Theory of filters on sets

A filter on a type α is a collection of sets of α which contains the whole α, is upwards-closed, and is stable under intersection. They are mostly used to abstract two related kinds of ideas:

• limits, including finite or infinite limits of sequences, finite or infinite limits of functions at a point or at infinity, etc...
• things happening eventually, including things happening for large enough n : ℕ, or near enough a point x, or for close enough pairs of points, or things happening almost everywhere in the sense of measure theory. Dually, filters can also express the idea of things happening often: for arbitrarily large n, or at a point in any neighborhood of given a point etc...

Main definitions

In this file, we endow Filter α it with a complete lattice structure. This structure is lifted from the lattice structure on Set (Set X) using the Galois insertion which maps a filter to its elements in one direction, and an arbitrary set of sets to the smallest filter containing it in the other direction. We also prove Filter is a monadic functor, with a push-forward operation Filter.map and a pull-back operation Filter.comap that form a Galois connections for the order on filters.

The examples of filters appearing in the description of the two motivating ideas are:

• (Filter.atTop : Filter ℕ) : made of sets of ℕ containing {n | n ≥ N} for some N
• 𝓝 x : made of neighborhoods of x in a topological space (defined in topology.basic)
• 𝓤 X : made of entourages of a uniform space (those space are generalizations of metric spaces defined in Mathlib/Topology/UniformSpace/Basic.lean)
• MeasureTheory.ae : made of sets whose complement has zero measure with respect to μ (defined in Mathlib/MeasureTheory/OuterMeasure/AE)

The predicate "happening eventually" is Filter.Eventually, and "happening often" is Filter.Frequently, whose definitions are immediate after Filter is defined (but they come rather late in this file in order to immediately relate them to the lattice structure).

Notations

• ∀ᶠ x in f, p x : f.Eventually p;
• ∃ᶠ x in f, p x : f.Frequently p;
• f =ᶠ[l] g : ∀ᶠ x in l, f x = g x;
• f ≤ᶠ[l] g : ∀ᶠ x in l, f x ≤ g x;
• 𝓟 s : Filter.Principal s, localized in Filter.

References

• [N. Bourbaki, General Topology][bourbaki1966]

Important note: Bourbaki requires that a filter on X cannot contain all sets of X, which we do not require. This gives Filter X better formal properties, in particular a bottom element ⊥ for its lattice structure, at the cost of including the assumption [NeBot f] in a number of lemmas and definitions.

instance Filter.inhabitedMem {α : Type u} {f : Filter α} : Inhabited { s : Set α // s ∈ f }

Equations

• Filter.inhabitedMem = { default := ⟨Set.univ, ⋯⟩ }

theorem Filter.filter_eq_iff {α : Type u} {f : Filter α} {g : Filter α} : f = g ↔ f.sets = g.sets

@[simp]

theorem Filter.sets_subset_sets {α : Type u} {f : Filter α} {g : Filter α} : f.sets ⊆ g.sets ↔ g ≤ f

@[simp]

theorem Filter.sets_ssubset_sets {α : Type u} {f : Filter α} {g : Filter α} : f.sets ⊂ g.sets ↔ g < f

theorem Filter.coext {α : Type u} {f : Filter α} {g : Filter α} (h : ∀ (s : Set α), sᶜ ∈ f ↔ sᶜ ∈ g) : f = g

An extensionality lemma that is useful for filters with good lemmas about sᶜ ∈ f (e.g., Filter.comap, Filter.coprod, Filter.Coprod, Filter.cofinite).

instance Filter.instTransSetSupersetMem {α : Type u} : Trans (fun (x1 x2 : Set α) => x1 ⊇ x2) (fun (x1 : Set α) (x2 : Filter α) => x1 ∈ x2) fun (x1 : Set α) (x2 : Filter α) => x1 ∈ x2

Equations

• Filter.instTransSetSupersetMem = { trans := ⋯ }

instance Filter.instTransSetMemSubset {α : Type u} : Trans Membership.mem (fun (x1 x2 : Set α) => x1 ⊆ x2) Membership.mem

Equations

• Filter.instTransSetMemSubset = { trans := ⋯ }

@[simp]

theorem Filter.inter_mem_iff {α : Type u} {f : Filter α} {s : Set α} {t : Set α} : s ∩ t ∈ f ↔ s ∈ f ∧ t ∈ f

theorem Filter.diff_mem {α : Type u} {f : Filter α} {s : Set α} {t : Set α} (hs : s ∈ f) (ht : tᶜ ∈ f) : s \ t ∈ f

theorem Filter.congr_sets {α : Type u} {f : Filter α} {s : Set α} {t : Set α} (h : {x : α | x ∈ s ↔ x ∈ t} ∈ f) : s ∈ f ↔ t ∈ f

theorem Filter.copy_eq {α : Type u} {f : Filter α} {S : Set (Set α)} (hmem : ∀ (s : Set α), s ∈ S ↔ s ∈ f) : f.copy S hmem = f

@[simp]

theorem Filter.biInter_mem {α : Type u} {f : Filter α} {β : Type v} {s : β → Set α} {is : Set β} (hf : is.Finite) : ⋂ i ∈ is, s i ∈ f ↔ ∀ i ∈ is, s i ∈ f

@[simp]

theorem Filter.biInter_finset_mem {α : Type u} {f : Filter α} {β : Type v} {s : β → Set α} (is : Finset β) : ⋂ i ∈ is, s i ∈ f ↔ ∀ i ∈ is, s i ∈ f

theorem Finset.iInter_mem_sets {α : Type u} {f : Filter α} {β : Type v} {s : β → Set α} (is : Finset β) : ⋂ i ∈ is, s i ∈ f ↔ ∀ i ∈ is, s i ∈ f

Alias of Filter.biInter_finset_mem.

@[simp]

theorem Filter.sInter_mem {α : Type u} {f : Filter α} {s : Set (Set α)} (hfin : s.Finite) : ⋂₀ s ∈ f ↔ ∀ U ∈ s, U ∈ f

@[simp]

theorem Filter.iInter_mem {α : Type u} {f : Filter α} {β : Sort v} {s : β → Set α} [Finite β] : ⋂ (i : β), s i ∈ f ↔ ∀ (i : β), s i ∈ f

theorem Filter.exists_mem_subset_iff {α : Type u} {f : Filter α} {s : Set α} : (∃ t ∈ f, t ⊆ s) ↔ s ∈ f

theorem Filter.monotone_mem {α : Type u} {f : Filter α} : Monotone fun (s : Set α) => s ∈ f

theorem Filter.exists_mem_and_iff {α : Type u} {f : Filter α} {P : Set α → Prop} {Q : Set α → Prop} (hP : Antitone P) (hQ : Antitone Q) : ((∃ u ∈ f, P u) ∧ ∃ u ∈ f, Q u) ↔ ∃ u ∈ f, P u ∧ Q u

theorem Filter.forall_in_swap {α : Type u} {f : Filter α} {β : Type u_1} {p : Set α → β → Prop} : (∀ a ∈ f, ∀ (b : β), p a b) ↔ ∀ (b : β), ∀ a ∈ f, p a b

theorem Filter.mem_principal_self {α : Type u} (s : Set α) : s ∈ Filter.principal s

theorem Filter.not_le {α : Type u} {f : Filter α} {g : Filter α} : ¬f ≤ g ↔ ∃ s ∈ g, s ∉ f

inductive Filter.GenerateSets {α : Type u} (g : Set (Set α)) : Set α → Prop

GenerateSets g s: s is in the filter closure of g.

• basic: ∀ {α : Type u} {g : Set (Set α)} {s : Set α}, s ∈ g → Filter.GenerateSets g s
• univ: ∀ {α : Type u} {g : Set (Set α)}, Filter.GenerateSets g Set.univ
• superset: ∀ {α : Type u} {g : Set (Set α)} {s t : Set α}, Filter.GenerateSets g s → s ⊆ t → Filter.GenerateSets g t
• inter: ∀ {α : Type u} {g : Set (Set α)} {s t : Set α}, Filter.GenerateSets g s → Filter.GenerateSets g t → Filter.GenerateSets g (s ∩ t)

def Filter.generate {α : Type u} (g : Set (Set α)) : Filter α

generate g is the largest filter containing the sets g.

Equations

• Filter.generate g = { sets := {s : Set α | Filter.GenerateSets g s}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ }

theorem Filter.mem_generate_of_mem {α : Type u} {s : Set (Set α)} {U : Set α} (h : U ∈ s) : U ∈ Filter.generate s

theorem Filter.le_generate_iff {α : Type u} {s : Set (Set α)} {f : Filter α} : f ≤ Filter.generate s ↔ s ⊆ f.sets

theorem Filter.mem_generate_iff {α : Type u} {s : Set (Set α)} {U : Set α} : U ∈ Filter.generate s ↔ ∃ t ⊆ s, t.Finite ∧ ⋂₀ t ⊆ U

@[simp]

theorem Filter.generate_singleton {α : Type u} (s : Set α) : Filter.generate {s} = Filter.principal s

def Filter.mkOfClosure {α : Type u} (s : Set (Set α)) (hs : (Filter.generate s).sets = s) : Filter α

mkOfClosure s hs constructs a filter on α whose elements set is exactly s : Set (Set α), provided one gives the assumption hs : (generate s).sets = s.

Equations

• Filter.mkOfClosure s hs = { sets := s, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ }

theorem Filter.mkOfClosure_sets {α : Type u} {s : Set (Set α)} {hs : (Filter.generate s).sets = s} : Filter.mkOfClosure s hs = Filter.generate s

def Filter.giGenerate (α : Type u_2) : GaloisInsertion Filter.generate Filter.sets

Galois insertion from sets of sets into filters.

Equations

• Filter.giGenerate α = { choice := fun (s : Set (Set α)) (hs : (Filter.generate s).sets ≤ s) => Filter.mkOfClosure s ⋯, gc := ⋯, le_l_u := ⋯, choice_eq := ⋯ }

theorem Filter.mem_inf_iff {α : Type u} {f : Filter α} {g : Filter α} {s : Set α} : s ∈ f ⊓ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, s = t₁ ∩ t₂

theorem Filter.mem_inf_of_left {α : Type u} {f : Filter α} {g : Filter α} {s : Set α} (h : s ∈ f) : s ∈ f ⊓ g

theorem Filter.mem_inf_of_right {α : Type u} {f : Filter α} {g : Filter α} {s : Set α} (h : s ∈ g) : s ∈ f ⊓ g

theorem Filter.inter_mem_inf {α : Type u} {f : Filter α} {g : Filter α} {s : Set α} {t : Set α} (hs : s ∈ f) (ht : t ∈ g) : s ∩ t ∈ f ⊓ g

theorem Filter.mem_inf_of_inter {α : Type u} {f : Filter α} {g : Filter α} {s : Set α} {t : Set α} {u : Set α} (hs : s ∈ f) (ht : t ∈ g) (h : s ∩ t ⊆ u) : u ∈ f ⊓ g

theorem Filter.mem_inf_iff_superset {α : Type u} {f : Filter α} {g : Filter α} {s : Set α} : s ∈ f ⊓ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ∩ t₂ ⊆ s

instance Filter.instCompleteLatticeFilter {α : Type u} : CompleteLattice (Filter α)

Equations

• Filter.instCompleteLatticeFilter = CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance Filter.instInhabited {α : Type u} : Inhabited (Filter α)

Equations

• Filter.instInhabited = { default := ⊥ }

theorem Filter.NeBot.ne {α : Type u} {f : Filter α} (hf : f.NeBot) : f ≠ ⊥

@[simp]

theorem Filter.not_neBot {α : Type u} {f : Filter α} : ¬f.NeBot ↔ f = ⊥

theorem Filter.NeBot.mono {α : Type u} {f : Filter α} {g : Filter α} (hf : f.NeBot) (hg : f ≤ g) : g.NeBot

theorem Filter.neBot_of_le {α : Type u} {f : Filter α} {g : Filter α} [hf : f.NeBot] (hg : f ≤ g) : g.NeBot

@[simp]

theorem Filter.sup_neBot {α : Type u} {f : Filter α} {g : Filter α} : (f ⊔ g).NeBot ↔ f.NeBot ∨ g.NeBot

theorem Filter.not_disjoint_self_iff {α : Type u} {f : Filter α} : ¬Disjoint f f ↔ f.NeBot

theorem Filter.bot_sets_eq {α : Type u} : ⊥.sets = Set.univ

theorem Filter.eq_or_neBot {α : Type u} (f : Filter α) : f = ⊥ ∨ f.NeBot

Either f = ⊥ or Filter.NeBot f. This is a version of eq_or_ne that uses Filter.NeBot as the second alternative, to be used as an instance.

theorem Filter.sup_sets_eq {α : Type u} {f : Filter α} {g : Filter α} : (f ⊔ g).sets = f.sets ∩ g.sets

theorem Filter.sSup_sets_eq {α : Type u} {s : Set (Filter α)} : (sSup s).sets = ⋂ f ∈ s, f.sets

theorem Filter.iSup_sets_eq {α : Type u} {ι : Sort x} {f : ι → Filter α} : (iSup f).sets = ⋂ (i : ι), (f i).sets

theorem Filter.generate_empty {α : Type u} : Filter.generate ∅ = ⊤

theorem Filter.generate_univ {α : Type u} : Filter.generate Set.univ = ⊥

theorem Filter.generate_union {α : Type u} {s : Set (Set α)} {t : Set (Set α)} : Filter.generate (s ∪ t) = Filter.generate s ⊓ Filter.generate t

theorem Filter.generate_iUnion {α : Type u} {ι : Sort x} {s : ι → Set (Set α)} : Filter.generate (⋃ (i : ι), s i) = ⨅ (i : ι), Filter.generate (s i)

@[simp]

theorem Filter.mem_sup {α : Type u} {f : Filter α} {g : Filter α} {s : Set α} : s ∈ f ⊔ g ↔ s ∈ f ∧ s ∈ g

theorem Filter.union_mem_sup {α : Type u} {f : Filter α} {g : Filter α} {s : Set α} {t : Set α} (hs : s ∈ f) (ht : t ∈ g) : s ∪ t ∈ f ⊔ g

@[simp]

theorem Filter.mem_iSup {α : Type u} {ι : Sort x} {x : Set α} {f : ι → Filter α} : x ∈ iSup f ↔ ∀ (i : ι), x ∈ f i

