Zorn's lemmas #

This file proves several formulations of Zorn's Lemma.

Variants #

The primary statement of Zorn's lemma is exists_maximal_of_chains_bounded. Then it is specialized to particular relations:

(≤) with zorn_partial_order
(⊆) with zorn_subset
(⊇) with zorn_superset

Lemma names carry modifiers:

₀: Quantifies over a set, as opposed to over a type.
_nonempty: Doesn't ask to prove that the empty chain is bounded and lets you give an element that will be smaller than the maximal element found (the maximal element is no smaller than any other element, but it can also be incomparable to some).

How-to #

This file comes across as confusing to those who haven't yet used it, so here is a detailed walkthrough:

Know what relation on which type/set you're looking for. See Variants above. You can discharge some conditions to Zorn's lemma directly using a _nonempty variant.
Write down the definition of your type/set, put a suffices : ∃ m, ∀ a, m ≺ a → a ≺ m, { ... }, (or whatever you actually need) followed by a apply some_version_of_zorn.
Fill in the details. This is where you start talking about chains.

A typical proof using Zorn could look like this

lemma zorny_lemma : zorny_statement :=
begin
  let s : set α := {x | whatever x},
  suffices : ∃ x ∈ s, ∀ y ∈ s, y ⊆ x → y = x, -- or with another operator
  { exact proof_post_zorn },
  apply zorn_subset, -- or another variant
  rintro c hcs hc,
  obtain rfl | hcnemp := c.eq_empty_or_nonempty, -- you might need to disjunct on c empty or not
  { exact ⟨edge_case_construction,
      proof_that_edge_case_construction_respects_whatever,
      proof_that_edge_case_construction_contains_all_stuff_in_c⟩ },
  exact ⟨construction,
    proof_that_construction_respects_whatever,
    proof_that_construction_contains_all_stuff_in_c⟩,
end

Notes #

Originally ported from Isabelle/HOL. The original file was written by Jacques D. Fleuriot, Tobias Nipkow, Christian Sternagel.

source

theorem exists_maximal_of_chains_bounded {α : Type u_1} {r : α → α → Prop} (h : ∀ (c : set α), is_chain r c → (∃ (ub : α), ∀ (a : α), a ∈ c → r a ub)) (trans : ∀ {a b c : α}, r a b → r b c → r a c) :

∃ (m : α), ∀ (a : α), r m a → r a m

Zorn's lemma

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

source

theorem exists_maximal_of_nonempty_chains_bounded {α : Type u_1} {r : α → α → Prop} [nonempty α] (h : ∀ (c : set α), is_chain r c → c.nonempty → (∃ (ub : α), ∀ (a : α), a ∈ c → r a ub)) (trans : ∀ {a b c : α}, r a b → r b c → r a c) :

∃ (m : α), ∀ (a : α), r m a → r a m

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

source

theorem zorn_preorder {α : Type u_1} [preorder α] (h : ∀ (c : set α), is_chain has_le.le c → bdd_above c) :

∃ (m : α), ∀ (a : α), m ≤ a → a ≤ m

source

theorem zorn_nonempty_preorder {α : Type u_1} [preorder α] [nonempty α] (h : ∀ (c : set α), is_chain has_le.le c → c.nonempty → bdd_above c) :

∃ (m : α), ∀ (a : α), m ≤ a → a ≤ m

source

theorem zorn_preorder₀ {α : Type u_1} [preorder α] (s : set α) (ih : ∀ (c : set α), c ⊆ s → is_chain has_le.le c → (∃ (ub : α) (H : ub ∈ s), ∀ (z : α), z ∈ c → z ≤ ub)) :

∃ (m : α) (H : m ∈ s), ∀ (z : α), z ∈ s → m ≤ z → z ≤ m

source

theorem zorn_nonempty_preorder₀ {α : Type u_1} [preorder α] (s : set α) (ih : ∀ (c : set α), c ⊆ s → is_chain has_le.le c → ∀ (y : α), y ∈ c → (∃ (ub : α) (H : ub ∈ s), ∀ (z : α), z ∈ c → z ≤ ub)) (x : α) (hxs : x ∈ s) :

