, p x 
Arguments: {α : Type u} {f : Ultrafilter α} {p : α → Prop}
Goal: (∃ᶠ (x : α) in ↑f, p x) ↔ ∀ᶠ (x : α) in ↑f, p x
-- Ultrafilter.compl_mem_iff_not_mem
theorem compl_mem_iff_not_mem : sᶜ ∈ f ↔ s ∉ f 
Arguments: {α : Type u} {f : Ultrafilter α} {s : Set α}
Goal: sᶜ ∈ f ↔ s ∉ f
-- Ultrafilter.diff_mem_iff
theorem diff_mem_iff (f : Ultrafilter α) : s \ t ∈ f ↔ s ∈ f ∧ t ∉ f 
Arguments: {α : Type u} {s t : Set α} (f : Ultrafilter α)
Goal: s \ t ∈ f ↔ s ∈ f ∧ t ∉ f
-- Ultrafilter.nonempty_of_mem
theorem nonempty_of_mem (hs : s ∈ f) : s.Nonempty 
Arguments: {α : Type u} {f : Ultrafilter α} {s : Set α} (hs : s ∈ f)
Goal: s.Nonempty
-- Ultrafilter.ne_empty_of_mem
theorem ne_empty_of_mem (hs : s ∈ f) : s ≠ ∅ 
Arguments: {α : Type u} {f : Ultrafilter α} {s : Set α} (hs : s ∈ f)
Goal: s ≠ ∅
-- Ultrafilter.empty_not_mem
@[simp]
theorem empty_not_mem : ∅ ∉ f 
Arguments: {α : Type u} {f : Ultrafilter α}
Goal: ∅ ∉ f
-- Ultrafilter.le_sup_iff
@[simp]
theorem le_sup_iff {u : Ultrafilter α} {f g : Filter α} : ↑u ≤ f ⊔ g ↔ ↑u ≤ f ∨ ↑u ≤ g 
Arguments: {α : Type u} {u : Ultrafilter α} {f g : Filter α}
Goal: ↑u ≤ f ⊔ g ↔ ↑u ≤ f ∨ ↑u ≤ g
-- Ultrafilter.union_mem_iff
@[simp]
theorem union_mem_iff : s ∪ t ∈ f ↔ s ∈ f ∨ t ∈ f 
Arguments: {α : Type u} {f : Ultrafilter α} {s t : Set α}
Goal: s ∪ t ∈ f ↔ s ∈ f ∨ t ∈ f
-- Ultrafilter.mem_or_compl_mem
theorem mem_or_compl_mem (f : Ultrafilter α) (s : Set α) : s ∈ f ∨ sᶜ ∈ f 
Arguments: {α : Type u} (f : Ultrafilter α) (s : Set α)
Goal: s ∈ f ∨ sᶜ ∈ f
-- Ultrafilter.em
protected theorem em (f : Ultrafilter α) (p : α → Prop) : (∀ᶠ x in f, p x) ∨ ∀ᶠ x in f, ¬p x 
Arguments: {α : Type u} (f : Ultrafilter α) (p : α → Prop)
Goal: (∀ᶠ (x : α) in ↑f, p x) ∨ ∀ᶠ (x : α) in ↑f, ¬p x
-- Ultrafilter.eventually_or
theorem eventually_or : (∀ᶠ x in f, p x ∨ q x) ↔ (∀ᶠ x in f, p x) ∨ ∀ᶠ x in f, q x 
Arguments: {α : Type u} {f : Ultrafilter α} {p q : α → Prop}
Goal: (∀ᶠ (x : α) in ↑f, p x ∨ q x) ↔ (∀ᶠ (x : α) in ↑f, p x) ∨ ∀ᶠ (x : α) in ↑f, q x
-- Ultrafilter.eventually_not
theorem eventually_not : (∀ᶠ x in f, ¬p x) ↔ ¬∀ᶠ x in f, p x 
Arguments: {α : Type u} {f : Ultrafilter α} {p : α → Prop}
Goal: (∀ᶠ (x : α) in ↑f, ¬p x) ↔ ¬∀ᶠ (x : α) in ↑f, p x
-- Ultrafilter.eventually_imp
theorem eventually_imp : (∀ᶠ x in f, p x → q x) ↔ (∀ᶠ x in f, p x) → ∀ᶠ x in f, q x 
Arguments: {α : Type u} {f : Ultrafilter α} {p q : α → Prop}
Goal: (∀ᶠ (x : α) in ↑f, p x → q x) ↔ (∀ᶠ (x : α) in ↑f, p x) → ∀ᶠ (x : α) in ↑f, q x
-- Ultrafilter.finite_sUnion_mem_iff
theorem finite_sUnion_mem_iff {s : Set (Set α)} (hs : s.Finite) : ⋃₀ s ∈ f ↔ ∃ t ∈ s, t ∈ f 
Arguments: {α : Type u} {f : Ultrafilter α} {s : Set (Set α)} (hs : s.Finite)
Goal: ⋃₀ s ∈ f ↔ ∃ t ∈ s, t ∈ f
-- Ultrafilter.finite_biUnion_mem_iff
theorem finite_biUnion_mem_iff {is : Set β} {s : β → Set α} (his : is.Finite) :
    (⋃ i ∈ is, s i) ∈ f ↔ ∃ i ∈ is, s i ∈ f 
Arguments: {α : Type u} {β : Type v} {f : Ultrafilter α} {is : Set β} {s : β → Set α} (his : is.Finite)