@[simp]

theorem Filter.iSup_neBot {α : Type u} {ι : Sort x} {f : ι → Filter α} : (⨆ (i : ι), f i).NeBot ↔ ∃ (i : ι), (f i).NeBot

theorem Filter.iInf_eq_generate {α : Type u} {ι : Sort x} (s : ι → Filter α) : iInf s = Filter.generate (⋃ (i : ι), (s i).sets)

theorem Filter.mem_iInf_of_mem {α : Type u} {ι : Sort x} {f : ι → Filter α} (i : ι) {s : Set α} (hs : s ∈ f i) : s ∈ ⨅ (i : ι), f i

theorem Filter.mem_iInf_of_iInter {α : Type u} {ι : Type u_2} {s : ι → Filter α} {U : Set α} {I : Set ι} (I_fin : I.Finite) {V : ↑I → Set α} (hV : ∀ (i : ↑I), V i ∈ s ↑i) (hU : ⋂ (i : ↑I), V i ⊆ U) : U ∈ ⨅ (i : ι), s i

theorem Filter.mem_iInf {α : Type u} {ι : Type u_2} {s : ι → Filter α} {U : Set α} : U ∈ ⨅ (i : ι), s i ↔ ∃ (I : Set ι), I.Finite ∧ ∃ (V : ↑I → Set α), (∀ (i : ↑I), V i ∈ s ↑i) ∧ U = ⋂ (i : ↑I), V i

theorem Filter.mem_iInf' {α : Type u} {ι : Type u_2} {s : ι → Filter α} {U : Set α} : U ∈ ⨅ (i : ι), s i ↔ ∃ (I : Set ι), I.Finite ∧ ∃ (V : ι → Set α), (∀ (i : ι), V i ∈ s i) ∧ (∀ i ∉ I, V i = Set.univ) ∧ U = ⋂ i ∈ I, V i ∧ U = ⋂ (i : ι), V i

theorem Filter.exists_iInter_of_mem_iInf {ι : Type u_2} {α : Type u_3} {f : ι → Filter α} {s : Set α} (hs : s ∈ ⨅ (i : ι), f i) : ∃ (t : ι → Set α), (∀ (i : ι), t i ∈ f i) ∧ s = ⋂ (i : ι), t i

theorem Filter.mem_iInf_of_finite {ι : Type u_2} [Finite ι] {α : Type u_3} {f : ι → Filter α} (s : Set α) : s ∈ ⨅ (i : ι), f i ↔ ∃ (t : ι → Set α), (∀ (i : ι), t i ∈ f i) ∧ s = ⋂ (i : ι), t i

@[simp]

theorem Filter.le_principal_iff {α : Type u} {s : Set α} {f : Filter α} : f ≤ Filter.principal s ↔ s ∈ f

theorem Filter.Iic_principal {α : Type u} (s : Set α) : Set.Iic (Filter.principal s) = {l : Filter α | s ∈ l}

theorem Filter.principal_mono {α : Type u} {s : Set α} {t : Set α} : Filter.principal s ≤ Filter.principal t ↔ s ⊆ t

theorem GCongr.filter_principal_mono {α : Type u} {s : Set α} {t : Set α} : s ⊆ t → Filter.principal s ≤ Filter.principal t

Alias of the reverse direction of Filter.principal_mono.

theorem Filter.monotone_principal {α : Type u} : Monotone Filter.principal

@[simp]

theorem Filter.principal_eq_iff_eq {α : Type u} {s : Set α} {t : Set α} : Filter.principal s = Filter.principal t ↔ s = t

@[simp]

theorem Filter.join_principal_eq_sSup {α : Type u} {s : Set (Filter α)} : (Filter.principal s).join = sSup s

@[simp]

theorem Filter.principal_univ {α : Type u} : Filter.principal Set.univ = ⊤

@[simp]

theorem Filter.principal_empty {α : Type u} : Filter.principal ∅ = ⊥

theorem Filter.generate_eq_biInf {α : Type u} (S : Set (Set α)) : Filter.generate S = ⨅ s ∈ S, Filter.principal s

Lattice equations

theorem Filter.empty_mem_iff_bot {α : Type u} {f : Filter α} : ∅ ∈ f ↔ f = ⊥

theorem Filter.nonempty_of_mem {α : Type u} {f : Filter α} [hf : f.NeBot] {s : Set α} (hs : s ∈ f) : s.Nonempty

theorem Filter.NeBot.nonempty_of_mem {α : Type u} {f : Filter α} (hf : f.NeBot) {s : Set α} (hs : s ∈ f) : s.Nonempty

@[simp]

theorem Filter.empty_not_mem {α : Type u} (f : Filter α) [f.NeBot] : ∅ ∉ f

theorem Filter.nonempty_of_neBot {α : Type u} (f : Filter α) [f.NeBot] : Nonempty α

theorem Filter.compl_not_mem {α : Type u} {f : Filter α} {s : Set α} [f.NeBot] (h : s ∈ f) : sᶜ ∉ f

theorem Filter.filter_eq_bot_of_isEmpty {α : Type u} [IsEmpty α] (f : Filter α) : f = ⊥

theorem Filter.disjoint_iff {α : Type u} {f : Filter α} {g : Filter α} : Disjoint f g ↔ ∃ s ∈ f, ∃ t ∈ g, Disjoint s t

theorem Filter.disjoint_of_disjoint_of_mem {α : Type u} {f : Filter α} {g : Filter α} {s : Set α} {t : Set α} (h : Disjoint s t) (hs : s ∈ f) (ht : t ∈ g) : Disjoint f g

theorem Filter.NeBot.not_disjoint {α : Type u} {f : Filter α} {s : Set α} {t : Set α} (hf : f.NeBot) (hs : s ∈ f) (ht : t ∈ f) : ¬Disjoint s t

theorem Filter.inf_eq_bot_iff {α : Type u} {f : Filter α} {g : Filter α} : f ⊓ g = ⊥ ↔ ∃ U ∈ f, ∃ V ∈ g, U ∩ V = ∅

theorem Pairwise.exists_mem_filter_of_disjoint {α : Type u} {ι : Type u_2} [Finite ι] {l : ι → Filter α} (hd : Pairwise (Disjoint on l)) : ∃ (s : ι → Set α), (∀ (i : ι), s i ∈ l i) ∧ Pairwise (Disjoint on s)

theorem Set.PairwiseDisjoint.exists_mem_filter {α : Type u} {ι : Type u_2} {l : ι → Filter α} {t : Set ι} (hd : t.PairwiseDisjoint l) (ht : t.Finite) : ∃ (s : ι → Set α), (∀ (i : ι), s i ∈ l i) ∧ t.PairwiseDisjoint s

instance Filter.unique {α : Type u} [IsEmpty α] : Unique (Filter α)

There is exactly one filter on an empty type.

Equations

• Filter.unique = { default := ⊥, uniq := ⋯ }

theorem Filter.NeBot.nonempty {α : Type u} (f : Filter α) [hf : f.NeBot] : Nonempty α

theorem Filter.eq_top_of_neBot {α : Type u} [Subsingleton α] (l : Filter α) [l.NeBot] : l = ⊤

There are only two filters on a Subsingleton: ⊥ and ⊤. If the type is empty, then they are equal.

theorem Filter.forall_mem_nonempty_iff_neBot {α : Type u} {f : Filter α} : (∀ s ∈ f, s.Nonempty) ↔ f.NeBot

instance Filter.instNontrivialFilter {α : Type u} [Nonempty α] : Nontrivial (Filter α)

Equations

• ⋯ = ⋯

theorem Filter.nontrivial_iff_nonempty {α : Type u} : Nontrivial (Filter α) ↔ Nonempty α

theorem Filter.eq_sInf_of_mem_iff_exists_mem {α : Type u} {S : Set (Filter α)} {l : Filter α} (h : ∀ {s : Set α}, s ∈ l ↔ ∃ f ∈ S, s ∈ f) : l = sInf S

theorem Filter.eq_iInf_of_mem_iff_exists_mem {α : Type u} {ι : Sort x} {f : ι → Filter α} {l : Filter α} (h : ∀ {s : Set α}, s ∈ l ↔ ∃ (i : ι), s ∈ f i) : l = iInf f

theorem Filter.eq_biInf_of_mem_iff_exists_mem {α : Type u} {ι : Sort x} {f : ι → Filter α} {p : ι → Prop} {l : Filter α} (h : ∀ {s : Set α}, s ∈ l ↔ ∃ (i : ι), p i ∧ s ∈ f i) : l = ⨅ (i : ι), ⨅ (_ : p i), f i

theorem Filter.iInf_sets_eq {α : Type u} {ι : Sort x} {f : ι → Filter α} (h : Directed (fun (x1 x2 : Filter α) => x1 ≥ x2) f) [ne : Nonempty ι] : (iInf f).sets = ⋃ (i : ι), (f i).sets

theorem Filter.mem_iInf_of_directed {α : Type u} {ι : Sort x} {f : ι → Filter α} (h : Directed (fun (x1 x2 : Filter α) => x1 ≥ x2) f) [Nonempty ι] (s : Set α) : s ∈ iInf f ↔ ∃ (i : ι), s ∈ f i

theorem Filter.mem_biInf_of_directed {α : Type u} {β : Type v} {f : β → Filter α} {s : Set β} (h : DirectedOn (f ⁻¹'o fun (x1 x2 : Filter α) => x1 ≥ x2) s) (ne : s.Nonempty) {t : Set α} : t ∈ ⨅ i ∈ s, f i ↔ ∃ i ∈ s, t ∈ f i

theorem Filter.biInf_sets_eq {α : Type u} {β : Type v} {f : β → Filter α} {s : Set β} (h : DirectedOn (f ⁻¹'o fun (x1 x2 : Filter α) => x1 ≥ x2) s) (ne : s.Nonempty) : (⨅ i ∈ s, f i).sets = ⋃ i ∈ s, (f i).sets

theorem Filter.iInf_sets_eq_finite {α : Type u} {ι : Type u_2} (f : ι → Filter α) : (⨅ (i : ι), f i).sets = ⋃ (t : Finset ι), (⨅ i ∈ t, f i).sets

theorem Filter.iInf_sets_eq_finite' {α : Type u} {ι : Sort x} (f : ι → Filter α) : (⨅ (i : ι), f i).sets = ⋃ (t : Finset (PLift ι)), (⨅ i ∈ t, f i.down).sets

theorem Filter.mem_iInf_finite {α : Type u} {ι : Type u_2} {f : ι → Filter α} (s : Set α) : s ∈ iInf f ↔ ∃ (t : Finset ι), s ∈ ⨅ i ∈ t, f i

theorem Filter.mem_iInf_finite' {α : Type u} {ι : Sort x} {f : ι → Filter α} (s : Set α) : s ∈ iInf f ↔ ∃ (t : Finset (PLift ι)), s ∈ ⨅ i ∈ t, f i.down

@[simp]

theorem Filter.sup_join {α : Type u} {f₁ : Filter (Filter α)} {f₂ : Filter (Filter α)} : f₁.join ⊔ f₂.join = (f₁ ⊔ f₂).join

@[simp]

theorem Filter.iSup_join {α : Type u} {ι : Sort w} {f : ι → Filter (Filter α)} : ⨆ (x : ι), (f x).join = (⨆ (x : ι), f x).join

instance Filter.instDistribLattice {α : Type u} : DistribLattice (Filter α)

Equations

• Filter.instDistribLattice = DistribLattice.mk ⋯

@[reducible, inline]

abbrev Filter.coframeMinimalAxioms {α : Type u} : Order.Coframe.MinimalAxioms (Filter α)

The dual version does not hold! Filter α is not a CompleteDistribLattice.

Equations

• Filter.coframeMinimalAxioms = Order.Coframe.MinimalAxioms.mk ⋯

instance Filter.instCoframe {α : Type u} : Order.Coframe (Filter α)

Equations

• Filter.instCoframe = Order.Coframe.ofMinimalAxioms Filter.coframeMinimalAxioms

theorem Filter.mem_iInf_finset {α : Type u} {β : Type v} {s : Finset α} {f : α → Filter β} {t : Set β} : t ∈ ⨅ a ∈ s, f a ↔ ∃ (p : α → Set β), (∀ a ∈ s, p a ∈ f a) ∧ t = ⋂ a ∈ s, p a

theorem Filter.iInf_neBot_of_directed' {α : Type u} {ι : Sort x} {f : ι → Filter α} [Nonempty ι] (hd : Directed (fun (x1 x2 : Filter α) => x1 ≥ x2) f) : (∀ (i : ι), (f i).NeBot) → (iInf f).NeBot

If f : ι → Filter α is directed, ι is not empty, and ∀ i, f i ≠ ⊥, then iInf f ≠ ⊥. See also iInf_neBot_of_directed for a version assuming Nonempty α instead of Nonempty ι.

theorem Filter.iInf_neBot_of_directed {α : Type u} {ι : Sort x} {f : ι → Filter α} [hn : Nonempty α] (hd : Directed (fun (x1 x2 : Filter α) => x1 ≥ x2) f) (hb : ∀ (i : ι), (f i).NeBot) : (iInf f).NeBot

If f : ι → Filter α is directed, α is not empty, and ∀ i, f i ≠ ⊥, then iInf f ≠ ⊥. See also iInf_neBot_of_directed' for a version assuming Nonempty ι instead of Nonempty α.

theorem Filter.sInf_neBot_of_directed' {α : Type u} {s : Set (Filter α)} (hne : s.Nonempty) (hd : DirectedOn (fun (x1 x2 : Filter α) => x1 ≥ x2) s) (hbot : ⊥ ∉ s) : (sInf s).NeBot

theorem Filter.sInf_neBot_of_directed {α : Type u} [Nonempty α] {s : Set (Filter α)} (hd : DirectedOn (fun (x1 x2 : Filter α) => x1 ≥ x2) s) (hbot : ⊥ ∉ s) : (sInf s).NeBot

theorem Filter.iInf_neBot_iff_of_directed' {α : Type u} {ι : Sort x} {f : ι → Filter α} [Nonempty ι] (hd : Directed (fun (x1 x2 : Filter α) => x1 ≥ x2) f) : (iInf f).NeBot ↔ ∀ (i : ι), (f i).NeBot

theorem Filter.iInf_neBot_iff_of_directed {α : Type u} {ι : Sort x} {f : ι → Filter α} [Nonempty α] (hd : Directed (fun (x1 x2 : Filter α) => x1 ≥ x2) f) : (iInf f).NeBot ↔ ∀ (i : ι), (f i).NeBot

theorem Filter.iInf_sets_induct {α : Type u} {ι : Sort x} {f : ι → Filter α} {s : Set α} (hs : s ∈ iInf f) {p : Set α → Prop} (uni : p Set.univ) (ins : ∀ {i : ι} {s₁ s₂ : Set α}, s₁ ∈ f i → p s₂ → p (s₁ ∩ s₂)) : p s

principal equations

@[simp]

theorem Filter.inf_principal {α : Type u} {s : Set α} {t : Set α} : Filter.principal s ⊓ Filter.principal t = Filter.principal (s ∩ t)

@[simp]

theorem Filter.sup_principal {α : Type u} {s : Set α} {t : Set α} : Filter.principal s ⊔ Filter.principal t = Filter.principal (s ∪ t)

@[simp]

theorem Filter.iSup_principal {α : Type u} {ι : Sort w} {s : ι → Set α} : ⨆ (x : ι), Filter.principal (s x) = Filter.principal (⋃ (i : ι), s i)

@[simp]

theorem Filter.principal_eq_bot_iff {α : Type u} {s : Set α} : Filter.principal s = ⊥ ↔ s = ∅