∃ (m : α) (H : m ∈ s), x ≤ m ∧ ∀ (z : α), z ∈ s → m ≤ z → z ≤ m

source

theorem zorn_partial_order {α : Type u_1} [partial_order α] (h : ∀ (c : set α), is_chain has_le.le c → bdd_above c) :

∃ (m : α), ∀ (a : α), m ≤ a → a = m

source

theorem zorn_nonempty_partial_order {α : Type u_1} [partial_order α] [nonempty α] (h : ∀ (c : set α), is_chain has_le.le c → c.nonempty → bdd_above c) :

∃ (m : α), ∀ (a : α), m ≤ a → a = m

source

theorem zorn_partial_order₀ {α : Type u_1} [partial_order α] (s : set α) (ih : ∀ (c : set α), c ⊆ s → is_chain has_le.le c → (∃ (ub : α) (H : ub ∈ s), ∀ (z : α), z ∈ c → z ≤ ub)) :

∃ (m : α) (H : m ∈ s), ∀ (z : α), z ∈ s → m ≤ z → z = m

source

theorem zorn_nonempty_partial_order₀ {α : Type u_1} [partial_order α] (s : set α) (ih : ∀ (c : set α), c ⊆ s → is_chain has_le.le c → ∀ (y : α), y ∈ c → (∃ (ub : α) (H : ub ∈ s), ∀ (z : α), z ∈ c → z ≤ ub)) (x : α) (hxs : x ∈ s) :

∃ (m : α) (H : m ∈ s), x ≤ m ∧ ∀ (z : α), z ∈ s → m ≤ z → z = m

source

theorem zorn_subset {α : Type u_1} (S : set (set α)) (h : ∀ (c : set (set α)), c ⊆ S → is_chain has_subset.subset c → (∃ (ub : set α) (H : ub ∈ S), ∀ (s : set α), s ∈ c → s ⊆ ub)) :

∃ (m : set α) (H : m ∈ S), ∀ (a : set α), a ∈ S → m ⊆ a → a = m

source

theorem zorn_subset_nonempty {α : Type u_1} (S : set (set α)) (H : ∀ (c : set (set α)), c ⊆ S → is_chain has_subset.subset c → c.nonempty → (∃ (ub : set α) (H : ub ∈ S), ∀ (s : set α), s ∈ c → s ⊆ ub)) (x : set α) (hx : x ∈ S) :

∃ (m : set α) (H : m ∈ S), x ⊆ m ∧ ∀ (a : set α), a ∈ S → m ⊆ a → a = m

source

theorem zorn_superset {α : Type u_1} (S : set (set α)) (h : ∀ (c : set (set α)), c ⊆ S → is_chain has_subset.subset c → (∃ (lb : set α) (H : lb ∈ S), ∀ (s : set α), s ∈ c → lb ⊆ s)) :

∃ (m : set α) (H : m ∈ S), ∀ (a : set α), a ∈ S → a ⊆ m → a = m

source

theorem zorn_superset_nonempty {α : Type u_1} (S : set (set α)) (H : ∀ (c : set (set α)), c ⊆ S → is_chain has_subset.subset c → c.nonempty → (∃ (lb : set α) (H : lb ∈ S), ∀ (s : set α), s ∈ c → lb ⊆ s)) (x : set α) (hx : x ∈ S) :

∃ (m : set α) (H : m ∈ S), m ⊆ x ∧ ∀ (a : set α), a ∈ S → a ⊆ m → a = m

source

theorem is_chain.exists_max_chain {α : Type u_1} {r : α → α → Prop} {c : set α} (hc : is_chain r c) :

∃ (M : set α), is_max_chain r M ∧ c ⊆ M

Every chain is contained in a maximal chain. This generalizes Hausdorff's maximality principle.