Goal: ⋃ i ∈ is, s i ∈ f ↔ ∃ i ∈ is, s i ∈ f
-- Ultrafilter.coe_map
@[simp, norm_cast]
theorem coe_map (m : α → β) (f : Ultrafilter α) : (map m f : Filter β) = Filter.map m ↑f 
Arguments: {α : Type u} {β : Type v} (m : α → β) (f : Ultrafilter α)
Goal: ↑(Ultrafilter.map m f) = Filter.map m ↑f
-- Ultrafilter.mem_map
@[simp]
theorem mem_map {m : α → β} {f : Ultrafilter α} {s : Set β} : s ∈ map m f ↔ m ⁻¹' s ∈ f 
Arguments: {α : Type u} {β : Type v} {m : α → β} {f : Ultrafilter α} {s : Set β}
Goal: s ∈ Ultrafilter.map m f ↔ m ⁻¹' s ∈ f
-- Ultrafilter.map_id
@[simp]
nonrec theorem map_id (f : Ultrafilter α) : f.map id = f 
Arguments: {α : Type u} (f : Ultrafilter α)
Goal: Ultrafilter.map id f = f
-- Ultrafilter.map_id'
@[simp]
theorem map_id' (f : Ultrafilter α) : (f.map fun x => x) = f 
Arguments: {α : Type u} (f : Ultrafilter α)
Goal: Ultrafilter.map (fun x => x) f = f
-- Ultrafilter.map_map
@[simp]
nonrec theorem map_map (f : Ultrafilter α) (m : α → β) (n : β → γ) :
  (f.map m).map n = f.map (n ∘ m) 
Arguments: {α : Type u} {β : Type v} {γ : Type u_1} (f : Ultrafilter α) (m : α → β) (n : β → γ)
Goal: Ultrafilter.map n (Ultrafilter.map m f) = Ultrafilter.map (n ∘ m) f
-- Ultrafilter.mem_comap
@[simp]
theorem mem_comap {m : α → β} (u : Ultrafilter β) (inj : Injective m) (large : Set.range m ∈ u)
    {s : Set α} : s ∈ u.comap inj large ↔ m '' s ∈ u 
Arguments: {α : Type u} {β : Type v} {m : α → β} (u : Ultrafilter β) (inj : Function.Injective m) (large : Set.range m ∈ u) {s : Set α}
Goal: s ∈ u.comap inj large ↔ m '' s ∈ u
-- Ultrafilter.coe_comap
@[simp, norm_cast]
theorem coe_comap {m : α → β} (u : Ultrafilter β) (inj : Injective m) (large : Set.range m ∈ u) :
    (u.comap inj large : Filter α) = Filter.comap m u 
Arguments: {α : Type u} {β : Type v} {m : α → β} (u : Ultrafilter β) (inj : Function.Injective m) (large : Set.range m ∈ u)
Goal: ↑(u.comap inj large) = Filter.comap m ↑u
-- Ultrafilter.comap_id
@[simp]
nonrec theorem comap_id (f : Ultrafilter α) (h₀ : Injective (id : α → α) := injective_id)
    (h₁ : range id ∈ f := (by rw [range_id]; exact univ_mem)) :
    f.comap h₀ h₁ = f 
Arguments: {α : Type u} (f : Ultrafilter α) (h₀ : Function.Injective id := ⋯) (h₁ : Set.range id ∈ f := ⋯)
Goal: f.comap h₀ h₁ = f
-- Ultrafilter.comap_comap
@[simp]
nonrec theorem comap_comap (f : Ultrafilter γ) {m : α → β} {n : β → γ} (inj₀ : Injective n)
    (large₀ : range n ∈ f) (inj₁ : Injective m) (large₁ : range m ∈ f.comap inj₀ large₀)
    (inj₂ : Injective (n ∘ m) := inj₀.comp inj₁)
    (large₂ : range (n ∘ m) ∈ f :=
      (by rw [range_comp]; exact image_mem_of_mem_comap large₀ large₁)) :
    (f.comap inj₀ large₀).comap inj₁ large₁ = f.comap inj₂ large₂ 
Arguments: {α : Type u} {β : Type v} {γ : Type u_1} (f : Ultrafilter γ) {m : α → β} {n : β → γ} (inj₀ : Function.Injective n) (large₀ : Set.range n ∈ f) (inj₁ : Function.Injective m) (large₁ : Set.range m ∈ f.comap inj₀ large₀) (inj₂ : Function.Injective (n ∘ m) := ⋯) (large₂ : Set.range (n ∘ m) ∈ f := ⋯)
