mathlib documentation

order.directed

Directed indexed families and sets #

This file defines directed indexed families and directed sets. An indexed family/set is directed iff each pair of elements has a shared upper bound.

Main declarations #

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def directed {α : Type u} {ι : Sort w} (r : α → α → Prop) (f : ι → α) :
Prop

A family of elements of α is directed (with respect to a relation ≼ on α) if there is a member of the family ≼-above any pair in the family.

Equations
  • directed r f = ∀ (x y : ι), ∃ (z : ι), r (f x) (f z) ∧ r (f y) (f z)
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def directed_on {α : Type u} (r : α → α → Prop) (s : set α) :
Prop

A subset of α is directed if there is an element of the set ≼-above any pair of elements in the set.

Equations
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theorem directed_on_iff_directed {α : Type u} {r : α → α → Prop} {s : set α} :
directed_on r s ↔ directed r coe
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theorem directed_on.directed_coe {α : Type u} {r : α → α → Prop} {s : set α} :
directed_on r s → directed r coe

Alias of directed_on_iff_directed.

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theorem directed_on_image {α : Type u} {β : Type v} {r : α → α → Prop} {s : set β} {f : β → α} :
directed_on r (f '' s) ↔ directed_on (f ⁻¹'o r) s
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theorem directed_on.mono {α : Type u} {r : α → α → Prop} {s : set α} (h : directed_on r s) {r' : α → α → Prop} (H : ∀ {a b : α}, r a b → r' a b) :
directed_on r' s
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theorem directed_comp {α : Type u} {β : Type v} {r : α → α → Prop} {ι : Sort u_1} {f : ι → β} {g : β → α} :
directed r (g ∘ f) ↔ directed (g ⁻¹'o r) f
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theorem directed.mono {α : Type u} {r s : α → α → Prop} {ι : Sort u_1} {f : ι → α} (H : ∀ (a b : α), r a b → s a b) (h : directed r f) :
directed s f
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theorem directed.mono_comp {α : Type u} {β : Type v} (r : α → α → Prop) {ι : Sort u_1} {rb : β → β → Prop} {g : α → β} {f : ι → α} (hg : ∀ ⦃x y : α⦄, r x y → rb (g x) (g y)) (hf : directed r f) :
directed rb (g ∘ f)
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theorem directed_of_sup {α : Type u} {β : Type v} [semilattice_sup α] {f : α → β} {r : β → β → Prop} (H : ∀ ⦃i j : α⦄, i ≤ j → r (f i) (f j)) :
directed r f

A monotone function on a sup-semilattice is directed.

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theorem monotone.directed_le {α : Type u} {β : Type v} [semilattice_sup α] [preorder β] {f : α → β} :
monotone f → directed has_le.le f
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theorem directed_of_inf {α : Type u} {β : Type v} [semilattice_inf α] {r : β → β → Prop} {f : α → β} (hf : ∀ (a₁ a₂ : α), a₁ ≤ a₂ → r (f a₂) (f a₁)) :
directed r f

An antitone function on an inf-semilattice is directed.

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@[class]
structure directed_order (α : Type u) :
Type u
  • to_preorder : preorder α
  • directed : ∀ (i j : α), ∃ (k : α), i ≤ k ∧ j ≤ k

A preorder is a directed_order if for any two elements i, j there is an element k such that i ≤ k and j ≤ k.

Instances
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@[protected, instance]
def linear_order.to_directed_order (α : Type u_1) [linear_order α] :
directed_order α
Equations