@[simp]

theorem Filter.principal_neBot_iff {α : Type u} {s : Set α} : (Filter.principal s).NeBot ↔ s.Nonempty

theorem Set.Nonempty.principal_neBot {α : Type u} {s : Set α} : s.Nonempty → (Filter.principal s).NeBot

Alias of the reverse direction of Filter.principal_neBot_iff.

theorem Filter.isCompl_principal {α : Type u} (s : Set α) : IsCompl (Filter.principal s) (Filter.principal sᶜ)

theorem Filter.mem_inf_principal' {α : Type u} {f : Filter α} {s : Set α} {t : Set α} : s ∈ f ⊓ Filter.principal t ↔ tᶜ ∪ s ∈ f

theorem Filter.mem_inf_principal {α : Type u} {f : Filter α} {s : Set α} {t : Set α} : s ∈ f ⊓ Filter.principal t ↔ {x : α | x ∈ t → x ∈ s} ∈ f

theorem Filter.iSup_inf_principal {α : Type u} {ι : Sort x} (f : ι → Filter α) (s : Set α) : ⨆ (i : ι), f i ⊓ Filter.principal s = (⨆ (i : ι), f i) ⊓ Filter.principal s

theorem Filter.inf_principal_eq_bot {α : Type u} {f : Filter α} {s : Set α} : f ⊓ Filter.principal s = ⊥ ↔ sᶜ ∈ f

theorem Filter.mem_of_eq_bot {α : Type u} {f : Filter α} {s : Set α} (h : f ⊓ Filter.principal sᶜ = ⊥) : s ∈ f

theorem Filter.diff_mem_inf_principal_compl {α : Type u} {f : Filter α} {s : Set α} (hs : s ∈ f) (t : Set α) : s \ t ∈ f ⊓ Filter.principal tᶜ

theorem Filter.principal_le_iff {α : Type u} {s : Set α} {f : Filter α} : Filter.principal s ≤ f ↔ ∀ V ∈ f, s ⊆ V

@[simp]

theorem Filter.iInf_principal_finset {α : Type u} {ι : Type w} (s : Finset ι) (f : ι → Set α) : ⨅ i ∈ s, Filter.principal (f i) = Filter.principal (⋂ i ∈ s, f i)

theorem Filter.iInf_principal {α : Type u} {ι : Sort w} [Finite ι] (f : ι → Set α) : ⨅ (i : ι), Filter.principal (f i) = Filter.principal (⋂ (i : ι), f i)

@[simp]

theorem Filter.iInf_principal' {α : Type u} {ι : Type w} [Finite ι] (f : ι → Set α) : ⨅ (i : ι), Filter.principal (f i) = Filter.principal (⋂ (i : ι), f i)

A special case of iInf_principal that is safe to mark simp.

theorem Filter.iInf_principal_finite {α : Type u} {ι : Type w} {s : Set ι} (hs : s.Finite) (f : ι → Set α) : ⨅ i ∈ s, Filter.principal (f i) = Filter.principal (⋂ i ∈ s, f i)

theorem Filter.join_mono {α : Type u} {f₁ : Filter (Filter α)} {f₂ : Filter (Filter α)} (h : f₁ ≤ f₂) : f₁.join ≤ f₂.join

Eventually

theorem Filter.eventually_iff {α : Type u} {f : Filter α} {P : α → Prop} : (∀ᶠ (x : α) in f, P x) ↔ {x : α | P x} ∈ f

@[simp]

theorem Filter.eventually_mem_set {α : Type u} {s : Set α} {l : Filter α} : (∀ᶠ (x : α) in l, x ∈ s) ↔ s ∈ l

theorem Filter.ext' {α : Type u} {f₁ : Filter α} {f₂ : Filter α} (h : ∀ (p : α → Prop), (∀ᶠ (x : α) in f₁, p x) ↔ ∀ᶠ (x : α) in f₂, p x) : f₁ = f₂

theorem Filter.Eventually.filter_mono {α : Type u} {f₁ : Filter α} {f₂ : Filter α} (h : f₁ ≤ f₂) {p : α → Prop} (hp : ∀ᶠ (x : α) in f₂, p x) : ∀ᶠ (x : α) in f₁, p x

theorem Filter.eventually_of_mem {α : Type u} {f : Filter α} {P : α → Prop} {U : Set α} (hU : U ∈ f) (h : ∀ x ∈ U, P x) : ∀ᶠ (x : α) in f, P x

theorem Filter.Eventually.and {α : Type u} {p : α → Prop} {q : α → Prop} {f : Filter α} : Filter.Eventually p f → Filter.Eventually q f → ∀ᶠ (x : α) in f, p x ∧ q x

@[simp]

theorem Filter.eventually_true {α : Type u} (f : Filter α) : ∀ᶠ (x : α) in f, True

theorem Filter.Eventually.of_forall {α : Type u} {p : α → Prop} {f : Filter α} (hp : ∀ (x : α), p x) : ∀ᶠ (x : α) in f, p x

@[deprecated Filter.Eventually.of_forall]

theorem Filter.eventually_of_forall {α : Type u} {p : α → Prop} {f : Filter α} (hp : ∀ (x : α), p x) : ∀ᶠ (x : α) in f, p x

Alias of Filter.Eventually.of_forall.

@[simp]

theorem Filter.eventually_false_iff_eq_bot {α : Type u} {f : Filter α} : (∀ᶠ (x : α) in f, False) ↔ f = ⊥

@[simp]

theorem Filter.eventually_const {α : Type u} {f : Filter α} [t : f.NeBot] {p : Prop} : (∀ᶠ (x : α) in f, p) ↔ p

theorem Filter.eventually_iff_exists_mem {α : Type u} {p : α → Prop} {f : Filter α} : (∀ᶠ (x : α) in f, p x) ↔ ∃ v ∈ f, ∀ y ∈ v, p y

theorem Filter.Eventually.exists_mem {α : Type u} {p : α → Prop} {f : Filter α} (hp : ∀ᶠ (x : α) in f, p x) : ∃ v ∈ f, ∀ y ∈ v, p y

theorem Filter.Eventually.mp {α : Type u} {p : α → Prop} {q : α → Prop} {f : Filter α} (hp : ∀ᶠ (x : α) in f, p x) (hq : ∀ᶠ (x : α) in f, p x → q x) : ∀ᶠ (x : α) in f, q x

theorem Filter.Eventually.mono {α : Type u} {p : α → Prop} {q : α → Prop} {f : Filter α} (hp : ∀ᶠ (x : α) in f, p x) (hq : ∀ (x : α), p x → q x) : ∀ᶠ (x : α) in f, q x

theorem Filter.forall_eventually_of_eventually_forall {α : Type u} {β : Type v} {f : Filter α} {p : α → β → Prop} (h : ∀ᶠ (x : α) in f, ∀ (y : β), p x y) (y : β) : ∀ᶠ (x : α) in f, p x y

@[simp]

theorem Filter.eventually_and {α : Type u} {p : α → Prop} {q : α → Prop} {f : Filter α} : (∀ᶠ (x : α) in f, p x ∧ q x) ↔ (∀ᶠ (x : α) in f, p x) ∧ ∀ᶠ (x : α) in f, q x

theorem Filter.Eventually.congr {α : Type u} {f : Filter α} {p : α → Prop} {q : α → Prop} (h' : ∀ᶠ (x : α) in f, p x) (h : ∀ᶠ (x : α) in f, p x ↔ q x) : ∀ᶠ (x : α) in f, q x

theorem Filter.eventually_congr {α : Type u} {f : Filter α} {p : α → Prop} {q : α → Prop} (h : ∀ᶠ (x : α) in f, p x ↔ q x) : (∀ᶠ (x : α) in f, p x) ↔ ∀ᶠ (x : α) in f, q x

@[simp]

theorem Filter.eventually_all {α : Type u} {ι : Sort u_2} [Finite ι] {l : Filter α} {p : ι → α → Prop} : (∀ᶠ (x : α) in l, ∀ (i : ι), p i x) ↔ ∀ (i : ι), ∀ᶠ (x : α) in l, p i x

@[simp]

theorem Filter.eventually_all_finite {α : Type u} {ι : Type u_2} {I : Set ι} (hI : I.Finite) {l : Filter α} {p : ι → α → Prop} : (∀ᶠ (x : α) in l, ∀ i ∈ I, p i x) ↔ ∀ i ∈ I, ∀ᶠ (x : α) in l, p i x

theorem Set.Finite.eventually_all {α : Type u} {ι : Type u_2} {I : Set ι} (hI : I.Finite) {l : Filter α} {p : ι → α → Prop} : (∀ᶠ (x : α) in l, ∀ i ∈ I, p i x) ↔ ∀ i ∈ I, ∀ᶠ (x : α) in l, p i x

Alias of Filter.eventually_all_finite.

@[simp]

theorem Filter.eventually_all_finset {α : Type u} {ι : Type u_2} (I : Finset ι) {l : Filter α} {p : ι → α → Prop} : (∀ᶠ (x : α) in l, ∀ i ∈ I, p i x) ↔ ∀ i ∈ I, ∀ᶠ (x : α) in l, p i x

theorem Finset.eventually_all {α : Type u} {ι : Type u_2} (I : Finset ι) {l : Filter α} {p : ι → α → Prop} : (∀ᶠ (x : α) in l, ∀ i ∈ I, p i x) ↔ ∀ i ∈ I, ∀ᶠ (x : α) in l, p i x

Alias of Filter.eventually_all_finset.

@[simp]

theorem Filter.eventually_or_distrib_left {α : Type u} {f : Filter α} {p : Prop} {q : α → Prop} : (∀ᶠ (x : α) in f, p ∨ q x) ↔ p ∨ ∀ᶠ (x : α) in f, q x

@[simp]

theorem Filter.eventually_or_distrib_right {α : Type u} {f : Filter α} {p : α → Prop} {q : Prop} : (∀ᶠ (x : α) in f, p x ∨ q) ↔ (∀ᶠ (x : α) in f, p x) ∨ q

theorem Filter.eventually_imp_distrib_left {α : Type u} {f : Filter α} {p : Prop} {q : α → Prop} : (∀ᶠ (x : α) in f, p → q x) ↔ p → ∀ᶠ (x : α) in f, q x

@[simp]

theorem Filter.eventually_bot {α : Type u} {p : α → Prop} : ∀ᶠ (x : α) in ⊥, p x

@[simp]

theorem Filter.eventually_top {α : Type u} {p : α → Prop} : (∀ᶠ (x : α) in ⊤, p x) ↔ ∀ (x : α), p x

@[simp]

theorem Filter.eventually_sup {α : Type u} {p : α → Prop} {f : Filter α} {g : Filter α} : (∀ᶠ (x : α) in f ⊔ g, p x) ↔ (∀ᶠ (x : α) in f, p x) ∧ ∀ᶠ (x : α) in g, p x

@[simp]

theorem Filter.eventually_sSup {α : Type u} {p : α → Prop} {fs : Set (Filter α)} : (∀ᶠ (x : α) in sSup fs, p x) ↔ ∀ f ∈ fs, ∀ᶠ (x : α) in f, p x

@[simp]

theorem Filter.eventually_iSup {α : Type u} {ι : Sort x} {p : α → Prop} {fs : ι → Filter α} : (∀ᶠ (x : α) in ⨆ (b : ι), fs b, p x) ↔ ∀ (b : ι), ∀ᶠ (x : α) in fs b, p x

@[simp]

theorem Filter.eventually_principal {α : Type u} {a : Set α} {p : α → Prop} : (∀ᶠ (x : α) in Filter.principal a, p x) ↔ ∀ x ∈ a, p x

theorem Filter.Eventually.forall_mem {α : Type u_2} {f : Filter α} {s : Set α} {P : α → Prop} (hP : ∀ᶠ (x : α) in f, P x) (hf : Filter.principal s ≤ f) (x : α) : x ∈ s → P x

theorem Filter.eventually_inf {α : Type u} {f : Filter α} {g : Filter α} {p : α → Prop} : (∀ᶠ (x : α) in f ⊓ g, p x) ↔ ∃ s ∈ f, ∃ t ∈ g, ∀ x ∈ s ∩ t, p x

theorem Filter.eventually_inf_principal {α : Type u} {f : Filter α} {p : α → Prop} {s : Set α} : (∀ᶠ (x : α) in f ⊓ Filter.principal s, p x) ↔ ∀ᶠ (x : α) in f, x ∈ s → p x

Frequently

theorem Filter.Eventually.frequently {α : Type u} {f : Filter α} [f.NeBot] {p : α → Prop} (h : ∀ᶠ (x : α) in f, p x) : ∃ᶠ (x : α) in f, p x

theorem Filter.Frequently.of_forall {α : Type u} {f : Filter α} [f.NeBot] {p : α → Prop} (h : ∀ (x : α), p x) : ∃ᶠ (x : α) in f, p x

@[deprecated Filter.Frequently.of_forall]

theorem Filter.frequently_of_forall {α : Type u} {f : Filter α} [f.NeBot] {p : α → Prop} (h : ∀ (x : α), p x) : ∃ᶠ (x : α) in f, p x

Alias of Filter.Frequently.of_forall.

theorem Filter.Frequently.mp {α : Type u} {p : α → Prop} {q : α → Prop} {f : Filter α} (h : ∃ᶠ (x : α) in f, p x) (hpq : ∀ᶠ (x : α) in f, p x → q x) : ∃ᶠ (x : α) in f, q x

theorem Filter.Frequently.filter_mono {α : Type u} {p : α → Prop} {f : Filter α} {g : Filter α} (h : ∃ᶠ (x : α) in f, p x) (hle : f ≤ g) : ∃ᶠ (x : α) in g, p x

theorem Filter.Frequently.mono {α : Type u} {p : α → Prop} {q : α → Prop} {f : Filter α} (h : ∃ᶠ (x : α) in f, p x) (hpq : ∀ (x : α), p x → q x) : ∃ᶠ (x : α) in f, q x

theorem Filter.Frequently.and_eventually {α : Type u} {p : α → Prop} {q : α → Prop} {f : Filter α} (hp : ∃ᶠ (x : α) in f, p x) (hq : ∀ᶠ (x : α) in f, q x) : ∃ᶠ (x : α) in f, p x ∧ q x

theorem Filter.Eventually.and_frequently {α : Type u} {p : α → Prop} {q : α → Prop} {f : Filter α} (hp : ∀ᶠ (x : α) in f, p x) (hq : ∃ᶠ (x : α) in f, q x) : ∃ᶠ (x : α) in f, p x ∧ q x

theorem Filter.Frequently.exists {α : Type u} {p : α → Prop} {f : Filter α} (hp : ∃ᶠ (x : α) in f, p x) : ∃ (x : α), p x

theorem Filter.Eventually.exists {α : Type u} {p : α → Prop} {f : Filter α} [f.NeBot] (hp : ∀ᶠ (x : α) in f, p x) : ∃ (x : α), p x

theorem Filter.frequently_iff_neBot {α : Type u} {l : Filter α} {p : α → Prop} : (∃ᶠ (x : α) in l, p x) ↔ (l ⊓ Filter.principal {x : α | p x}).NeBot

theorem Filter.frequently_mem_iff_neBot {α : Type u} {l : Filter α} {s : Set α} : (∃ᶠ (x : α) in l, x ∈ s) ↔ (l ⊓ Filter.principal s).NeBot

theorem Filter.frequently_iff_forall_eventually_exists_and {α : Type u} {p : α → Prop} {f : Filter α} : (∃ᶠ (x : α) in f, p x) ↔ ∀ {q : α → Prop}, (∀ᶠ (x : α) in f, q x) → ∃ (x : α), p x ∧ q x

theorem Filter.frequently_iff {α : Type u} {f : Filter α} {P : α → Prop} : (∃ᶠ (x : α) in f, P x) ↔ ∀ {U : Set α}, U ∈ f → ∃ x ∈ U, P x

