mathlib documentation

order.filter.ultrafilter

Ultrafilters #

An ultrafilter is a minimal (maximal in the set order) proper filter. In this file we define

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structure ultrafilter (α : Type u_1) :
Type u_1

An ultrafilter is a minimal (maximal in the set order) proper filter.

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@[protected, instance]
def ultrafilter.filter.has_coe_t {α : Type u} :
has_coe_t (ultrafilter α) (filter α)
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@[protected, instance]
def ultrafilter.has_mem {α : Type u} :
has_mem (set α) (ultrafilter α)
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theorem ultrafilter.unique {α : Type u} (f : ultrafilter α) {g : filter α} (h : g ≤ ↑f) (hne : g.ne_bot . "apply_instance") :
g = ↑f
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@[protected, instance]
def ultrafilter.ne_bot {α : Type u} (f : ultrafilter α) :
↑f.ne_bot
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@[simp, norm_cast]
theorem ultrafilter.mem_coe {α : Type u} {f : ultrafilter α} {s : set α} :
s ∈ ↑f ↔ s ∈ f
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theorem ultrafilter.coe_injective {α : Type u} :
function.injective coe
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theorem ultrafilter.eq_of_le {α : Type u} {f g : ultrafilter α} (h : ↑f ≤ ↑g) :
f = g
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@[simp, norm_cast]
theorem ultrafilter.coe_le_coe {α : Type u} {f g : ultrafilter α} :
↑f ≤ ↑g ↔ f = g
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@[simp, norm_cast]
theorem ultrafilter.coe_inj {α : Type u} {f g : ultrafilter α} :
↑f = ↑g ↔ f = g
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@[ext]
theorem ultrafilter.ext {α : Type u} ⦃f g : ultrafilter α⦄ (h : ∀ (s : set α), s ∈ f ↔ s ∈ g) :
f = g
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theorem ultrafilter.le_of_inf_ne_bot {α : Type u} (f : ultrafilter α) {g : filter α} (hg : (↑f ⊓ g).ne_bot) :
↑f ≤ g
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theorem ultrafilter.le_of_inf_ne_bot' {α : Type u} (f : ultrafilter α) {g : filter α} (hg : (g ⊓ ↑f).ne_bot) :
↑f ≤ g
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@[simp]
theorem ultrafilter.compl_not_mem_iff {α : Type u} {f : ultrafilter α} {s : set α} :
sᶜ ∉ f ↔ s ∈ f
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@[simp]
theorem ultrafilter.frequently_iff_eventually {α : Type u} {f : ultrafilter α} {p : α → Prop} :
(∃ᶠ (x : α) in ↑f, p x) ↔ ∀ᶠ (x : α) in ↑f, p x
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theorem filter.frequently.eventually {α : Type u} {f : ultrafilter α} {p : α → Prop} :
(∃ᶠ (x : α) in ↑f, p x) → (∀ᶠ (x : α) in ↑f, p x)

Alias of frequently_iff_eventually.

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theorem ultrafilter.compl_mem_iff_not_mem {α : Type u} {f : ultrafilter α} {s : set α} :
sᶜ ∈ f ↔ s ∉ f
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theorem ultrafilter.diff_mem_iff {α : Type u} {s t : set α} (f : ultrafilter α) :
s \ t ∈ f ↔ s ∈ f ∧ t ∉ f
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def ultrafilter.of_compl_not_mem_iff {α : Type u} (f : filter α) (h : ∀ (s : set α), sᶜ ∉ f ↔ s ∈ f) :
ultrafilter α

If sᶜ ∉ f ↔ s ∈ f, then f is an ultrafilter. The other implication is given by ultrafilter.compl_not_mem_iff.