Goal: (f.comap inj₀ large₀).comap inj₁ large₁ = f.comap inj₂ large₂
-- Ultrafilter.mem_pure
@[simp]
theorem mem_pure {a : α} {s : Set α} : s ∈ (pure a : Ultrafilter α) ↔ a ∈ s 
Arguments: {α : Type u} {a : α} {s : Set α}
Goal: s ∈ pure a ↔ a ∈ s
-- Ultrafilter.coe_pure
@[simp]
theorem coe_pure (a : α) : ↑(pure a : Ultrafilter α) = (pure a : Filter α) 
Arguments: {α : Type u} (a : α)
Goal: ↑(pure a) = pure a
-- Ultrafilter.map_pure
@[simp]
theorem map_pure (m : α → β) (a : α) : map m (pure a) = pure (m a) 
Arguments: {α : Type u} {β : Type v} (m : α → β) (a : α)
Goal: Ultrafilter.map m (pure a) = pure (m a)
-- Ultrafilter.comap_pure
@[simp]
theorem comap_pure {m : α → β} (a : α) (inj : Injective m) (large) :
    comap (pure <| m a) inj large = pure a 
Arguments: {α : Type u} {β : Type v} {m : α → β} (a : α) (inj : Function.Injective m) (large : Set.range m ∈ pure (m a))
Goal: (pure (m a)).comap inj large = pure a
-- Ultrafilter.pure_injective
theorem pure_injective : Injective (pure : α → Ultrafilter α) 
Arguments: {α : Type u}
Goal: Function.Injective pure
-- Ultrafilter.eq_pure_of_finite_mem
theorem eq_pure_of_finite_mem (h : s.Finite) (h' : s ∈ f) : ∃ x ∈ s, f = pure x 
Arguments: {α : Type u} {f : Ultrafilter α} {s : Set α} (h : s.Finite) (h' : s ∈ f)
Goal: ∃ x ∈ s, f = pure x
-- Ultrafilter.eq_pure_of_finite
theorem eq_pure_of_finite [Finite α] (f : Ultrafilter α) : ∃ a, f = pure a 
Arguments: {α : Type u} [Finite α] (f : Ultrafilter α)
Goal: ∃ a, f = pure a
-- Ultrafilter.le_cofinite_or_eq_pure
theorem le_cofinite_or_eq_pure (f : Ultrafilter α) : (f : Filter α) ≤ cofinite ∨ ∃ a, f = pure a 
Arguments: {α : Type u} (f : Ultrafilter α)
Goal: ↑f ≤ Filter.cofinite ∨ ∃ a, f = pure a
-- Ultrafilter.exists_le
theorem exists_le (f : Filter α) [h : NeBot f] : ∃ u : Ultrafilter α, ↑u ≤ f 
Arguments: {α : Type u} (f : Filter α) [h : f.NeBot]
Goal: ∃ u, ↑u ≤ f
-- Ultrafilter.of_le
theorem of_le (f : Filter α) [NeBot f] : ↑(of f) ≤ f 
Arguments: {α : Type u} (f : Filter α) [f.NeBot]
Goal: ↑(Ultrafilter.of f) ≤ f
-- Ultrafilter.of_coe
theorem of_coe (f : Ultrafilter α) : of ↑f = f 
Arguments: {α : Type u} (f : Ultrafilter α)
Goal: Ultrafilter.of ↑f = f
-- Ultrafilter.exists_ultrafilter_of_finite_inter_nonempty
theorem exists_ultrafilter_of_finite_inter_nonempty (S : Set (Set α))
    (cond : ∀ T : Finset (Set α), (↑T : Set (Set α)) ⊆ S → (⋂₀ (↑T : Set (Set α))).Nonempty) :
    ∃ F : Ultrafilter α, S ⊆ F.sets 
Arguments: {α : Type u} (S : Set (Set α)) (cond : ∀ (T : Finset (Set α)), ↑T ⊆ S → (⋂₀ ↑T).Nonempty)
Goal: ∃ F, S ⊆ (↑F).sets
-- Filter.isAtom_pure
theorem isAtom_pure : IsAtom (pure a : Filter α) 
Arguments: {α : Type u} {a : α}
Goal: IsAtom (pure a)
-- Filter.NeBot.le_pure_iff
protected theorem NeBot.le_pure_iff (hf : f.NeBot) : f ≤ pure a ↔ f = pure a 