@[simp]

theorem Filter.not_eventually {α : Type u} {p : α → Prop} {f : Filter α} : (¬∀ᶠ (x : α) in f, p x) ↔ ∃ᶠ (x : α) in f, ¬p x

@[simp]

theorem Filter.not_frequently {α : Type u} {p : α → Prop} {f : Filter α} : (¬∃ᶠ (x : α) in f, p x) ↔ ∀ᶠ (x : α) in f, ¬p x

@[simp]

theorem Filter.frequently_true_iff_neBot {α : Type u} (f : Filter α) : (∃ᶠ (x : α) in f, True) ↔ f.NeBot

@[simp]

theorem Filter.frequently_false {α : Type u} (f : Filter α) : ¬∃ᶠ (x : α) in f, False

@[simp]

theorem Filter.frequently_const {α : Type u} {f : Filter α} [f.NeBot] {p : Prop} : (∃ᶠ (x : α) in f, p) ↔ p

@[simp]

theorem Filter.frequently_or_distrib {α : Type u} {f : Filter α} {p : α → Prop} {q : α → Prop} : (∃ᶠ (x : α) in f, p x ∨ q x) ↔ (∃ᶠ (x : α) in f, p x) ∨ ∃ᶠ (x : α) in f, q x

theorem Filter.frequently_or_distrib_left {α : Type u} {f : Filter α} [f.NeBot] {p : Prop} {q : α → Prop} : (∃ᶠ (x : α) in f, p ∨ q x) ↔ p ∨ ∃ᶠ (x : α) in f, q x

theorem Filter.frequently_or_distrib_right {α : Type u} {f : Filter α} [f.NeBot] {p : α → Prop} {q : Prop} : (∃ᶠ (x : α) in f, p x ∨ q) ↔ (∃ᶠ (x : α) in f, p x) ∨ q

theorem Filter.frequently_imp_distrib {α : Type u} {f : Filter α} {p : α → Prop} {q : α → Prop} : (∃ᶠ (x : α) in f, p x → q x) ↔ (∀ᶠ (x : α) in f, p x) → ∃ᶠ (x : α) in f, q x

theorem Filter.frequently_imp_distrib_left {α : Type u} {f : Filter α} [f.NeBot] {p : Prop} {q : α → Prop} : (∃ᶠ (x : α) in f, p → q x) ↔ p → ∃ᶠ (x : α) in f, q x

theorem Filter.frequently_imp_distrib_right {α : Type u} {f : Filter α} [f.NeBot] {p : α → Prop} {q : Prop} : (∃ᶠ (x : α) in f, p x → q) ↔ (∀ᶠ (x : α) in f, p x) → q

theorem Filter.eventually_imp_distrib_right {α : Type u} {f : Filter α} {p : α → Prop} {q : Prop} : (∀ᶠ (x : α) in f, p x → q) ↔ (∃ᶠ (x : α) in f, p x) → q

@[simp]

theorem Filter.frequently_and_distrib_left {α : Type u} {f : Filter α} {p : Prop} {q : α → Prop} : (∃ᶠ (x : α) in f, p ∧ q x) ↔ p ∧ ∃ᶠ (x : α) in f, q x

@[simp]

theorem Filter.frequently_and_distrib_right {α : Type u} {f : Filter α} {p : α → Prop} {q : Prop} : (∃ᶠ (x : α) in f, p x ∧ q) ↔ (∃ᶠ (x : α) in f, p x) ∧ q

@[simp]

theorem Filter.frequently_bot {α : Type u} {p : α → Prop} : ¬∃ᶠ (x : α) in ⊥, p x

@[simp]

theorem Filter.frequently_top {α : Type u} {p : α → Prop} : (∃ᶠ (x : α) in ⊤, p x) ↔ ∃ (x : α), p x

@[simp]

theorem Filter.frequently_principal {α : Type u} {a : Set α} {p : α → Prop} : (∃ᶠ (x : α) in Filter.principal a, p x) ↔ ∃ x ∈ a, p x

theorem Filter.frequently_inf_principal {α : Type u} {f : Filter α} {s : Set α} {p : α → Prop} : (∃ᶠ (x : α) in f ⊓ Filter.principal s, p x) ↔ ∃ᶠ (x : α) in f, x ∈ s ∧ p x

theorem Filter.Frequently.of_inf_principal {α : Type u} {f : Filter α} {s : Set α} {p : α → Prop} : (∃ᶠ (x : α) in f ⊓ Filter.principal s, p x) → ∃ᶠ (x : α) in f, x ∈ s ∧ p x

Alias of the forward direction of Filter.frequently_inf_principal.

theorem Filter.Frequently.inf_principal {α : Type u} {f : Filter α} {s : Set α} {p : α → Prop} : (∃ᶠ (x : α) in f, x ∈ s ∧ p x) → ∃ᶠ (x : α) in f ⊓ Filter.principal s, p x

Alias of the reverse direction of Filter.frequently_inf_principal.

theorem Filter.frequently_sup {α : Type u} {p : α → Prop} {f : Filter α} {g : Filter α} : (∃ᶠ (x : α) in f ⊔ g, p x) ↔ (∃ᶠ (x : α) in f, p x) ∨ ∃ᶠ (x : α) in g, p x

@[simp]

theorem Filter.frequently_sSup {α : Type u} {p : α → Prop} {fs : Set (Filter α)} : (∃ᶠ (x : α) in sSup fs, p x) ↔ ∃ f ∈ fs, ∃ᶠ (x : α) in f, p x

@[simp]

theorem Filter.frequently_iSup {α : Type u} {β : Type v} {p : α → Prop} {fs : β → Filter α} : (∃ᶠ (x : α) in ⨆ (b : β), fs b, p x) ↔ ∃ (b : β), ∃ᶠ (x : α) in fs b, p x

theorem Filter.Eventually.choice {α : Type u} {β : Type v} {r : α → β → Prop} {l : Filter α} [l.NeBot] (h : ∀ᶠ (x : α) in l, ∃ (y : β), r x y) : ∃ (f : α → β), ∀ᶠ (x : α) in l, r x (f x)

Relation “eventually equal”

theorem Filter.EventuallyEq.eventually {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} (h : f =ᶠ[l] g) : ∀ᶠ (x : α) in l, f x = g x

@[simp]

theorem Filter.eventuallyEq_top {α : Type u} {β : Type v} {f : α → β} {g : α → β} : f =ᶠ[⊤] g ↔ f = g

theorem Filter.EventuallyEq.rw {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} (h : f =ᶠ[l] g) (p : α → β → Prop) (hf : ∀ᶠ (x : α) in l, p x (f x)) : ∀ᶠ (x : α) in l, p x (g x)

theorem Filter.eventuallyEq_set {α : Type u} {s : Set α} {t : Set α} {l : Filter α} : s =ᶠ[l] t ↔ ∀ᶠ (x : α) in l, x ∈ s ↔ x ∈ t

theorem Filter.EventuallyEq.mem_iff {α : Type u} {s : Set α} {t : Set α} {l : Filter α} : s =ᶠ[l] t → ∀ᶠ (x : α) in l, x ∈ s ↔ x ∈ t

Alias of the forward direction of Filter.eventuallyEq_set.

theorem Filter.Eventually.set_eq {α : Type u} {s : Set α} {t : Set α} {l : Filter α} : (∀ᶠ (x : α) in l, x ∈ s ↔ x ∈ t) → s =ᶠ[l] t

Alias of the reverse direction of Filter.eventuallyEq_set.

@[simp]

theorem Filter.eventuallyEq_univ {α : Type u} {s : Set α} {l : Filter α} : s =ᶠ[l] Set.univ ↔ s ∈ l

theorem Filter.EventuallyEq.exists_mem {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} (h : f =ᶠ[l] g) : ∃ s ∈ l, Set.EqOn f g s

theorem Filter.eventuallyEq_of_mem {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} {s : Set α} (hs : s ∈ l) (h : Set.EqOn f g s) : f =ᶠ[l] g

theorem Filter.eventuallyEq_iff_exists_mem {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} : f =ᶠ[l] g ↔ ∃ s ∈ l, Set.EqOn f g s

theorem Filter.EventuallyEq.filter_mono {α : Type u} {β : Type v} {l : Filter α} {l' : Filter α} {f : α → β} {g : α → β} (h₁ : f =ᶠ[l] g) (h₂ : l' ≤ l) : f =ᶠ[l'] g

@[simp]

theorem Filter.EventuallyEq.refl {α : Type u} {β : Type v} (l : Filter α) (f : α → β) : f =ᶠ[l] f

theorem Filter.EventuallyEq.rfl {α : Type u} {β : Type v} {l : Filter α} {f : α → β} : f =ᶠ[l] f

theorem Filter.EventuallyEq.of_eq {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} (h : f = g) : f =ᶠ[l] g

theorem Eq.eventuallyEq {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} (h : f = g) : f =ᶠ[l] g

Alias of Filter.EventuallyEq.of_eq.

theorem Filter.EventuallyEq.symm {α : Type u} {β : Type v} {f : α → β} {g : α → β} {l : Filter α} (H : f =ᶠ[l] g) : g =ᶠ[l] f

theorem Filter.eventuallyEq_comm {α : Type u} {β : Type v} {f : α → β} {g : α → β} {l : Filter α} : f =ᶠ[l] g ↔ g =ᶠ[l] f

theorem Filter.EventuallyEq.trans {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} {h : α → β} (H₁ : f =ᶠ[l] g) (H₂ : g =ᶠ[l] h) : f =ᶠ[l] h

theorem Filter.EventuallyEq.congr_left {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} {h : α → β} (H : f =ᶠ[l] g) : f =ᶠ[l] h ↔ g =ᶠ[l] h

theorem Filter.EventuallyEq.congr_right {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} {h : α → β} (H : g =ᶠ[l] h) : f =ᶠ[l] g ↔ f =ᶠ[l] h

instance Filter.instTransForallEventuallyEq {α : Type u} {β : Type v} {l : Filter α} : Trans (fun (x1 x2 : α → β) => x1 =ᶠ[l] x2) (fun (x1 x2 : α → β) => x1 =ᶠ[l] x2) fun (x1 x2 : α → β) => x1 =ᶠ[l] x2

Equations

• Filter.instTransForallEventuallyEq = { trans := ⋯ }

theorem Filter.EventuallyEq.prod_mk {α : Type u} {β : Type v} {γ : Type w} {l : Filter α} {f : α → β} {f' : α → β} (hf : f =ᶠ[l] f') {g : α → γ} {g' : α → γ} (hg : g =ᶠ[l] g') : (fun (x : α) => (f x, g x)) =ᶠ[l] fun (x : α) => (f' x, g' x)

theorem Filter.EventuallyEq.fun_comp {α : Type u} {β : Type v} {γ : Type w} {f : α → β} {g : α → β} {l : Filter α} (H : f =ᶠ[l] g) (h : β → γ) : h ∘ f =ᶠ[l] h ∘ g

theorem Filter.EventuallyEq.comp₂ {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_2} {f : α → β} {f' : α → β} {g : α → γ} {g' : α → γ} {l : Filter α} (Hf : f =ᶠ[l] f') (h : β → γ → δ) (Hg : g =ᶠ[l] g') : (fun (x : α) => h (f x) (g x)) =ᶠ[l] fun (x : α) => h (f' x) (g' x)

theorem Filter.EventuallyEq.mul {α : Type u} {β : Type v} [Mul β] {f : α → β} {f' : α → β} {g : α → β} {g' : α → β} {l : Filter α} (h : f =ᶠ[l] g) (h' : f' =ᶠ[l] g') : (fun (x : α) => f x * f' x) =ᶠ[l] fun (x : α) => g x * g' x

theorem Filter.EventuallyEq.add {α : Type u} {β : Type v} [Add β] {f : α → β} {f' : α → β} {g : α → β} {g' : α → β} {l : Filter α} (h : f =ᶠ[l] g) (h' : f' =ᶠ[l] g') : (fun (x : α) => f x + f' x) =ᶠ[l] fun (x : α) => g x + g' x

theorem Filter.EventuallyEq.pow_const {α : Type u} {β : Type v} {γ : Type u_2} [Pow β γ] {f : α → β} {g : α → β} {l : Filter α} (h : f =ᶠ[l] g) (c : γ) : (fun (x : α) => f x ^ c) =ᶠ[l] fun (x : α) => g x ^ c

theorem Filter.EventuallyEq.const_smul {α : Type u} {β : Type v} {γ : Type u_2} [SMul γ β] {f : α → β} {g : α → β} {l : Filter α} (h : f =ᶠ[l] g) (c : γ) : (fun (x : α) => c • f x) =ᶠ[l] fun (x : α) => c • g x

theorem Filter.EventuallyEq.inv {α : Type u} {β : Type v} [Inv β] {f : α → β} {g : α → β} {l : Filter α} (h : f =ᶠ[l] g) : (fun (x : α) => (f x)⁻¹) =ᶠ[l] fun (x : α) => (g x)⁻¹

theorem Filter.EventuallyEq.neg {α : Type u} {β : Type v} [Neg β] {f : α → β} {g : α → β} {l : Filter α} (h : f =ᶠ[l] g) : (fun (x : α) => -f x) =ᶠ[l] fun (x : α) => -g x

theorem Filter.EventuallyEq.div {α : Type u} {β : Type v} [Div β] {f : α → β} {f' : α → β} {g : α → β} {g' : α → β} {l : Filter α} (h : f =ᶠ[l] g) (h' : f' =ᶠ[l] g') : (fun (x : α) => f x / f' x) =ᶠ[l] fun (x : α) => g x / g' x

theorem Filter.EventuallyEq.sub {α : Type u} {β : Type v} [Sub β] {f : α → β} {f' : α → β} {g : α → β} {g' : α → β} {l : Filter α} (h : f =ᶠ[l] g) (h' : f' =ᶠ[l] g') : (fun (x : α) => f x - f' x) =ᶠ[l] fun (x : α) => g x - g' x

theorem Filter.EventuallyEq.const_vadd {α : Type u} {β : Type v} {γ : Type u_2} [VAdd γ β] {f : α → β} {g : α → β} {l : Filter α} (h : f =ᶠ[l] g) (c : γ) : (fun (x : α) => c +ᵥ f x) =ᶠ[l] fun (x : α) => c +ᵥ g x

