File: Mathlib/Order/Filter/Cocardinal.lean

full_name: Filter.cocardinal
code: /-- The filter defined by all sets that have a complement with at most cardinality `c`. For a union
of `c` sets of `c` elements to have `c` elements, we need that `c` is a regular cardinal. -/
def cocardinal (hreg : c.IsRegular) : Filter α 
goal: Filter α

full_name: Filter.mem_cocardinal
code: @[simp]
theorem mem_cocardinal {s : Set α} :
    s ∈ cocardinal α hreg ↔ Cardinal.mk (sᶜ : Set α) < c 
goal: s ∈ Filter.cocardinal α hreg ↔ #↑sᶜ < c

full_name: Filter.cocardinal_aleph0_eq_cofinite
code: @[simp] lemma cocardinal_aleph0_eq_cofinite :
    cocardinal (α := α) isRegular_aleph0 = cofinite 
goal: Filter.cocardinal α Cardinal.isRegular_aleph0 = Filter.cofinite

full_name: Filter.instCardinalInterFilter_cocardinal
code: instance instCardinalInterFilter_cocardinal : CardinalInterFilter (cocardinal (α := α) hreg) c 
goal: CardinalInterFilter (Filter.cocardinal α hreg) c

full_name: Filter.eventually_cocardinal
code: @[simp]
theorem eventually_cocardinal {p : α → Prop} :
    (∀ᶠ x in cocardinal α hreg, p x) ↔ #{ x | ¬p x } < c 
goal: (∀ᶠ (x : α) in Filter.cocardinal α hreg, p x) ↔ #↑{x | ¬p x} < c

full_name: Filter.hasBasis_cocardinal
code: theorem hasBasis_cocardinal : HasBasis (cocardinal α hreg) {s : Set α | #s < c} compl 
goal: (Filter.cocardinal α hreg).HasBasis {s | #↑s < c} compl

full_name: Filter.frequently_cocardinal
code: theorem frequently_cocardinal {p : α → Prop} :
    (∃ᶠ x in cocardinal α hreg, p x) ↔ c ≤ # { x | p x } 
goal: (∃ᶠ (x : α) in Filter.cocardinal α hreg, p x) ↔ c ≤ #↑{x | p x}

full_name: Filter.frequently_cocardinal_mem
code: lemma frequently_cocardinal_mem {s : Set α} :
    (∃ᶠ x in cocardinal α hreg, x ∈ s) ↔ c ≤ #s 
goal: (∃ᶠ (x : α) in Filter.cocardinal α hreg, x ∈ s) ↔ c ≤ #↑s

full_name: Filter.cocardinal_inf_principal_neBot_iff
code: @[simp]
lemma cocardinal_inf_principal_neBot_iff {s : Set α} :
    (cocardinal α hreg ⊓ 𝓟 s).NeBot ↔ c ≤ #s 
goal: (Filter.cocardinal α hreg ⊓ 𝓟 s).NeBot ↔ c ≤ #↑s

full_name: Filter.compl_mem_cocardinal_of_card_lt
code: theorem compl_mem_cocardinal_of_card_lt {s : Set α} (hs : #s < c) :
    sᶜ ∈ cocardinal α hreg 
goal: sᶜ ∈ Filter.cocardinal α hreg

full_name: Set.Finite.compl_mem_cocardinal
code: theorem _root_.Set.Finite.compl_mem_cocardinal {s : Set α} (hs : s.Finite) :
    sᶜ ∈ cocardinal α hreg 
goal: sᶜ ∈ Filter.cocardinal α hreg

full_name: Filter.eventually_cocardinal_nmem_of_card_lt
code: theorem eventually_cocardinal_nmem_of_card_lt  {s : Set α} (hs : #s < c) :
    ∀ᶠ x in cocardinal α hreg, x ∉ s 
goal: ∀ᶠ (x : α) in Filter.cocardinal α hreg, x ∉ s

full_name: Finset.eventually_cocardinal_nmem
code: theorem _root_.Finset.eventually_cocardinal_nmem (s : Finset α) :
    ∀ᶠ x in cocardinal α hreg, x ∉ s 
goal: ∀ᶠ (x : α) in Filter.cocardinal α hreg, x ∉ s

full_name: Filter.eventually_cocardinal_ne
code: theorem eventually_cocardinal_ne (x : α) : ∀ᶠ a in cocardinal α hreg, a ≠ x 
goal: ∀ᶠ (a : α) in Filter.cocardinal α hreg, a ≠ x

full_name: Filter.cocountable
code: /-- The filter defined by all sets that have countable complements. -/
abbrev cocountable : Filter α 
goal: Filter α

full_name: Filter.mem_cocountable
code: theorem mem_cocountable {s : Set α} :
    s ∈ cocountable ↔ (sᶜ : Set α).Countable 
goal: s ∈ Filter.cocountable ↔ sᶜ.Countable