Arguments: {α : Type u} {f : Filter α} {a : α} (hf : f.NeBot)
Goal: f ≤ pure a ↔ f = pure a
-- Filter.NeBot.eq_pure_iff
protected theorem NeBot.eq_pure_iff (hf : f.NeBot) {x : α} :
    f = pure x ↔ {x} ∈ f 
Arguments: {α : Type u} {f : Filter α} (hf : f.NeBot) {x : α}
Goal: f = pure x ↔ {x} ∈ f
-- Filter.atTop_eq_pure_of_isTop
lemma atTop_eq_pure_of_isTop [LinearOrder α] {x : α} (hx : IsTop x) :
    (atTop : Filter α) = pure x 
Arguments: {α : Type u} [LinearOrder α] {x : α} (hx : IsTop x)
Goal: Filter.atTop = pure x
-- Filter.atBot_eq_pure_of_isBot
lemma atBot_eq_pure_of_isBot [LinearOrder α] {x : α} (hx : IsBot x) :
    (atBot : Filter α) = pure x 
Arguments: {α : Type u} [LinearOrder α] {x : α} (hx : IsBot x)
Goal: Filter.atBot = pure x
-- Filter.lt_pure_iff
@[simp]
theorem lt_pure_iff : f < pure a ↔ f = ⊥ 
Arguments: {α : Type u} {f : Filter α} {a : α}
Goal: f < pure a ↔ f = ⊥
-- Filter.le_pure_iff'
theorem le_pure_iff' : f ≤ pure a ↔ f = ⊥ ∨ f = pure a 
Arguments: {α : Type u} {f : Filter α} {a : α}
Goal: f ≤ pure a ↔ f = ⊥ ∨ f = pure a
-- Filter.Iic_pure
@[simp]
theorem Iic_pure (a : α) : Iic (pure a : Filter α) = {⊥, pure a} 
Arguments: {α : Type u} (a : α)
Goal: Set.Iic (pure a) = {⊥, pure a}
-- Filter.mem_iff_ultrafilter
theorem mem_iff_ultrafilter : s ∈ f ↔ ∀ g : Ultrafilter α, ↑g ≤ f → s ∈ g 
Arguments: {α : Type u} {f : Filter α} {s : Set α}
Goal: s ∈ f ↔ ∀ (g : Ultrafilter α), ↑g ≤ f → s ∈ g
-- Filter.le_iff_ultrafilter
theorem le_iff_ultrafilter {f₁ f₂ : Filter α} : f₁ ≤ f₂ ↔ ∀ g : Ultrafilter α, ↑g ≤ f₁ → ↑g ≤ f₂ 
Arguments: {α : Type u} {f₁ f₂ : Filter α}
Goal: f₁ ≤ f₂ ↔ ∀ (g : Ultrafilter α), ↑g ≤ f₁ → ↑g ≤ f₂
-- Filter.iSup_ultrafilter_le_eq
theorem iSup_ultrafilter_le_eq (f : Filter α) :
    ⨆ (g : Ultrafilter α) (_ : g ≤ f), (g : Filter α) = f 
Arguments: {α : Type u} (f : Filter α)
Goal: ⨆ g, ⨆ (_ : ↑g ≤ f), ↑g = f
-- Filter.tendsto_iff_ultrafilter
theorem tendsto_iff_ultrafilter (f : α → β) (l₁ : Filter α) (l₂ : Filter β) :
    Tendsto f l₁ l₂ ↔ ∀ g : Ultrafilter α, ↑g ≤ l₁ → Tendsto f g l₂ 
Arguments: {α : Type u} {β : Type v} (f : α → β) (l₁ : Filter α) (l₂ : Filter β)
Goal: Filter.Tendsto f l₁ l₂ ↔ ∀ (g : Ultrafilter α), ↑g ≤ l₁ → Filter.Tendsto f (↑g) l₂
-- Filter.exists_ultrafilter_iff
theorem exists_ultrafilter_iff {f : Filter α} : (∃ u : Ultrafilter α, ↑u ≤ f) ↔ NeBot f 
Arguments: {α : Type u} {f : Filter α}
Goal: (∃ u, ↑u ≤ f) ↔ f.NeBot
-- Filter.forall_neBot_le_iff
theorem forall_neBot_le_iff {g : Filter α} {p : Filter α → Prop} (hp : Monotone p) :
    (∀ f : Filter α, NeBot f → f ≤ g → p f) ↔ ∀ f : Ultrafilter α, ↑f ≤ g → p f 
Arguments: {α : Type u} {g : Filter α} {p : Filter α → Prop} (hp : Monotone p)
Goal: (∀ (f : Filter α), f.NeBot → f ≤ g → p f) ↔ ∀ (f : Ultrafilter α), ↑f ≤ g → p ↑f
-- Filter.hyperfilter_le_cofinite
theorem hyperfilter_le_cofinite : ↑(hyperfilter α) ≤ @cofinite α 
Arguments: {α : Type u} [Infinite α]