theorem Filter.EventuallyEq.smul {α : Type u} {β : Type v} {𝕜 : Type u_2} [SMul 𝕜 β] {l : Filter α} {f : α → 𝕜} {f' : α → 𝕜} {g : α → β} {g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') : (fun (x : α) => f x • g x) =ᶠ[l] fun (x : α) => f' x • g' x

theorem Filter.EventuallyEq.vadd {α : Type u} {β : Type v} {𝕜 : Type u_2} [VAdd 𝕜 β] {l : Filter α} {f : α → 𝕜} {f' : α → 𝕜} {g : α → β} {g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') : (fun (x : α) => f x +ᵥ g x) =ᶠ[l] fun (x : α) => f' x +ᵥ g' x

theorem Filter.EventuallyEq.sup {α : Type u} {β : Type v} [Max β] {l : Filter α} {f : α → β} {f' : α → β} {g : α → β} {g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') : (fun (x : α) => f x ⊔ g x) =ᶠ[l] fun (x : α) => f' x ⊔ g' x

theorem Filter.EventuallyEq.inf {α : Type u} {β : Type v} [Min β] {l : Filter α} {f : α → β} {f' : α → β} {g : α → β} {g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') : (fun (x : α) => f x ⊓ g x) =ᶠ[l] fun (x : α) => f' x ⊓ g' x

theorem Filter.EventuallyEq.preimage {α : Type u} {β : Type v} {l : Filter α} {f : α → β} {g : α → β} (h : f =ᶠ[l] g) (s : Set β) : f ⁻¹' s =ᶠ[l] g ⁻¹' s

theorem Filter.EventuallyEq.inter {α : Type u} {s : Set α} {t : Set α} {s' : Set α} {t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') : s ∩ s' =ᶠ[l] t ∩ t'

theorem Filter.EventuallyEq.union {α : Type u} {s : Set α} {t : Set α} {s' : Set α} {t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') : s ∪ s' =ᶠ[l] t ∪ t'

theorem Filter.EventuallyEq.compl {α : Type u} {s : Set α} {t : Set α} {l : Filter α} (h : s =ᶠ[l] t) : sᶜ =ᶠ[l] tᶜ

theorem Filter.EventuallyEq.diff {α : Type u} {s : Set α} {t : Set α} {s' : Set α} {t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') : s \ s' =ᶠ[l] t \ t'

theorem Filter.EventuallyEq.symmDiff {α : Type u} {s : Set α} {t : Set α} {s' : Set α} {t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') : symmDiff s s' =ᶠ[l] symmDiff t t'

theorem Filter.eventuallyEq_empty {α : Type u} {s : Set α} {l : Filter α} : s =ᶠ[l] ∅ ↔ ∀ᶠ (x : α) in l, x ∉ s

theorem Filter.inter_eventuallyEq_left {α : Type u} {s : Set α} {t : Set α} {l : Filter α} : s ∩ t =ᶠ[l] s ↔ ∀ᶠ (x : α) in l, x ∈ s → x ∈ t

theorem Filter.inter_eventuallyEq_right {α : Type u} {s : Set α} {t : Set α} {l : Filter α} : s ∩ t =ᶠ[l] t ↔ ∀ᶠ (x : α) in l, x ∈ t → x ∈ s

@[simp]

theorem Filter.eventuallyEq_principal {α : Type u} {β : Type v} {s : Set α} {f : α → β} {g : α → β} : f =ᶠ[Filter.principal s] g ↔ Set.EqOn f g s

theorem Filter.eventuallyEq_inf_principal_iff {α : Type u} {β : Type v} {F : Filter α} {s : Set α} {f : α → β} {g : α → β} : f =ᶠ[F ⊓ Filter.principal s] g ↔ ∀ᶠ (x : α) in F, x ∈ s → f x = g x

theorem Filter.EventuallyEq.sub_eq {α : Type u} {β : Type v} [AddGroup β] {f : α → β} {g : α → β} {l : Filter α} (h : f =ᶠ[l] g) : f - g =ᶠ[l] 0

theorem Filter.eventuallyEq_iff_sub {α : Type u} {β : Type v} [AddGroup β] {f : α → β} {g : α → β} {l : Filter α} : f =ᶠ[l] g ↔ f - g =ᶠ[l] 0

theorem Filter.EventuallyLE.congr {α : Type u} {β : Type v} [LE β] {l : Filter α} {f : α → β} {f' : α → β} {g : α → β} {g' : α → β} (H : f ≤ᶠ[l] g) (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') : f' ≤ᶠ[l] g'

theorem Filter.eventuallyLE_congr {α : Type u} {β : Type v} [LE β] {l : Filter α} {f : α → β} {f' : α → β} {g : α → β} {g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') : f ≤ᶠ[l] g ↔ f' ≤ᶠ[l] g'

theorem Filter.EventuallyEq.le {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f : α → β} {g : α → β} (h : f =ᶠ[l] g) : f ≤ᶠ[l] g

theorem Filter.EventuallyLE.refl {α : Type u} {β : Type v} [Preorder β] (l : Filter α) (f : α → β) : f ≤ᶠ[l] f

theorem Filter.EventuallyLE.rfl {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f : α → β} : f ≤ᶠ[l] f

theorem Filter.EventuallyLE.trans {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f : α → β} {g : α → β} {h : α → β} (H₁ : f ≤ᶠ[l] g) (H₂ : g ≤ᶠ[l] h) : f ≤ᶠ[l] h

instance Filter.instTransForallEventuallyLE {α : Type u} {β : Type v} [Preorder β] {l : Filter α} : Trans (fun (x1 x2 : α → β) => x1 ≤ᶠ[l] x2) (fun (x1 x2 : α → β) => x1 ≤ᶠ[l] x2) fun (x1 x2 : α → β) => x1 ≤ᶠ[l] x2

Equations

• Filter.instTransForallEventuallyLE = { trans := ⋯ }

theorem Filter.EventuallyEq.trans_le {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f : α → β} {g : α → β} {h : α → β} (H₁ : f =ᶠ[l] g) (H₂ : g ≤ᶠ[l] h) : f ≤ᶠ[l] h

instance Filter.instTransForallEventuallyEqEventuallyLE {α : Type u} {β : Type v} [Preorder β] {l : Filter α} : Trans (fun (x1 x2 : α → β) => x1 =ᶠ[l] x2) (fun (x1 x2 : α → β) => x1 ≤ᶠ[l] x2) fun (x1 x2 : α → β) => x1 ≤ᶠ[l] x2

Equations

• Filter.instTransForallEventuallyEqEventuallyLE = { trans := ⋯ }

theorem Filter.EventuallyLE.trans_eq {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f : α → β} {g : α → β} {h : α → β} (H₁ : f ≤ᶠ[l] g) (H₂ : g =ᶠ[l] h) : f ≤ᶠ[l] h

instance Filter.instTransForallEventuallyLEEventuallyEq {α : Type u} {β : Type v} [Preorder β] {l : Filter α} : Trans (fun (x1 x2 : α → β) => x1 ≤ᶠ[l] x2) (fun (x1 x2 : α → β) => x1 =ᶠ[l] x2) fun (x1 x2 : α → β) => x1 ≤ᶠ[l] x2

Equations

• Filter.instTransForallEventuallyLEEventuallyEq = { trans := ⋯ }

theorem Filter.EventuallyLE.antisymm {α : Type u} {β : Type v} [PartialOrder β] {l : Filter α} {f : α → β} {g : α → β} (h₁ : f ≤ᶠ[l] g) (h₂ : g ≤ᶠ[l] f) : f =ᶠ[l] g

theorem Filter.eventuallyLE_antisymm_iff {α : Type u} {β : Type v} [PartialOrder β] {l : Filter α} {f : α → β} {g : α → β} : f =ᶠ[l] g ↔ f ≤ᶠ[l] g ∧ g ≤ᶠ[l] f

theorem Filter.EventuallyLE.le_iff_eq {α : Type u} {β : Type v} [PartialOrder β] {l : Filter α} {f : α → β} {g : α → β} (h : f ≤ᶠ[l] g) : g ≤ᶠ[l] f ↔ g =ᶠ[l] f

theorem Filter.Eventually.ne_of_lt {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f : α → β} {g : α → β} (h : ∀ᶠ (x : α) in l, f x < g x) : ∀ᶠ (x : α) in l, f x ≠ g x

theorem Filter.Eventually.ne_top_of_lt {α : Type u} {β : Type v} [PartialOrder β] [OrderTop β] {l : Filter α} {f : α → β} {g : α → β} (h : ∀ᶠ (x : α) in l, f x < g x) : ∀ᶠ (x : α) in l, f x ≠ ⊤

theorem Filter.Eventually.lt_top_of_ne {α : Type u} {β : Type v} [PartialOrder β] [OrderTop β] {l : Filter α} {f : α → β} (h : ∀ᶠ (x : α) in l, f x ≠ ⊤) : ∀ᶠ (x : α) in l, f x < ⊤

theorem Filter.Eventually.lt_top_iff_ne_top {α : Type u} {β : Type v} [PartialOrder β] [OrderTop β] {l : Filter α} {f : α → β} : (∀ᶠ (x : α) in l, f x < ⊤) ↔ ∀ᶠ (x : α) in l, f x ≠ ⊤

theorem Filter.EventuallyLE.inter {α : Type u} {s : Set α} {t : Set α} {s' : Set α} {t' : Set α} {l : Filter α} (h : s ≤ᶠ[l] t) (h' : s' ≤ᶠ[l] t') : s ∩ s' ≤ᶠ[l] t ∩ t'

theorem Filter.EventuallyLE.union {α : Type u} {s : Set α} {t : Set α} {s' : Set α} {t' : Set α} {l : Filter α} (h : s ≤ᶠ[l] t) (h' : s' ≤ᶠ[l] t') : s ∪ s' ≤ᶠ[l] t ∪ t'

theorem Filter.EventuallyLE.iUnion {α : Type u} {ι : Sort x} {l : Filter α} [Finite ι] {s : ι → Set α} {t : ι → Set α} (h : ∀ (i : ι), s i ≤ᶠ[l] t i) : ⋃ (i : ι), s i ≤ᶠ[l] ⋃ (i : ι), t i

theorem Filter.EventuallyEq.iUnion {α : Type u} {ι : Sort x} {l : Filter α} [Finite ι] {s : ι → Set α} {t : ι → Set α} (h : ∀ (i : ι), s i =ᶠ[l] t i) : ⋃ (i : ι), s i =ᶠ[l] ⋃ (i : ι), t i

theorem Filter.EventuallyLE.iInter {α : Type u} {ι : Sort x} {l : Filter α} [Finite ι] {s : ι → Set α} {t : ι → Set α} (h : ∀ (i : ι), s i ≤ᶠ[l] t i) : ⋂ (i : ι), s i ≤ᶠ[l] ⋂ (i : ι), t i

theorem Filter.EventuallyEq.iInter {α : Type u} {ι : Sort x} {l : Filter α} [Finite ι] {s : ι → Set α} {t : ι → Set α} (h : ∀ (i : ι), s i =ᶠ[l] t i) : ⋂ (i : ι), s i =ᶠ[l] ⋂ (i : ι), t i

theorem Set.Finite.eventuallyLE_iUnion {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f : ι → Set α} {g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i) : ⋃ i ∈ s, f i ≤ᶠ[l] ⋃ i ∈ s, g i

theorem Filter.EventuallyLE.biUnion {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f : ι → Set α} {g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i) : ⋃ i ∈ s, f i ≤ᶠ[l] ⋃ i ∈ s, g i

Alias of Set.Finite.eventuallyLE_iUnion.

theorem Set.Finite.eventuallyEq_iUnion {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f : ι → Set α} {g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i) : ⋃ i ∈ s, f i =ᶠ[l] ⋃ i ∈ s, g i

theorem Filter.EventuallyEq.biUnion {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f : ι → Set α} {g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i) : ⋃ i ∈ s, f i =ᶠ[l] ⋃ i ∈ s, g i

Alias of Set.Finite.eventuallyEq_iUnion.

theorem Set.Finite.eventuallyLE_iInter {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f : ι → Set α} {g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i) : ⋂ i ∈ s, f i ≤ᶠ[l] ⋂ i ∈ s, g i

theorem Filter.EventuallyLE.biInter {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f : ι → Set α} {g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i) : ⋂ i ∈ s, f i ≤ᶠ[l] ⋂ i ∈ s, g i

Alias of Set.Finite.eventuallyLE_iInter.

theorem Set.Finite.eventuallyEq_iInter {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f : ι → Set α} {g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i) : ⋂ i ∈ s, f i =ᶠ[l] ⋂ i ∈ s, g i

theorem Filter.EventuallyEq.biInter {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f : ι → Set α} {g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i) : ⋂ i ∈ s, f i =ᶠ[l] ⋂ i ∈ s, g i

Alias of Set.Finite.eventuallyEq_iInter.

theorem Finset.eventuallyLE_iUnion {α : Type u} {l : Filter α} {ι : Type u_2} (s : Finset ι) {f : ι → Set α} {g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i) : ⋃ i ∈ s, f i ≤ᶠ[l] ⋃ i ∈ s, g i

theorem Finset.eventuallyEq_iUnion {α : Type u} {l : Filter α} {ι : Type u_2} (s : Finset ι) {f : ι → Set α} {g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i) : ⋃ i ∈ s, f i =ᶠ[l] ⋃ i ∈ s, g i

theorem Finset.eventuallyLE_iInter {α : Type u} {l : Filter α} {ι : Type u_2} (s : Finset ι) {f : ι → Set α} {g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i) : ⋂ i ∈ s, f i ≤ᶠ[l] ⋂ i ∈ s, g i

theorem Finset.eventuallyEq_iInter {α : Type u} {l : Filter α} {ι : Type u_2} (s : Finset ι) {f : ι → Set α} {g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i) : ⋂ i ∈ s, f i =ᶠ[l] ⋂ i ∈ s, g i

theorem Filter.EventuallyLE.compl {α : Type u} {s : Set α} {t : Set α} {l : Filter α} (h : s ≤ᶠ[l] t) : tᶜ ≤ᶠ[l] sᶜ

theorem Filter.EventuallyLE.diff {α : Type u} {s : Set α} {t : Set α} {s' : Set α} {t' : Set α} {l : Filter α} (h : s ≤ᶠ[l] t) (h' : t' ≤ᶠ[l] s') : s \ s' ≤ᶠ[l] t \ t'

theorem Filter.set_eventuallyLE_iff_mem_inf_principal {α : Type u} {s : Set α} {t : Set α} {l : Filter α} : s ≤ᶠ[l] t ↔ t ∈ l ⊓ Filter.principal s

theorem Filter.set_eventuallyLE_iff_inf_principal_le {α : Type u} {s : Set α} {t : Set α} {l : Filter α} : s ≤ᶠ[l] t ↔ l ⊓ Filter.principal s ≤ l ⊓ Filter.principal t

theorem Filter.set_eventuallyEq_iff_inf_principal {α : Type u} {s : Set α} {t : Set α} {l : Filter α} : s =ᶠ[l] t ↔ l ⊓ Filter.principal s = l ⊓ Filter.principal t

theorem Filter.EventuallyLE.sup {α : Type u} {β : Type v} [SemilatticeSup β] {l : Filter α} {f₁ : α → β} {f₂ : α → β} {g₁ : α → β} {g₂ : α → β} (hf : f₁ ≤ᶠ[l] f₂) (hg : g₁ ≤ᶠ[l] g₂) : f₁ ⊔ g₁ ≤ᶠ[l] f₂ ⊔ g₂

theorem Filter.EventuallyLE.sup_le {α : Type u} {β : Type v} [SemilatticeSup β] {l : Filter α} {f : α → β} {g : α → β} {h : α → β} (hf : f ≤ᶠ[l] h) (hg : g ≤ᶠ[l] h) : f ⊔ g ≤ᶠ[l] h

theorem Filter.EventuallyLE.le_sup_of_le_left {α : Type u} {β : Type v} [SemilatticeSup β] {l : Filter α} {f : α → β} {g : α → β} {h : α → β} (hf : h ≤ᶠ[l] f) : h ≤ᶠ[l] f ⊔ g

theorem Filter.EventuallyLE.le_sup_of_le_right {α : Type u} {β : Type v} [SemilatticeSup β] {l : Filter α} {f : α → β} {g : α → β} {h : α → β} (hg : h ≤ᶠ[l] g) : h ≤ᶠ[l] f ⊔ g

theorem Filter.join_le {α : Type u} {f : Filter (Filter α)} {l : Filter α} (h : ∀ᶠ (m : Filter α) in f, m ≤ l) : f.join ≤ l

