This repository has been archived by the owner on Jul 24, 2024. It is now read-only.
-
Notifications
You must be signed in to change notification settings - Fork 298
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
feat(order/filter/ultrafilter): an ultrafilter is an atom in the latt…
…ice of filters (#17116) * add `filter.nontrivial`, `filter.Inf_ne_bot_of_directed`, and `filter.Inf_ne_bot_of_directed'`; * add `is_atom_iff` and `is_coatom_iff`; * generalize `ultrafilter.exists_le` to any partial order; * prove that a filter is an ultrafilter if and only if it is an atom in the lattice of filters.
- Loading branch information
Showing
4 changed files
with
88 additions
and
34 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,39 @@ | ||
/- | ||
Copyright (c) 2022 Yury Kudryashov. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Yury Kudryashov | ||
-/ | ||
import order.zorn | ||
import order.atoms | ||
|
||
/-! | ||
# Zorn lemma for (co)atoms | ||
In this file we use Zorn's lemma to prove that a partial order is atomic if every nonempty chain | ||
`c`, `⊥ ∉ c`, has a lower bound not equal to `⊥`. We also prove the order dual version of this | ||
statement. | ||
-/ | ||
|
||
open set | ||
|
||
/-- **Zorn's lemma**: A partial order is coatomic if every nonempty chain `c`, `⊤ ∉ c`, has an upper | ||
bound not equal to `⊤`. -/ | ||
lemma is_coatomic.of_is_chain_bounded {α : Type*} [partial_order α] [order_top α] | ||
(h : ∀ c : set α, is_chain (≤) c → c.nonempty → ⊤ ∉ c → ∃ x ≠ ⊤, x ∈ upper_bounds c) : | ||
is_coatomic α := | ||
begin | ||
refine ⟨λ x, le_top.eq_or_lt.imp_right $ λ hx, _⟩, | ||
rcases zorn_nonempty_partial_order₀ (Ico x ⊤) (λ c hxc hc y hy, _) x (left_mem_Ico.2 hx) | ||
with ⟨y, ⟨hxy, hy⟩, -, hy'⟩, | ||
{ refine ⟨y, ⟨hy.ne, λ z hyz, le_top.eq_or_lt.resolve_right $ λ hz, _⟩, hxy⟩, | ||
exact hyz.ne' (hy' z ⟨hxy.trans hyz.le, hz⟩ hyz.le) }, | ||
{ rcases h c hc ⟨y, hy⟩ (λ h, (hxc h).2.ne rfl) with ⟨z, hz, hcz⟩, | ||
exact ⟨z, ⟨le_trans (hxc hy).1 (hcz hy), hz.lt_top⟩, hcz⟩ } | ||
end | ||
|
||
/-- **Zorn's lemma**: A partial order is atomic if every nonempty chain `c`, `⊥ ∉ c`, has an lower | ||
bound not equal to `⊥`. -/ | ||
lemma is_atomic.of_is_chain_bounded {α : Type*} [partial_order α] [order_bot α] | ||
(h : ∀ c : set α, is_chain (≤) c → c.nonempty → ⊥ ∉ c → ∃ x ≠ ⊥, x ∈ lower_bounds c) : | ||
is_atomic α := | ||
is_coatomic_dual_iff_is_atomic.mp $ is_coatomic.of_is_chain_bounded $ λ c hc, h c hc.symm |