mathlib documentation

order.zorn

def zorn.chain {α : Type u} (r : α → α → Prop) (c : 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} (hx : x c) (hy : y c) :
r 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} (H : zorn.chain r c) :
theorem zorn.chain_insert {α : Type u} {r : α → α → Prop} {c : set α} {a : α} (hc : zorn.chain r c) (ha : ∀ (b : α), b cb ar a b r b a) :
def zorn.super_chain {α : Type u} {r : α → α → Prop} (c₁ c₂ : set α) :
Prop

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

Equations
def zorn.is_max_chain {α : Type u} {r : α → α → Prop} (c : 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} {r : α → α → Prop} (c : set α) :
set 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} (h : ∃ (c' : set α), zorn.chain r c zorn.super_chain c c') :
theorem zorn.chain_succ {α : Type u} {r : α → α → Prop} {c : set α} (hc : zorn.chain r c) :
theorem zorn.super_of_not_max {α : Type u} {r : α → α → Prop} {c : set α} (hc₁ : zorn.chain r c) (hc₂ : ¬zorn.is_max_chain c) :
theorem zorn.succ_increasing {α : Type u} {r : α → α → Prop} {c : set α} :
inductive zorn.chain_closure {α : Type u} {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} (hc₁ : zorn.chain_closure c₁) (hc₂ : zorn.chain_closure c₂) :
c₁ c₂ c₂ c₁
theorem zorn.chain_closure_succ_fixpoint {α : Type u} {r : α → α → Prop} {c₁ c₂ : set α} (hc₁ : zorn.chain_closure c₁) (hc₂ : zorn.chain_closure c₂) (h_eq : zorn.succ_chain c₂ = c₂) :
c₁ c₂
theorem zorn.chain_closure_succ_fixpoint_iff {α : Type u} {r : α → α → Prop} {c : set α} (hc : zorn.chain_closure c) :
theorem zorn.chain_chain_closure {α : Type u} {r : α → α → Prop} {c : set α} (hc : zorn.chain_closure c) :
def zorn.max_chain {α : Type u} {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} (h : ∀ (c : set α), zorn.chain r c(∃ (ub : α), ∀ (a : α), a cr a ub)) (trans : ∀ {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.exists_maximal_of_nonempty_chains_bounded {α : Type u} {r : α → α → Prop} [nonempty α] (h : ∀ (c : set α), zorn.chain r cc.nonempty(∃ (ub : α), ∀ (a : α), a cr a ub)) (trans : ∀ {a b c : α}, r a br b cr a c) :
∃ (m : α), ∀ (a : α), r m ar a m

If every nonempty chain of a nonempty type has an upper bound, then there is a maximal element. (A variant of Zorn's lemma.)

theorem zorn.zorn_partial_order {α : Type u} [partial_order α] (h : ∀ (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_nonempty_partial_order {α : Type u} [partial_order α] [nonempty α] (h : ∀ (c : set α), zorn.chain has_le.le cc.nonempty(∃ (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) (hxs : 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)) (h : ∀ (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) (hx : 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} (hrc : zorn.chain r c) :
zorn.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} (h : zorn.chain (f ⁻¹'o r) c) :
directed r (λ (x : {a // a c}), f x)