Push-forwards, pull-backs, and the monad structure

@[simp]

theorem Filter.map_principal {α : Type u} {β : Type v} {s : Set α} {f : α → β} : Filter.map f (Filter.principal s) = Filter.principal (f '' s)

@[simp]

theorem Filter.eventually_map {α : Type u} {β : Type v} {f : Filter α} {m : α → β} {P : β → Prop} : (∀ᶠ (b : β) in Filter.map m f, P b) ↔ ∀ᶠ (a : α) in f, P (m a)

@[simp]

theorem Filter.frequently_map {α : Type u} {β : Type v} {f : Filter α} {m : α → β} {P : β → Prop} : (∃ᶠ (b : β) in Filter.map m f, P b) ↔ ∃ᶠ (a : α) in f, P (m a)

@[simp]

theorem Filter.mem_map {α : Type u} {β : Type v} {f : Filter α} {m : α → β} {t : Set β} : t ∈ Filter.map m f ↔ m ⁻¹' t ∈ f

theorem Filter.mem_map' {α : Type u} {β : Type v} {f : Filter α} {m : α → β} {t : Set β} : t ∈ Filter.map m f ↔ {x : α | m x ∈ t} ∈ f

theorem Filter.image_mem_map {α : Type u} {β : Type v} {f : Filter α} {m : α → β} {s : Set α} (hs : s ∈ f) : m '' s ∈ Filter.map m f

@[simp]

theorem Filter.image_mem_map_iff {α : Type u} {β : Type v} {f : Filter α} {m : α → β} {s : Set α} (hf : Function.Injective m) : m '' s ∈ Filter.map m f ↔ s ∈ f

theorem Filter.range_mem_map {α : Type u} {β : Type v} {f : Filter α} {m : α → β} : Set.range m ∈ Filter.map m f

theorem Filter.mem_map_iff_exists_image {α : Type u} {β : Type v} {f : Filter α} {m : α → β} {t : Set β} : t ∈ Filter.map m f ↔ ∃ s ∈ f, m '' s ⊆ t

@[simp]

theorem Filter.map_id {α : Type u} {f : Filter α} : Filter.map id f = f

@[simp]

theorem Filter.map_id' {α : Type u} {f : Filter α} : Filter.map (fun (x : α) => x) f = f

@[simp]

theorem Filter.map_compose {α : Type u} {β : Type v} {γ : Type w} {m : α → β} {m' : β → γ} : Filter.map m' ∘ Filter.map m = Filter.map (m' ∘ m)

@[simp]

theorem Filter.map_map {α : Type u} {β : Type v} {γ : Type w} {f : Filter α} {m : α → β} {m' : β → γ} : Filter.map m' (Filter.map m f) = Filter.map (m' ∘ m) f

theorem Filter.map_congr {α : Type u} {β : Type v} {m₁ : α → β} {m₂ : α → β} {f : Filter α} (h : m₁ =ᶠ[f] m₂) : Filter.map m₁ f = Filter.map m₂ f

If functions m₁ and m₂ are eventually equal at a filter f, then they map this filter to the same filter.

theorem Filter.mem_comap' {α : Type u} {β : Type v} {f : α → β} {l : Filter β} {s : Set α} : s ∈ Filter.comap f l ↔ {y : β | ∀ ⦃x : α⦄, f x = y → x ∈ s} ∈ l

theorem Filter.mem_comap'' {α : Type u} {β : Type v} {f : α → β} {l : Filter β} {s : Set α} : s ∈ Filter.comap f l ↔ Set.kernImage f s ∈ l

theorem Filter.mem_comap_prod_mk {α : Type u} {β : Type v} {x : α} {s : Set β} {F : Filter (α × β)} : s ∈ Filter.comap (Prod.mk x) F ↔ {p : α × β | p.1 = x → p.2 ∈ s} ∈ F

RHS form is used, e.g., in the definition of UniformSpace.

@[simp]

theorem Filter.eventually_comap {α : Type u} {β : Type v} {f : α → β} {l : Filter β} {p : α → Prop} : (∀ᶠ (a : α) in Filter.comap f l, p a) ↔ ∀ᶠ (b : β) in l, ∀ (a : α), f a = b → p a

@[simp]

theorem Filter.frequently_comap {α : Type u} {β : Type v} {f : α → β} {l : Filter β} {p : α → Prop} : (∃ᶠ (a : α) in Filter.comap f l, p a) ↔ ∃ᶠ (b : β) in l, ∃ (a : α), f a = b ∧ p a

theorem Filter.mem_comap_iff_compl {α : Type u} {β : Type v} {f : α → β} {l : Filter β} {s : Set α} : s ∈ Filter.comap f l ↔ (f '' sᶜ)ᶜ ∈ l

theorem Filter.compl_mem_comap {α : Type u} {β : Type v} {f : α → β} {l : Filter β} {s : Set α} : sᶜ ∈ Filter.comap f l ↔ (f '' s)ᶜ ∈ l

def Filter.kernMap {α : Type u} {β : Type v} (m : α → β) (f : Filter α) : Filter β

The analog of kernImage for filters. A set s belongs to Filter.kernMap m f if either of the following equivalent conditions hold.

1. There exists a set t ∈ f such that s = kernImage m t. This is used as a definition.
2. There exists a set t such that tᶜ ∈ f and sᶜ = m '' t, see Filter.mem_kernMap_iff_compl and Filter.compl_mem_kernMap.

This definition because it gives a right adjoint to Filter.comap, and because it has a nice interpretation when working with co- filters (Filter.cocompact, Filter.cofinite, ...). For example, kernMap m (cocompact α) is the filter generated by the complements of the sets m '' K where K is a compact subset of α.

Equations

• Filter.kernMap m f = { sets := Set.kernImage m '' f.sets, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ }

theorem Filter.mem_kernMap {α : Type u} {β : Type v} {m : α → β} {f : Filter α} {s : Set β} : s ∈ Filter.kernMap m f ↔ ∃ t ∈ f, Set.kernImage m t = s

theorem Filter.mem_kernMap_iff_compl {α : Type u} {β : Type v} {m : α → β} {f : Filter α} {s : Set β} : s ∈ Filter.kernMap m f ↔ ∃ (t : Set α), tᶜ ∈ f ∧ m '' t = sᶜ

theorem Filter.compl_mem_kernMap {α : Type u} {β : Type v} {m : α → β} {f : Filter α} {s : Set β} : sᶜ ∈ Filter.kernMap m f ↔ ∃ (t : Set α), tᶜ ∈ f ∧ m '' t = s

instance Filter.instLawfulFunctor : LawfulFunctor Filter

Equations

• Filter.instLawfulFunctor = ⋯

theorem Filter.pure_sets {α : Type u} (a : α) : (pure a).sets = {s : Set α | a ∈ s}

@[simp]

theorem Filter.eventually_pure {α : Type u} {a : α} {p : α → Prop} : (∀ᶠ (x : α) in pure a, p x) ↔ p a

@[simp]

theorem Filter.principal_singleton {α : Type u} (a : α) : Filter.principal {a} = pure a

@[simp]

theorem Filter.map_pure {α : Type u} {β : Type v} (f : α → β) (a : α) : Filter.map f (pure a) = pure (f a)

theorem Filter.pure_le_principal {α : Type u} {s : Set α} (a : α) : pure a ≤ Filter.principal s ↔ a ∈ s

@[simp]

theorem Filter.join_pure {α : Type u} (f : Filter α) : (pure f).join = f

@[simp]

theorem Filter.pure_bind {α : Type u} {β : Type v} (a : α) (m : α → Filter β) : (pure a).bind m = m a

theorem Filter.map_bind {γ : Type w} {α : Type u_2} {β : Type u_3} (m : β → γ) (f : Filter α) (g : α → Filter β) : Filter.map m (f.bind g) = f.bind (Filter.map m ∘ g)

theorem Filter.bind_map {γ : Type w} {α : Type u_2} {β : Type u_3} (m : α → β) (f : Filter α) (g : β → Filter γ) : (Filter.map m f).bind g = f.bind (g ∘ m)

Filter as a Monad

In this section we define Filter.monad, a Monad structure on Filters. This definition is not an instance because its Seq projection is not equal to the Filter.seq function we use in the Applicative instance on Filter.

def Filter.monad : Monad Filter

The monad structure on filters.

Equations

• Filter.monad = Monad.mk

theorem Filter.lawfulMonad : LawfulMonad Filter

instance Filter.instAlternative : Alternative Filter

Equations

• Filter.instAlternative = Alternative.mk (fun {α : Type ?u.27} => ⊥) fun {α : Type ?u.27} (x : Filter α) (y : Unit → Filter α) => x ⊔ y ()

@[simp]

theorem Filter.map_def {α : Type u_2} {β : Type u_2} (m : α → β) (f : Filter α) : m <$> f = Filter.map m f

@[simp]

theorem Filter.bind_def {α : Type u_2} {β : Type u_2} (f : Filter α) (m : α → Filter β) : f >>= m = f.bind m

map and comap equations

@[simp]

theorem Filter.mem_comap {α : Type u} {β : Type v} {g : Filter β} {m : α → β} {s : Set α} : s ∈ Filter.comap m g ↔ ∃ t ∈ g, m ⁻¹' t ⊆ s

theorem Filter.preimage_mem_comap {α : Type u} {β : Type v} {g : Filter β} {m : α → β} {t : Set β} (ht : t ∈ g) : m ⁻¹' t ∈ Filter.comap m g

theorem Filter.Eventually.comap {α : Type u} {β : Type v} {g : Filter β} {p : β → Prop} (hf : ∀ᶠ (b : β) in g, p b) (f : α → β) : ∀ᶠ (a : α) in Filter.comap f g, p (f a)

theorem Filter.comap_id {α : Type u} {f : Filter α} : Filter.comap id f = f

theorem Filter.comap_id' {α : Type u} {f : Filter α} : Filter.comap (fun (x : α) => x) f = f

theorem Filter.comap_const_of_not_mem {α : Type u} {β : Type v} {g : Filter β} {t : Set β} {x : β} (ht : t ∈ g) (hx : x ∉ t) : Filter.comap (fun (x_1 : α) => x) g = ⊥

theorem Filter.comap_const_of_mem {α : Type u} {β : Type v} {g : Filter β} {x : β} (h : ∀ t ∈ g, x ∈ t) : Filter.comap (fun (x_1 : α) => x) g = ⊤

theorem Filter.map_const {α : Type u} {β : Type v} {f : Filter α} [f.NeBot] {c : β} : Filter.map (fun (x : α) => c) f = pure c

theorem Filter.comap_comap {α : Type u} {β : Type v} {γ : Type w} {f : Filter α} {m : γ → β} {n : β → α} : Filter.comap m (Filter.comap n f) = Filter.comap (n ∘ m) f

The variables in the following lemmas are used as in this diagram:

    φ
  α → β
θ ↓   ↓ ψ
  γ → δ
    ρ

theorem Filter.map_comm {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_1} {φ : α → β} {θ : α → γ} {ψ : β → δ} {ρ : γ → δ} (H : ψ ∘ φ = ρ ∘ θ) (F : Filter α) : Filter.map ψ (Filter.map φ F) = Filter.map ρ (Filter.map θ F)

theorem Filter.comap_comm {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_1} {φ : α → β} {θ : α → γ} {ψ : β → δ} {ρ : γ → δ} (H : ψ ∘ φ = ρ ∘ θ) (G : Filter δ) : Filter.comap φ (Filter.comap ψ G) = Filter.comap θ (Filter.comap ρ G)

theorem Function.Semiconj.filter_map {α : Type u} {β : Type v} {f : α → β} {ga : α → α} {gb : β → β} (h : Function.Semiconj f ga gb) : Function.Semiconj (Filter.map f) (Filter.map ga) (Filter.map gb)

theorem Function.Commute.filter_map {α : Type u} {f : α → α} {g : α → α} (h : Function.Commute f g) : Function.Commute (Filter.map f) (Filter.map g)

theorem Function.Semiconj.filter_comap {α : Type u} {β : Type v} {f : α → β} {ga : α → α} {gb : β → β} (h : Function.Semiconj f ga gb) : Function.Semiconj (Filter.comap f) (Filter.comap gb) (Filter.comap ga)

theorem Function.Commute.filter_comap {α : Type u} {f : α → α} {g : α → α} (h : Function.Commute f g) : Function.Commute (Filter.comap f) (Filter.comap g)

theorem Function.LeftInverse.filter_map {α : Type u} {β : Type v} {f : α → β} {g : β → α} (hfg : Function.LeftInverse g f) : Function.LeftInverse (Filter.map g) (Filter.map f)

theorem Function.LeftInverse.filter_comap {α : Type u} {β : Type v} {f : α → β} {g : β → α} (hfg : Function.LeftInverse g f) : Function.RightInverse (Filter.comap g) (Filter.comap f)

theorem Function.RightInverse.filter_map {α : Type u} {β : Type v} {f : α → β} {g : β → α} (hfg : Function.RightInverse g f) : Function.RightInverse (Filter.map g) (Filter.map f)

theorem Function.RightInverse.filter_comap {α : Type u} {β : Type v} {f : α → β} {g : β → α} (hfg : Function.RightInverse g f) : Function.LeftInverse (Filter.comap g) (Filter.comap f)

theorem Set.LeftInvOn.filter_map_Iic {α : Type u} {β : Type v} {s : Set α} {f : α → β} {g : β → α} (hfg : Set.LeftInvOn g f s) : Set.LeftInvOn (Filter.map g) (Filter.map f) (Set.Iic (Filter.principal s))

theorem Set.RightInvOn.filter_map_Iic {α : Type u} {β : Type v} {t : Set β} {f : α → β} {g : β → α} (hfg : Set.RightInvOn g f t) : Set.RightInvOn (Filter.map g) (Filter.map f) (Set.Iic (Filter.principal t))