Goal: ↑(Filter.hyperfilter α) ≤ Filter.cofinite
-- Nat.hyperfilter_le_atTop
theorem _root_.Nat.hyperfilter_le_atTop : (hyperfilter ℕ).toFilter ≤ atTop 
Goal: ↑(Filter.hyperfilter ℕ) ≤ Filter.atTop
-- Filter.bot_ne_hyperfilter
@[simp]
theorem bot_ne_hyperfilter : (⊥ : Filter α) ≠ hyperfilter α 
Arguments: {α : Type u} [Infinite α]
Goal: ⊥ ≠ ↑(Filter.hyperfilter α)
-- Filter.nmem_hyperfilter_of_finite
theorem nmem_hyperfilter_of_finite {s : Set α} (hf : s.Finite) : s ∉ hyperfilter α 
Arguments: {α : Type u} [Infinite α] {s : Set α} (hf : s.Finite)
Goal: s ∉ Filter.hyperfilter α
-- Filter.compl_mem_hyperfilter_of_finite
theorem compl_mem_hyperfilter_of_finite {s : Set α} (hf : Set.Finite s) : sᶜ ∈ hyperfilter α 
Arguments: {α : Type u} [Infinite α] {s : Set α} (hf : s.Finite)
Goal: sᶜ ∈ Filter.hyperfilter α
-- Filter.mem_hyperfilter_of_finite_compl
theorem mem_hyperfilter_of_finite_compl {s : Set α} (hf : Set.Finite sᶜ) : s ∈ hyperfilter α 
Arguments: {α : Type u} [Infinite α] {s : Set α} (hf : sᶜ.Finite)
Goal: s ∈ Filter.hyperfilter α
-- Ultrafilter.comap_inf_principal_neBot_of_image_mem
theorem comap_inf_principal_neBot_of_image_mem (h : m '' s ∈ g) : (Filter.comap m g ⊓ 𝓟 s).NeBot 
Arguments: {α : Type u} {β : Type v} {m : α → β} {s : Set α} {g : Ultrafilter β} (h : m '' s ∈ g)
Goal: (Filter.comap m ↑g ⊓ 𝓟 s).NeBot
-- Ultrafilter.ofComapInfPrincipal_mem
theorem ofComapInfPrincipal_mem (h : m '' s ∈ g) : s ∈ ofComapInfPrincipal h 
Arguments: {α : Type u} {β : Type v} {m : α → β} {s : Set α} {g : Ultrafilter β} (h : m '' s ∈ g)
Goal: s ∈ Ultrafilter.ofComapInfPrincipal h
-- Ultrafilter.ofComapInfPrincipal_eq_of_map
theorem ofComapInfPrincipal_eq_of_map (h : m '' s ∈ g) : (ofComapInfPrincipal h).map m = g 
Arguments: {α : Type u} {β : Type v} {m : α → β} {s : Set α} {g : Ultrafilter β} (h : m '' s ∈ g)
Goal: Ultrafilter.map m (Ultrafilter.ofComapInfPrincipal h) = g
-- GeneralizedBooleanAlgebra.inf_inf_sdiff
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
    sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
    inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
Arguments: {α : Type u} [self : GeneralizedBooleanAlgebra α] (a b : α)
Goal: a ⊓ b ⊓ a \ b = ⊥
-- GeneralizedBooleanAlgebra.sup_inf_sdiff
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
    sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
    inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
Arguments: {α : Type u} [self : GeneralizedBooleanAlgebra α] (a b : α)
Goal: a ⊓ b ⊔ a \ b = a
-- sup_inf_sdiff
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x 
Arguments: {α : Type u} [GeneralizedBooleanAlgebra α] (x y : α)
Goal: x ⊓ y ⊔ x \ y = x
-- inf_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ 
Arguments: {α : Type u} [GeneralizedBooleanAlgebra α] (x y : α)
Goal: x ⊓ y ⊓ x \ y = ⊥
-- sup_sdiff_inf
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x 