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theorem ultrafilter.nonempty_of_mem {α : Type u} {f : ultrafilter α} {s : set α} (hs : s ∈ f) :
s.nonempty
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theorem ultrafilter.ne_empty_of_mem {α : Type u} {f : ultrafilter α} {s : set α} (hs : s ∈ f) :
s ≠ ∅
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@[simp]
theorem ultrafilter.empty_not_mem {α : Type u} {f : ultrafilter α} :
∅ ∉ f
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theorem ultrafilter.mem_or_compl_mem {α : Type u} (f : ultrafilter α) (s : set α) :
s ∈ f ∨ sᶜ ∈ f
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@[protected]
theorem ultrafilter.em {α : Type u} (f : ultrafilter α) (p : α → Prop) :
(∀ᶠ (x : α) in ↑f, p x) ∨ ∀ᶠ (x : α) in ↑f, ¬p x
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theorem ultrafilter.eventually_or {α : Type u} {f : ultrafilter α} {p 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
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theorem ultrafilter.union_mem_iff {α : Type u} {f : ultrafilter α} {s t : set α} :
s ∪ t ∈ f ↔ s ∈ f ∨ t ∈ f
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theorem ultrafilter.eventually_not {α : Type u} {f : ultrafilter α} {p : α → Prop} :
(∀ᶠ (x : α) in ↑f, ¬p x) ↔ ¬∀ᶠ (x : α) in ↑f, p x
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theorem ultrafilter.eventually_imp {α : Type u} {f : ultrafilter α} {p 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)
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theorem ultrafilter.finite_sUnion_mem_iff {α : Type u} {f : ultrafilter α} {s : set (set α)} (hs : s.finite) :
⋃₀s ∈ f ↔ ∃ (t : set α) (H : t ∈ s), t ∈ f
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theorem ultrafilter.finite_bUnion_mem_iff {α : Type u} {β : Type v} {f : ultrafilter α} {is : set β} {s : β → set α} (his : is.finite) :
(⋃ (i : β) (H : i ∈ is), s i) ∈ f ↔ ∃ (i : β) (H : i ∈ is), s i ∈ f
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def ultrafilter.map {α : Type u} {β : Type v} (m : α → β) (f : ultrafilter α) :
ultrafilter β

Pushforward for ultrafilters.

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@[simp, norm_cast]
theorem ultrafilter.coe_map {α : Type u} {β : Type v} (m : α → β) (f : ultrafilter α) :
↑(ultrafilter.map m f) = filter.map m ↑f
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@[simp]
theorem ultrafilter.mem_map {α : Type u} {β : Type v} {m : α → β} {f : ultrafilter α} {s : set β} :
s ∈ ultrafilter.map m f ↔ m ⁻¹' s ∈ f
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def ultrafilter.comap {α : Type u} {β : Type v} {m : α → β} (u : ultrafilter β) (inj : function.injective m) (large : set.range m ∈ u) :
ultrafilter α

The pullback of an ultrafilter along an injection whose range is large with respect to the given ultrafilter.

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@[simp]
theorem ultrafilter.mem_comap {α : Type u} {β : Type v} {m : α → β} (u : ultrafilter β) (inj : function.injective m) (large : set.range m ∈ u) {s : set α} :
s ∈ u.comap inj large ↔ m '' s ∈ u
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@[simp]
theorem ultrafilter.coe_comap {α : Type u} {β : Type v} {m : α → β} (u : ultrafilter β) (inj : function.injective m) (large : set.range m ∈ u) :
↑(u.comap inj large) = filter.comap m ↑u
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@[protected, instance]
def ultrafilter.has_pure  :
has_pure ultrafilter

The principal ultrafilter associated to a point x.

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@[simp]
theorem ultrafilter.mem_pure {α : Type u} {a : α} {s : set α} :
s ∈ pure a ↔ a ∈ s
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@[protected, instance]
def ultrafilter.inhabited {α : Type u} [inhabited α] :
inhabited (ultrafilter α)
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@[protected, instance]
def ultrafilter.nonempty {α : Type u} [nonempty α] :
nonempty (ultrafilter α)
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theorem ultrafilter.eq_principal_of_finite_mem {α : Type u} {f : ultrafilter α} {s : set α} (h : s.finite) (h' : s ∈ f) :
∃ (x : α) (H : x ∈ s), ↑f = pure x
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def ultrafilter.bind {α : Type u} {β : Type v} (f : ultrafilter α) (m : α → ultrafilter β) :
ultrafilter β

Monadic bind for ultrafilters, coming from the one on filters defined in terms of map and join.

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@[protected, instance]
def ultrafilter.has_bind  :
has_bind ultrafilter
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@[protected, instance]
def ultrafilter.functor  :
functor ultrafilter
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@[protected, instance]
def ultrafilter.monad  :
monad ultrafilter
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@[protected, instance]
def ultrafilter.is_lawful_monad  :
is_lawful_monad ultrafilter
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theorem ultrafilter.exists_le {α : Type u} (f : filter α) [h : f.ne_bot] :
∃ (u : ultrafilter α), ↑u ≤ f

The ultrafilter lemma: Any proper filter is contained in an ultrafilter.

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theorem filter.exists_ultrafilter_le {α : Type u} (f : filter α) [h : f.ne_bot] :
∃ (u : ultrafilter α), ↑u ≤ f

Alias of exists_le.

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noncomputable def ultrafilter.of {α : Type u} (f : filter α) [f.ne_bot] :
ultrafilter α

Construct an ultrafilter extending a given filter. The ultrafilter lemma is the assertion that such a filter exists; we use the axiom of choice to pick one.