@[simp]

theorem Filter.comap_principal {α : Type u} {β : Type v} {m : α → β} {t : Set β} : Filter.comap m (Filter.principal t) = Filter.principal (m ⁻¹' t)

theorem Filter.principal_subtype {α : Type u_2} (s : Set α) (t : Set ↑s) : Filter.principal t = Filter.comap Subtype.val (Filter.principal (Subtype.val '' t))

@[simp]

theorem Filter.comap_pure {α : Type u} {β : Type v} {m : α → β} {b : β} : Filter.comap m (pure b) = Filter.principal (m ⁻¹' {b})

theorem Filter.map_le_iff_le_comap {α : Type u} {β : Type v} {f : Filter α} {g : Filter β} {m : α → β} : Filter.map m f ≤ g ↔ f ≤ Filter.comap m g

theorem Filter.gc_map_comap {α : Type u} {β : Type v} (m : α → β) : GaloisConnection (Filter.map m) (Filter.comap m)

theorem Filter.comap_le_iff_le_kernMap {α : Type u} {β : Type v} {f : Filter α} {g : Filter β} {m : α → β} : Filter.comap m g ≤ f ↔ g ≤ Filter.kernMap m f

theorem Filter.gc_comap_kernMap {α : Type u} {β : Type v} (m : α → β) : GaloisConnection (Filter.comap m) (Filter.kernMap m)

theorem Filter.kernMap_principal {α : Type u} {β : Type v} {m : α → β} {s : Set α} : Filter.kernMap m (Filter.principal s) = Filter.principal (Set.kernImage m s)

theorem Filter.map_mono {α : Type u} {β : Type v} {m : α → β} : Monotone (Filter.map m)

theorem Filter.comap_mono {α : Type u} {β : Type v} {m : α → β} : Monotone (Filter.comap m)

theorem GCongr.Filter.map_le_map {α : Type u} {β : Type v} {m : α → β} {F : Filter α} {G : Filter α} (h : F ≤ G) : Filter.map m F ≤ Filter.map m G

Temporary lemma that we can tag with gcongr

theorem GCongr.Filter.comap_le_comap {α : Type u} {β : Type v} {m : α → β} {F : Filter β} {G : Filter β} (h : F ≤ G) : Filter.comap m F ≤ Filter.comap m G

Temporary lemma that we can tag with gcongr

@[simp]

theorem Filter.map_bot {α : Type u} {β : Type v} {m : α → β} : Filter.map m ⊥ = ⊥

@[simp]

theorem Filter.map_sup {α : Type u} {β : Type v} {f₁ : Filter α} {f₂ : Filter α} {m : α → β} : Filter.map m (f₁ ⊔ f₂) = Filter.map m f₁ ⊔ Filter.map m f₂

@[simp]

theorem Filter.map_iSup {α : Type u} {β : Type v} {ι : Sort x} {m : α → β} {f : ι → Filter α} : Filter.map m (⨆ (i : ι), f i) = ⨆ (i : ι), Filter.map m (f i)

@[simp]

theorem Filter.map_top {α : Type u} {β : Type v} (f : α → β) : Filter.map f ⊤ = Filter.principal (Set.range f)

@[simp]

theorem Filter.comap_top {α : Type u} {β : Type v} {m : α → β} : Filter.comap m ⊤ = ⊤

@[simp]

theorem Filter.comap_inf {α : Type u} {β : Type v} {g₁ : Filter β} {g₂ : Filter β} {m : α → β} : Filter.comap m (g₁ ⊓ g₂) = Filter.comap m g₁ ⊓ Filter.comap m g₂

@[simp]

theorem Filter.comap_iInf {α : Type u} {β : Type v} {ι : Sort x} {m : α → β} {f : ι → Filter β} : Filter.comap m (⨅ (i : ι), f i) = ⨅ (i : ι), Filter.comap m (f i)

theorem Filter.le_comap_top {α : Type u} {β : Type v} (f : α → β) (l : Filter α) : l ≤ Filter.comap f ⊤

theorem Filter.map_comap_le {α : Type u} {β : Type v} {g : Filter β} {m : α → β} : Filter.map m (Filter.comap m g) ≤ g

theorem Filter.le_comap_map {α : Type u} {β : Type v} {f : Filter α} {m : α → β} : f ≤ Filter.comap m (Filter.map m f)

@[simp]

theorem Filter.comap_bot {α : Type u} {β : Type v} {m : α → β} : Filter.comap m ⊥ = ⊥

theorem Filter.neBot_of_comap {α : Type u} {β : Type v} {g : Filter β} {m : α → β} (h : (Filter.comap m g).NeBot) : g.NeBot

theorem Filter.comap_inf_principal_range {α : Type u} {β : Type v} {g : Filter β} {m : α → β} : Filter.comap m (g ⊓ Filter.principal (Set.range m)) = Filter.comap m g

theorem Filter.disjoint_comap {α : Type u} {β : Type v} {g₁ : Filter β} {g₂ : Filter β} {m : α → β} (h : Disjoint g₁ g₂) : Disjoint (Filter.comap m g₁) (Filter.comap m g₂)

theorem Filter.comap_iSup {α : Type u} {β : Type v} {ι : Sort u_2} {f : ι → Filter β} {m : α → β} : Filter.comap m (iSup f) = ⨆ (i : ι), Filter.comap m (f i)

theorem Filter.comap_sSup {α : Type u} {β : Type v} {s : Set (Filter β)} {m : α → β} : Filter.comap m (sSup s) = ⨆ f ∈ s, Filter.comap m f

theorem Filter.comap_sup {α : Type u} {β : Type v} {g₁ : Filter β} {g₂ : Filter β} {m : α → β} : Filter.comap m (g₁ ⊔ g₂) = Filter.comap m g₁ ⊔ Filter.comap m g₂

theorem Filter.map_comap {α : Type u} {β : Type v} (f : Filter β) (m : α → β) : Filter.map m (Filter.comap m f) = f ⊓ Filter.principal (Set.range m)

theorem Filter.map_comap_setCoe_val {β : Type v} (f : Filter β) (s : Set β) : Filter.map Subtype.val (Filter.comap Subtype.val f) = f ⊓ Filter.principal s

theorem Filter.map_comap_of_mem {α : Type u} {β : Type v} {f : Filter β} {m : α → β} (hf : Set.range m ∈ f) : Filter.map m (Filter.comap m f) = f

instance Filter.canLift {α : Type u} {β : Type v} (c : β → α) (p : α → Prop) [CanLift α β c p] : CanLift (Filter α) (Filter β) (Filter.map c) fun (f : Filter α) => ∀ᶠ (x : α) in f, p x

Equations

• ⋯ = ⋯

theorem Filter.comap_le_comap_iff {α : Type u} {β : Type v} {f : Filter β} {g : Filter β} {m : α → β} (hf : Set.range m ∈ f) : Filter.comap m f ≤ Filter.comap m g ↔ f ≤ g

theorem Filter.map_comap_of_surjective {α : Type u} {β : Type v} {f : α → β} (hf : Function.Surjective f) (l : Filter β) : Filter.map f (Filter.comap f l) = l

theorem Filter.comap_injective {α : Type u} {β : Type v} {f : α → β} (hf : Function.Surjective f) : Function.Injective (Filter.comap f)

theorem Function.Surjective.filter_map_top {α : Type u} {β : Type v} {f : α → β} (hf : Function.Surjective f) : Filter.map f ⊤ = ⊤

theorem Filter.subtype_coe_map_comap {α : Type u} (s : Set α) (f : Filter α) : Filter.map Subtype.val (Filter.comap Subtype.val f) = f ⊓ Filter.principal s

theorem Filter.image_mem_of_mem_comap {α : Type u} {β : Type v} {f : Filter α} {c : β → α} (h : Set.range c ∈ f) {W : Set β} (W_in : W ∈ Filter.comap c f) : c '' W ∈ f

theorem Filter.image_coe_mem_of_mem_comap {α : Type u} {f : Filter α} {U : Set α} (h : U ∈ f) {W : Set ↑U} (W_in : W ∈ Filter.comap Subtype.val f) : Subtype.val '' W ∈ f

theorem Filter.comap_map {α : Type u} {β : Type v} {f : Filter α} {m : α → β} (h : Function.Injective m) : Filter.comap m (Filter.map m f) = f

theorem Filter.mem_comap_iff {α : Type u} {β : Type v} {f : Filter β} {m : α → β} (inj : Function.Injective m) (large : Set.range m ∈ f) {S : Set α} : S ∈ Filter.comap m f ↔ m '' S ∈ f

theorem Filter.map_le_map_iff_of_injOn {α : Type u} {β : Type v} {l₁ : Filter α} {l₂ : Filter α} {f : α → β} {s : Set α} (h₁ : s ∈ l₁) (h₂ : s ∈ l₂) (hinj : Set.InjOn f s) : Filter.map f l₁ ≤ Filter.map f l₂ ↔ l₁ ≤ l₂

theorem Filter.map_le_map_iff {α : Type u} {β : Type v} {f : Filter α} {g : Filter α} {m : α → β} (hm : Function.Injective m) : Filter.map m f ≤ Filter.map m g ↔ f ≤ g

theorem Filter.map_eq_map_iff_of_injOn {α : Type u} {β : Type v} {f : Filter α} {g : Filter α} {m : α → β} {s : Set α} (hsf : s ∈ f) (hsg : s ∈ g) (hm : Set.InjOn m s) : Filter.map m f = Filter.map m g ↔ f = g

theorem Filter.map_inj {α : Type u} {β : Type v} {f : Filter α} {g : Filter α} {m : α → β} (hm : Function.Injective m) : Filter.map m f = Filter.map m g ↔ f = g

theorem Filter.map_injective {α : Type u} {β : Type v} {m : α → β} (hm : Function.Injective m) : Function.Injective (Filter.map m)

theorem Filter.comap_neBot_iff {α : Type u} {β : Type v} {f : Filter β} {m : α → β} : (Filter.comap m f).NeBot ↔ ∀ t ∈ f, ∃ (a : α), m a ∈ t

theorem Filter.comap_neBot {α : Type u} {β : Type v} {f : Filter β} {m : α → β} (hm : ∀ t ∈ f, ∃ (a : α), m a ∈ t) : (Filter.comap m f).NeBot

theorem Filter.comap_neBot_iff_frequently {α : Type u} {β : Type v} {f : Filter β} {m : α → β} : (Filter.comap m f).NeBot ↔ ∃ᶠ (y : β) in f, y ∈ Set.range m

theorem Filter.comap_neBot_iff_compl_range {α : Type u} {β : Type v} {f : Filter β} {m : α → β} : (Filter.comap m f).NeBot ↔ (Set.range m)ᶜ ∉ f

theorem Filter.comap_eq_bot_iff_compl_range {α : Type u} {β : Type v} {f : Filter β} {m : α → β} : Filter.comap m f = ⊥ ↔ (Set.range m)ᶜ ∈ f

theorem Filter.comap_surjective_eq_bot {α : Type u} {β : Type v} {f : Filter β} {m : α → β} (hm : Function.Surjective m) : Filter.comap m f = ⊥ ↔ f = ⊥

theorem Filter.disjoint_comap_iff {α : Type u} {β : Type v} {g₁ : Filter β} {g₂ : Filter β} {m : α → β} (h : Function.Surjective m) : Disjoint (Filter.comap m g₁) (Filter.comap m g₂) ↔ Disjoint g₁ g₂

theorem Filter.NeBot.comap_of_range_mem {α : Type u} {β : Type v} {f : Filter β} {m : α → β} : f.NeBot → Set.range m ∈ f → (Filter.comap m f).NeBot

@[simp]

theorem Filter.comap_fst_neBot_iff {α : Type u} {β : Type v} {f : Filter α} : (Filter.comap Prod.fst f).NeBot ↔ f.NeBot ∧ Nonempty β

theorem Filter.comap_fst_neBot {α : Type u} {β : Type v} [Nonempty β] {f : Filter α} [f.NeBot] : (Filter.comap Prod.fst f).NeBot

@[simp]

theorem Filter.comap_snd_neBot_iff {α : Type u} {β : Type v} {f : Filter β} : (Filter.comap Prod.snd f).NeBot ↔ Nonempty α ∧ f.NeBot

theorem Filter.comap_snd_neBot {α : Type u} {β : Type v} [Nonempty α] {f : Filter β} [f.NeBot] : (Filter.comap Prod.snd f).NeBot

theorem Filter.comap_eval_neBot_iff' {ι : Type u_2} {α : ι → Type u_3} {i : ι} {f : Filter (α i)} : (Filter.comap (Function.eval i) f).NeBot ↔ (∀ (j : ι), Nonempty (α j)) ∧ f.NeBot

@[simp]

theorem Filter.comap_eval_neBot_iff {ι : Type u_2} {α : ι → Type u_3} [∀ (j : ι), Nonempty (α j)] {i : ι} {f : Filter (α i)} : (Filter.comap (Function.eval i) f).NeBot ↔ f.NeBot

theorem Filter.comap_eval_neBot {ι : Type u_2} {α : ι → Type u_3} [∀ (j : ι), Nonempty (α j)] (i : ι) (f : Filter (α i)) [f.NeBot] : (Filter.comap (Function.eval i) f).NeBot

theorem Filter.comap_coe_neBot_of_le_principal {γ : Type w} {s : Set γ} {l : Filter γ} [h : l.NeBot] (h' : l ≤ Filter.principal s) : (Filter.comap Subtype.val l).NeBot

theorem Filter.NeBot.comap_of_surj {α : Type u} {β : Type v} {f : Filter β} {m : α → β} (hf : f.NeBot) (hm : Function.Surjective m) : (Filter.comap m f).NeBot

theorem Filter.NeBot.comap_of_image_mem {α : Type u} {β : Type v} {f : Filter β} {m : α → β} (hf : f.NeBot) {s : Set α} (hs : m '' s ∈ f) : (Filter.comap m f).NeBot

@[simp]

theorem Filter.map_eq_bot_iff {α : Type u} {β : Type v} {f : Filter α} {m : α → β} : Filter.map m f = ⊥ ↔ f = ⊥

theorem Filter.map_neBot_iff {α : Type u} {β : Type v} (f : α → β) {F : Filter α} : (Filter.map f F).NeBot ↔ F.NeBot

theorem Filter.NeBot.map {α : Type u} {β : Type v} {f : Filter α} (hf : f.NeBot) (m : α → β) : (Filter.map m f).NeBot

theorem Filter.NeBot.of_map {α : Type u} {β : Type v} {f : Filter α} {m : α → β} : (Filter.map m f).NeBot → f.NeBot

instance Filter.map_neBot {α : Type u} {β : Type v} {f : Filter α} {m : α → β} [hf : f.NeBot] : (Filter.map m f).NeBot