Arguments: {α : Type u} [GeneralizedBooleanAlgebra α] (x y : α)
Goal: x \ y ⊔ x ⊓ y = x
-- inf_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ 
Arguments: {α : Type u} [GeneralizedBooleanAlgebra α] (x y : α)
Goal: x \ y ⊓ (x ⊓ y) = ⊥
-- disjoint_inf_sdiff
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: Disjoint (x ⊓ y) (x \ y)
-- sdiff_unique
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥)
Goal: x \ y = z
-- _private.Mathlib.Order.BooleanAlgebra.0.sdiff_le'
private theorem sdiff_le' : x \ y ≤ x 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x \ y ≤ x
-- _private.Mathlib.Order.BooleanAlgebra.0.sdiff_sup_self'
private theorem sdiff_sup_self' : y \ x ⊔ x = y ⊔ x 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: y \ x ⊔ x = y ⊔ x
-- sdiff_inf_sdiff
@[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x \ y ⊓ y \ x = ⊥
-- disjoint_sdiff_sdiff
theorem disjoint_sdiff_sdiff : Disjoint (x \ y) (y \ x) 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: Disjoint (x \ y) (y \ x)
-- inf_sdiff_self_right
@[simp]
theorem inf_sdiff_self_right : x ⊓ y \ x = ⊥ 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x ⊓ y \ x = ⊥
-- inf_sdiff_self_left
@[simp]
theorem inf_sdiff_self_left : y \ x ⊓ x = ⊥ 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: y \ x ⊓ x = ⊥
-- disjoint_sdiff_self_left
theorem disjoint_sdiff_self_left : Disjoint (y \ x) x 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: Disjoint (y \ x) x
-- disjoint_sdiff_self_right
theorem disjoint_sdiff_self_right : Disjoint x (y \ x) 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: Disjoint x (y \ x)
-- le_sdiff
lemma le_sdiff : x ≤ y \ z ↔ x ≤ y ∧ Disjoint x z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: x ≤ y \ z ↔ x ≤ y ∧ Disjoint x z
-- sdiff_eq_left
@[simp] lemma sdiff_eq_left : x \ y = x ↔ Disjoint x y 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x \ y = x ↔ Disjoint x y
-- Disjoint.sdiff_eq_of_sup_eq
theorem Disjoint.sdiff_eq_of_sup_eq (hi : Disjoint x z) (hs : x ⊔ z = y) : y \ x = z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hi : Disjoint x z) (hs : x ⊔ z = y)
Goal: y \ x = z
-- Disjoint.sdiff_unique
protected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :
    y \ x = z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z)
Goal: y \ x = z
-- disjoint_sdiff_iff_le
theorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \ x) ↔ z ≤ x 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hz : z ≤ y) (hx : x ≤ y)
Goal: Disjoint z (y \ x) ↔ z ≤ x
-- le_iff_disjoint_sdiff
theorem le_iff_disjoint_sdiff (hz : z ≤ y) (hx : x ≤ y) : z ≤ x ↔ Disjoint z (y \ x) 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hz : z ≤ y) (hx : x ≤ y)
Goal: z ≤ x ↔ Disjoint z (y \ x)
-- inf_sdiff_eq_bot_iff
theorem inf_sdiff_eq_bot_iff (hz : z ≤ y) (hx : x ≤ y) : z ⊓ y \ x = ⊥ ↔ z ≤ x 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hz : z ≤ y) (hx : x ≤ y)
Goal: z ⊓ y \ x = ⊥ ↔ z ≤ x