mathlib documentation

order.zorn

def zorn.chain {α : Type u} :
(α → α → Prop)set α → Prop

A chain is a subset c satisfying x ≺ y ∨ x = y ∨ y ≺ x for all x y ∈ c.

Equations
theorem zorn.chain.total_of_refl {α : Type u} {r : α → α → Prop} [is_refl α r] {c : set α} (H : zorn.chain r c) {x y : α} :
x cy cr x y r y x

theorem zorn.chain.mono {α : Type u} {r : α → α → Prop} {c c' : set α} :
c' czorn.chain r czorn.chain r c'

theorem zorn.chain.directed_on {α : Type u} {r : α → α → Prop} [is_refl α r] {c : set α} :

theorem zorn.chain_insert {α : Type u} {r : α → α → Prop} {c : set α} {a : α} :
zorn.chain r c(∀ (b : α), b cb ar a b r b a)zorn.chain r (insert a c)

def zorn.super_chain {α : Type u} :
Π {r : α → α → Prop}, set αset α → Prop

super_chain c₁ c₂ means that c₂ is a chain that strictly includesc₁`.

Equations
def zorn.is_max_chain {α : Type u} :
Π {r : α → α → Prop}, set α → Prop

A chain c is a maximal chain if there does not exists a chain strictly including c.

Equations
def zorn.succ_chain {α : Type u} :
Π {r : α → α → Prop}, set αset α

Given a set c, if there exists a chain c' strictly including c, then succ_chain c is one of these chains. Otherwise it is c.

Equations
theorem zorn.succ_spec {α : Type u} {r : α → α → Prop} {c : set α} :

theorem zorn.chain_succ {α : Type u} {r : α → α → Prop} {c : set α} :

theorem zorn.super_of_not_max {α : Type u} {r : α → α → Prop} {c : set α} :

theorem zorn.succ_increasing {α : Type u} {r : α → α → Prop} {c : set α} :

inductive zorn.chain_closure {α : Type u} :
Π {r : α → α → Prop}, set α → Prop

Set of sets reachable from using succ_chain and ⋃₀.

theorem zorn.chain_closure_empty {α : Type u} {r : α → α → Prop} :

theorem zorn.chain_closure_closure {α : Type u} {r : α → α → Prop} :

theorem zorn.chain_closure_total {α : Type u} {r : α → α → Prop} {c₁ c₂ : set α} :
zorn.chain_closure c₁zorn.chain_closure c₂c₁ c₂ c₂ c₁

theorem zorn.chain_closure_succ_fixpoint {α : Type u} {r : α → α → Prop} {c₁ c₂ : set α} :
zorn.chain_closure c₁zorn.chain_closure c₂zorn.succ_chain c₂ = c₂c₁ c₂

theorem zorn.chain_closure_succ_fixpoint_iff {α : Type u} {r : α → α → Prop} {c : set α} :

theorem zorn.chain_chain_closure {α : Type u} {r : α → α → Prop} {c : set α} :

def zorn.max_chain {α : Type u} :
Π {r : α → α → Prop}, set α

max_chain is the union of all sets in the chain closure.

Equations
theorem zorn.max_chain_spec {α : Type u} {r : α → α → Prop} :

Hausdorff's maximality principle

There exists a maximal totally ordered subset of α. Note that we do not require α to be partially ordered by r.

theorem zorn.exists_maximal_of_chains_bounded {α : Type u} {r : α → α → Prop} :
(∀ (c : set α), zorn.chain r c(∃ (ub : α), ∀ (a : α), a cr a ub))(∀ {a b c : α}, r a br b cr a c)(∃ (m : α), ∀ (a : α), r m ar a m)

Zorn's lemma

If every chain has an upper bound, then there is a maximal element

theorem zorn.zorn_partial_order {α : Type u} [partial_order α] :
(∀ (c : set α), zorn.chain has_le.le c(∃ (ub : α), ∀ (a : α), a ca ub))(∃ (m : α), ∀ (a : α), m aa = m)

theorem zorn.zorn_partial_order₀ {α : Type u} [partial_order α] (s : set α) (ih : ∀ (c : set α), c szorn.chain has_le.le c∀ (y : α), y c(∃ (ub : α) (H : ub s), ∀ (z : α), z cz ub)) (x : α) :
x s(∃ (m : α) (H : m s), x m ∀ (z : α), z sm zz = m)

theorem zorn.zorn_subset {α : Type u} (S : set (set α)) :
(∀ (c : set (set α)), c Szorn.chain has_subset.subset c(∃ (ub : set α) (H : ub S), ∀ (s : set α), s cs ub))(∃ (m : set α) (H : m S), ∀ (a : set α), a Sm aa = m)

theorem zorn.zorn_subset₀ {α : Type u} (S : set (set α)) (H : ∀ (c : set (set α)), c Szorn.chain has_subset.subset cc.nonempty(∃ (ub : set α) (H : ub S), ∀ (s : set α), s cs ub)) (x : set α) :
x S(∃ (m : set α) (H : m S), x m ∀ (a : set α), a Sm aa = m)

theorem zorn.chain.total {α : Type u} [preorder α] {c : set α} (H : zorn.chain has_le.le c) {x y : α} :
x cy cx y y x

theorem zorn.chain.image {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) (f : α → β) (h : ∀ (x y : α), r x ys (f x) (f y)) {c : set α} :
zorn.chain r czorn.chain s (f '' c)

theorem directed_of_chain {α : Type u_1} {β : Type u_2} {r : β → β → Prop} [is_refl β r] {f : α → β} {c : set α} :
zorn.chain (f ⁻¹'o r) cdirected r (λ (x : {a // a c}), f x)