Equations

• ⋯ = ⋯

theorem Filter.sInter_comap_sets {α : Type u} {β : Type v} (f : α → β) (F : Filter β) : ⋂₀ (Filter.comap f F).sets = ⋂ U ∈ F, f ⁻¹' U

theorem Filter.map_iInf_le {α : Type u} {β : Type v} {ι : Sort x} {f : ι → Filter α} {m : α → β} : Filter.map m (iInf f) ≤ ⨅ (i : ι), Filter.map m (f i)

theorem Filter.map_iInf_eq {α : Type u} {β : Type v} {ι : Sort x} {f : ι → Filter α} {m : α → β} (hf : Directed (fun (x1 x2 : Filter α) => x1 ≥ x2) f) [Nonempty ι] : Filter.map m (iInf f) = ⨅ (i : ι), Filter.map m (f i)

theorem Filter.map_biInf_eq {α : Type u} {β : Type v} {ι : Type w} {f : ι → Filter α} {m : α → β} {p : ι → Prop} (h : DirectedOn (f ⁻¹'o fun (x1 x2 : Filter α) => x1 ≥ x2) {x : ι | p x}) (ne : ∃ (i : ι), p i) : Filter.map m (⨅ (i : ι), ⨅ (_ : p i), f i) = ⨅ (i : ι), ⨅ (_ : p i), Filter.map m (f i)

theorem Filter.map_inf_le {α : Type u} {β : Type v} {f : Filter α} {g : Filter α} {m : α → β} : Filter.map m (f ⊓ g) ≤ Filter.map m f ⊓ Filter.map m g

theorem Filter.map_inf {α : Type u} {β : Type v} {f : Filter α} {g : Filter α} {m : α → β} (h : Function.Injective m) : Filter.map m (f ⊓ g) = Filter.map m f ⊓ Filter.map m g

theorem Filter.map_inf' {α : Type u} {β : Type v} {f : Filter α} {g : Filter α} {m : α → β} {t : Set α} (htf : t ∈ f) (htg : t ∈ g) (h : Set.InjOn m t) : Filter.map m (f ⊓ g) = Filter.map m f ⊓ Filter.map m g

theorem Filter.disjoint_of_map {α : Type u_2} {β : Type u_3} {F : Filter α} {G : Filter α} {f : α → β} (h : Disjoint (Filter.map f F) (Filter.map f G)) : Disjoint F G

theorem Filter.disjoint_map {α : Type u} {β : Type v} {m : α → β} (hm : Function.Injective m) {f₁ : Filter α} {f₂ : Filter α} : Disjoint (Filter.map m f₁) (Filter.map m f₂) ↔ Disjoint f₁ f₂

theorem Filter.map_equiv_symm {α : Type u} {β : Type v} (e : α ≃ β) (f : Filter β) : Filter.map (⇑e.symm) f = Filter.comap (⇑e) f

theorem Filter.map_eq_comap_of_inverse {α : Type u} {β : Type v} {f : Filter α} {m : α → β} {n : β → α} (h₁ : m ∘ n = id) (h₂ : n ∘ m = id) : Filter.map m f = Filter.comap n f

theorem Filter.comap_equiv_symm {α : Type u} {β : Type v} (e : α ≃ β) (f : Filter α) : Filter.comap (⇑e.symm) f = Filter.map (⇑e) f

theorem Filter.map_swap_eq_comap_swap {α : Type u} {β : Type v} {f : Filter (α × β)} : Prod.swap <$> f = Filter.comap Prod.swap f

theorem Filter.map_swap4_eq_comap {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_1} {f : Filter ((α × β) × γ × δ)} : Filter.map (fun (p : (α × β) × γ × δ) => ((p.1.1, p.2.1), p.1.2, p.2.2)) f = Filter.comap (fun (p : (α × γ) × β × δ) => ((p.1.1, p.2.1), p.1.2, p.2.2)) f

A useful lemma when dealing with uniformities.

theorem Filter.le_map {α : Type u} {β : Type v} {f : Filter α} {m : α → β} {g : Filter β} (h : ∀ s ∈ f, m '' s ∈ g) : g ≤ Filter.map m f

theorem Filter.le_map_iff {α : Type u} {β : Type v} {f : Filter α} {m : α → β} {g : Filter β} : g ≤ Filter.map m f ↔ ∀ s ∈ f, m '' s ∈ g

theorem Filter.push_pull {α : Type u} {β : Type v} (f : α → β) (F : Filter α) (G : Filter β) : Filter.map f (F ⊓ Filter.comap f G) = Filter.map f F ⊓ G

theorem Filter.push_pull' {α : Type u} {β : Type v} (f : α → β) (F : Filter α) (G : Filter β) : Filter.map f (Filter.comap f G ⊓ F) = G ⊓ Filter.map f F

theorem Filter.disjoint_comap_iff_map {α : Type u} {β : Type v} {f : α → β} {F : Filter α} {G : Filter β} : Disjoint F (Filter.comap f G) ↔ Disjoint (Filter.map f F) G

theorem Filter.disjoint_comap_iff_map' {α : Type u} {β : Type v} {f : α → β} {F : Filter α} {G : Filter β} : Disjoint (Filter.comap f G) F ↔ Disjoint G (Filter.map f F)

theorem Filter.neBot_inf_comap_iff_map {α : Type u} {β : Type v} {f : α → β} {F : Filter α} {G : Filter β} : (F ⊓ Filter.comap f G).NeBot ↔ (Filter.map f F ⊓ G).NeBot

theorem Filter.neBot_inf_comap_iff_map' {α : Type u} {β : Type v} {f : α → β} {F : Filter α} {G : Filter β} : (Filter.comap f G ⊓ F).NeBot ↔ (G ⊓ Filter.map f F).NeBot

theorem Filter.comap_inf_principal_neBot_of_image_mem {α : Type u} {β : Type v} {f : Filter β} {m : α → β} (hf : f.NeBot) {s : Set α} (hs : m '' s ∈ f) : (Filter.comap m f ⊓ Filter.principal s).NeBot

theorem Filter.principal_eq_map_coe_top {α : Type u} (s : Set α) : Filter.principal s = Filter.map Subtype.val ⊤

theorem Filter.inf_principal_eq_bot_iff_comap {α : Type u} {F : Filter α} {s : Set α} : F ⊓ Filter.principal s = ⊥ ↔ Filter.comap Subtype.val F = ⊥

theorem Filter.singleton_mem_pure {α : Type u} {a : α} : {a} ∈ pure a

theorem Filter.pure_injective {α : Type u} : Function.Injective pure

instance Filter.pure_neBot {α : Type u} {a : α} : (pure a).NeBot

Equations

• ⋯ = ⋯

@[simp]

theorem Filter.le_pure_iff {α : Type u} {f : Filter α} {a : α} : f ≤ pure a ↔ {a} ∈ f

theorem Filter.mem_seq_def {α : Type u} {β : Type v} {f : Filter (α → β)} {g : Filter α} {s : Set β} : s ∈ f.seq g ↔ ∃ u ∈ f, ∃ t ∈ g, ∀ x ∈ u, ∀ y ∈ t, x y ∈ s

theorem Filter.mem_seq_iff {α : Type u} {β : Type v} {f : Filter (α → β)} {g : Filter α} {s : Set β} : s ∈ f.seq g ↔ ∃ u ∈ f, ∃ t ∈ g, u.seq t ⊆ s

theorem Filter.mem_map_seq_iff {α : Type u} {β : Type v} {γ : Type w} {f : Filter α} {g : Filter β} {m : α → β → γ} {s : Set γ} : s ∈ (Filter.map m f).seq g ↔ ∃ (t : Set β) (u : Set α), t ∈ g ∧ u ∈ f ∧ ∀ x ∈ u, ∀ y ∈ t, m x y ∈ s

theorem Filter.seq_mem_seq {α : Type u} {β : Type v} {f : Filter (α → β)} {g : Filter α} {s : Set (α → β)} {t : Set α} (hs : s ∈ f) (ht : t ∈ g) : s.seq t ∈ f.seq g

theorem Filter.le_seq {α : Type u} {β : Type v} {f : Filter (α → β)} {g : Filter α} {h : Filter β} (hh : ∀ t ∈ f, ∀ u ∈ g, t.seq u ∈ h) : h ≤ f.seq g

theorem Filter.seq_mono {α : Type u} {β : Type v} {f₁ : Filter (α → β)} {f₂ : Filter (α → β)} {g₁ : Filter α} {g₂ : Filter α} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : f₁.seq g₁ ≤ f₂.seq g₂

@[simp]

theorem Filter.pure_seq_eq_map {α : Type u} {β : Type v} (g : α → β) (f : Filter α) : (pure g).seq f = Filter.map g f

@[simp]

theorem Filter.seq_pure {α : Type u} {β : Type v} (f : Filter (α → β)) (a : α) : f.seq (pure a) = Filter.map (fun (g : α → β) => g a) f

@[simp]

theorem Filter.seq_assoc {α : Type u} {β : Type v} {γ : Type w} (x : Filter α) (g : Filter (α → β)) (h : Filter (β → γ)) : h.seq (g.seq x) = ((Filter.map (fun (x1 : β → γ) (x2 : α → β) => x1 ∘ x2) h).seq g).seq x

theorem Filter.prod_map_seq_comm {α : Type u} {β : Type v} (f : Filter α) (g : Filter β) : (Filter.map Prod.mk f).seq g = (Filter.map (fun (b : β) (a : α) => (a, b)) g).seq f

theorem Filter.seq_eq_filter_seq {α : Type u} {β : Type u} (f : Filter (α → β)) (g : Filter α) : f <*> g = f.seq g

instance Filter.instLawfulApplicative : LawfulApplicative Filter

Equations

• Filter.instLawfulApplicative = ⋯

instance Filter.instCommApplicative : CommApplicative Filter

Equations

• Filter.instCommApplicative = ⋯

bind equations

@[simp]

theorem Filter.eventually_bind {α : Type u} {β : Type v} {f : Filter α} {m : α → Filter β} {p : β → Prop} : (∀ᶠ (y : β) in f.bind m, p y) ↔ ∀ᶠ (x : α) in f, ∀ᶠ (y : β) in m x, p y

@[simp]

theorem Filter.eventuallyEq_bind {α : Type u} {β : Type v} {γ : Type w} {f : Filter α} {m : α → Filter β} {g₁ : β → γ} {g₂ : β → γ} : g₁ =ᶠ[f.bind m] g₂ ↔ ∀ᶠ (x : α) in f, g₁ =ᶠ[m x] g₂

@[simp]

theorem Filter.eventuallyLE_bind {α : Type u} {β : Type v} {γ : Type w} [LE γ] {f : Filter α} {m : α → Filter β} {g₁ : β → γ} {g₂ : β → γ} : g₁ ≤ᶠ[f.bind m] g₂ ↔ ∀ᶠ (x : α) in f, g₁ ≤ᶠ[m x] g₂

theorem Filter.mem_bind' {α : Type u} {β : Type v} {s : Set β} {f : Filter α} {m : α → Filter β} : s ∈ f.bind m ↔ {a : α | s ∈ m a} ∈ f

@[simp]

theorem Filter.mem_bind {α : Type u} {β : Type v} {s : Set β} {f : Filter α} {m : α → Filter β} : s ∈ f.bind m ↔ ∃ t ∈ f, ∀ x ∈ t, s ∈ m x

theorem Filter.bind_le {α : Type u} {β : Type v} {f : Filter α} {g : α → Filter β} {l : Filter β} (h : ∀ᶠ (x : α) in f, g x ≤ l) : f.bind g ≤ l

theorem Filter.bind_mono {α : Type u} {β : Type v} {f₁ : Filter α} {f₂ : Filter α} {g₁ : α → Filter β} {g₂ : α → Filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ᶠ[f₁] g₂) : f₁.bind g₁ ≤ f₂.bind g₂

theorem Filter.bind_inf_principal {α : Type u} {β : Type v} {f : Filter α} {g : α → Filter β} {s : Set β} : (f.bind fun (x : α) => g x ⊓ Filter.principal s) = f.bind g ⊓ Filter.principal s

theorem Filter.sup_bind {α : Type u} {β : Type v} {f : Filter α} {g : Filter α} {h : α → Filter β} : (f ⊔ g).bind h = f.bind h ⊔ g.bind h

theorem Filter.principal_bind {α : Type u} {β : Type v} {s : Set α} {f : α → Filter β} : (Filter.principal s).bind f = ⨆ x ∈ s, f x

theorem Set.EqOn.eventuallyEq {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} {g : α → β} (h : Set.EqOn f g s) : f =ᶠ[Filter.principal s] g

theorem Set.EqOn.eventuallyEq_of_mem {α : Type u_1} {β : Type u_2} {s : Set α} {l : Filter α} {f : α → β} {g : α → β} (h : Set.EqOn f g s) (hl : s ∈ l) : f =ᶠ[l] g

theorem HasSubset.Subset.eventuallyLE {α : Type u_1} {l : Filter α} {s : Set α} {t : Set α} (h : s ⊆ t) : s ≤ᶠ[l] t

theorem Filter.map_surjOn_Iic_iff_le_map {α : Type u_1} {β : Type u_2} {F : Filter α} {G : Filter β} {m : α → β} : Set.SurjOn (Filter.map m) (Set.Iic F) (Set.Iic G) ↔ G ≤ Filter.map m F

theorem Filter.map_surjOn_Iic_iff_surjOn {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β} : Set.SurjOn (Filter.map m) (Set.Iic (Filter.principal s)) (Set.Iic (Filter.principal t)) ↔ Set.SurjOn m s t

theorem Set.SurjOn.filter_map_Iic {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β} : Set.SurjOn m s t → Set.SurjOn (Filter.map m) (Set.Iic (Filter.principal s)) (Set.Iic (Filter.principal t))

Alias of the reverse direction of Filter.map_surjOn_Iic_iff_surjOn.

theorem Filter.filter_injOn_Iic_iff_injOn {α : Type u_1} {β : Type u_2} {s : Set α} {m : α → β} : Set.InjOn (Filter.map m) (Set.Iic (Filter.principal s)) ↔ Set.InjOn m s

theorem Set.InjOn.filter_map_Iic {α : Type u_1} {β : Type u_2} {s : Set α} {m : α → β} : Set.InjOn m s → Set.InjOn (Filter.map m) (Set.Iic (Filter.principal s))

Alias of the reverse direction of Filter.filter_injOn_Iic_iff_injOn.

theorem Filter.compl_mem_comk {α : Type u_1} {p : Set α → Prop} {he : p ∅} {hmono : ∀ (t : Set α), p t → ∀ s ⊆ t, p s} {hunion : ∀ (s : Set α), p s → ∀ (t : Set α), p t → p (s ∪ t)} {s : Set α} : sᶜ ∈ Filter.comk p he hmono hunion ↔ p s