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theorem ultrafilter.of_le {α : Type u} (f : filter α) [f.ne_bot] :
↑(ultrafilter.of f) ≤ f
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theorem ultrafilter.of_coe {α : Type u} (f : ultrafilter α) :
ultrafilter.of ↑f = f
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theorem ultrafilter.exists_ultrafilter_of_finite_inter_nonempty {α : Type u} (S : set (set α)) (cond : ∀ (T : finset (set α)), ↑T ⊆ S → (⋂₀ ↑T).nonempty) :
∃ (F : ultrafilter α), S ⊆ F.to_filter.sets
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theorem filter.mem_iff_ultrafilter {α : Type u} {s : set α} {f : filter α} :
s ∈ f ↔ ∀ (g : ultrafilter α), ↑g ≤ f → s ∈ g
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theorem filter.le_iff_ultrafilter {α : Type u} {f₁ f₂ : filter α} :
f₁ ≤ f₂ ↔ ∀ (g : ultrafilter α), ↑g ≤ f₁ → ↑g ≤ f₂
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theorem filter.supr_ultrafilter_le_eq {α : Type u} (f : filter α) :
(⨆ (g : ultrafilter α) (hg : ↑g ≤ f), ↑g) = f

A filter equals the intersection of all the ultrafilters which contain it.

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theorem filter.tendsto_iff_ultrafilter {α : Type u} {β : Type v} (f : α → β) (l₁ : filter α) (l₂ : filter β) :
filter.tendsto f l₁ l₂ ↔ ∀ (g : ultrafilter α), ↑g ≤ l₁ → filter.tendsto f ↑g l₂

The tendsto relation can be checked on ultrafilters.

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theorem filter.exists_ultrafilter_iff {α : Type u} {f : filter α} :
(∃ (u : ultrafilter α), ↑u ≤ f) ↔ f.ne_bot
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theorem filter.forall_ne_bot_le_iff {α : Type u} {g : filter α} {p : filter α → Prop} (hp : monotone p) :
(∀ (f : filter α), f.ne_bot → f ≤ g → p f) ↔ ∀ (f : ultrafilter α), ↑f ≤ g → p ↑f
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noncomputable def filter.hyperfilter (α : Type u) [infinite α] :
ultrafilter α

The ultrafilter extending the cofinite filter.

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theorem filter.hyperfilter_le_cofinite {α : Type u} [infinite α] :
↑(filter.hyperfilter α) ≤ filter.cofinite
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@[simp]
theorem filter.bot_ne_hyperfilter {α : Type u} [infinite α] :
⊥ ≠ ↑(filter.hyperfilter α)
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theorem filter.nmem_hyperfilter_of_finite {α : Type u} [infinite α] {s : set α} (hf : s.finite) :
s ∉ filter.hyperfilter α
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theorem set.finite.nmem_hyperfilter {α : Type u} [infinite α] {s : set α} (hf : s.finite) :
s ∉ filter.hyperfilter α

Alias of nmem_hyperfilter_of_finite.

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theorem filter.compl_mem_hyperfilter_of_finite {α : Type u} [infinite α] {s : set α} (hf : s.finite) :
sᶜ ∈ filter.hyperfilter α
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theorem set.finite.compl_mem_hyperfilter {α : Type u} [infinite α] {s : set α} (hf : s.finite) :
sᶜ ∈ filter.hyperfilter α

Alias of compl_mem_hyperfilter_of_finite.

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theorem filter.mem_hyperfilter_of_finite_compl {α : Type u} [infinite α] {s : set α} (hf : sᶜ.finite) :
s ∈ filter.hyperfilter α
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theorem ultrafilter.comap_inf_principal_ne_bot_of_image_mem {α : Type u} {β : Type v} {m : α → β} {s : set α} {g : ultrafilter β} (h : m '' s ∈ g) :
(filter.comap m ↑g ⊓ 𝓟 s).ne_bot
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noncomputable def ultrafilter.of_comap_inf_principal {α : Type u} {β : Type v} {m : α → β} {s : set α} {g : ultrafilter β} (h : m '' s ∈ g) :
ultrafilter α

Ultrafilter extending the inf of a comapped ultrafilter and a principal ultrafilter.

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theorem ultrafilter.of_comap_inf_principal_mem {α : Type u} {β : Type v} {m : α → β} {s : set α} {g : ultrafilter β} (h : m '' s ∈ g) :
s ∈ ultrafilter.of_comap_inf_principal h
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theorem ultrafilter.of_comap_inf_principal_eq_of_map {α : Type u} {β : Type v} {m : α → β} {s : set α} {g : ultrafilter β} (h : m '' s ∈ g) :
ultrafilter.map m (ultrafilter.of_comap_inf_principal h) = g