feat(order/complete_well_founded, data/finset/lattice): introduce compact elements of a complete lattice and characterize compact lattices as those with all elements compact (#5825)

Provide a definition of compact elements. Prove the equivalence of two characterizations of compact elements. Add "all elements are compact" to the equivalent characterizations of compact lattices. Introduce an auxiliary lemma about finite sups and directed sets.

<!--
A reference for the two equivalent definitions of compact element can be found [here](https://ncatlab.org/nlab/show/compact+element+in+a+lattice)
-->

Author: cemulate

src/algebra/lie/basic.lean

@@ -8,7 +8,7 @@ import algebra.algebra.basic
 import linear_algebra.bilinear_form
 import linear_algebra.matrix
 import order.preorder_hom
-import order.complete_well_founded
+import order.compactly_generated
 import tactic.noncomm_ring
 
 /-!

src/data/finset/lattice.lean

@@ -119,6 +119,27 @@ begin
     { rw [finset.sup_insert, id.def], exact h h_subset.1 (h₁ htne h_subset.2), }, },
 end
 
+lemma sup_le_of_le_directed {α : Type*} [semilattice_sup_bot α] (s : set α)
+  (hs : s.nonempty) (hdir : directed_on (≤) s) (t : finset α):
+  (∀ x ∈ t, ∃ y ∈ s, x ≤ y) → ∃ x, x ∈ s ∧ t.sup id ≤ x :=
+begin
+  classical,
+  apply finset.induction_on t,
+  { simpa only [forall_prop_of_true, and_true, forall_prop_of_false, bot_le, not_false_iff,
+      sup_empty, forall_true_iff, not_mem_empty], },
+  { intros a r har ih h,
+    have incs : ↑r ⊆ ↑(insert a r), by { rw finset.coe_subset, apply finset.subset_insert, },
+    -- x ∈ s is above the sup of r
+    obtain ⟨x, ⟨hxs, hsx_sup⟩⟩ := ih (λ x hx, h x $ incs hx),
+    -- y ∈ s is above a
+    obtain ⟨y, hys, hay⟩ := h a (finset.mem_insert_self a r),
+    -- z ∈ s is above x and y
+    obtain ⟨z, hzs, ⟨hxz, hyz⟩⟩ := hdir x hxs y hys,
+    use [z, hzs],
+    rw [sup_insert, id.def, _root_.sup_le_iff],
+    exact ⟨le_trans hay hyz, le_trans hsx_sup hxz⟩, },
+end
+
 end sup
 
 lemma sup_eq_supr [complete_lattice β] (s : finset α) (f : α → β) : s.sup f = (⨆a∈s, f a) :=

src/order/compactly_generated.lean

@@ -0,0 +1,218 @@
+/-
+Copyright (c) 2021 Oliver Nash. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Oliver Nash
+-/
+import order.well_founded
+import order.order_iso_nat
+import data.set.finite
+import tactic.tfae
+
+/-!
+# Compactness properties for complete lattices
+
+For complete lattices, there are numerous equivalent ways to express the fact that the relation `>`
+is well-founded. In this file we define three especially-useful characterisations and provide
+proofs that they are indeed equivalent to well-foundedness.
+
+## Main definitions
+ * `complete_lattice.is_sup_closed_compact`
+ * `complete_lattice.is_Sup_finite_compact`
+ * `complete_lattice.is_compact_element`
+ * `complete_lattice.is_compactly_generated`
+
+## Main results
+The main result is that the following four conditions are equivalent for a complete lattice:
+ * `well_founded (>)`
+ * `complete_lattice.is_sup_closed_compact`
+ * `complete_lattice.is_Sup_finite_compact`
+ * `∀ k, complete_lattice.is_compact_element k`
+
+This is demonstrated by means of the following four lemmas:
+ * `complete_lattice.well_founded.is_Sup_finite_compact`
+ * `complete_lattice.is_Sup_finite_compact.is_sup_closed_compact`
+ * `complete_lattice.is_sup_closed_compact.well_founded`
+ * `complete_lattice.is_Sup_finite_compact_iff_all_elements_compact`
+
+ We also show well-founded lattices are compactly generated
+ (`complete_lattice.compactly_generated_of_well_founded`).
+
+## Tags
+
+complete lattice, well-founded, compact
+-/
+
+namespace complete_lattice
+
+variables (α : Type*) [complete_lattice α]
+
+/-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset
+contains its `Sup`. -/
+def is_sup_closed_compact : Prop :=
+  ∀ (s : set α) (h : s.nonempty), (∀ a b, a ∈ s → b ∈ s → a ⊔ b ∈ s) → (Sup s) ∈ s
+
+/-- A compactness property for a complete lattice is that any subset has a finite subset with the
+same `Sup`. -/
+def is_Sup_finite_compact : Prop :=
+∀ (s : set α), ∃ (t : finset α), ↑t ⊆ s ∧ Sup s = t.sup id
+
+/-- An element `k` of a complete lattice is said to be compact if any set with `Sup`
+above `k` has a finite subset with `Sup` above `k`.  Such an element is also called
+"finite" or "S-compact". -/
+def is_compact_element {α : Type*} [complete_lattice α] (k : α) :=
+∀ s : set α, k ≤ Sup s → ∃ t : finset α, ↑t ⊆ s ∧ k ≤ t.sup id
+
+/-- A complete lattice is said to be compactly generated if any
+element is the `Sup` of compact elements. -/
+def is_compactly_generated : Prop :=
+∀ (x : α), ∃ (s : set α), (∀ x ∈ s, is_compact_element x) ∧ Sup s = x
+
+/-- An element `k` is compact if and only if any directed set with `Sup` above
+`k` already got above `k` at some point in the set. -/
+theorem is_compact_element_iff_le_of_directed_Sup_le (k : α) :
+  is_compact_element k ↔
+  ∀ s : set α, s.nonempty → directed_on (≤) s → k ≤ Sup s → ∃ x : α, x ∈ s ∧ k ≤ x :=
+begin
+  classical,
+  split,
+  { by_cases hbot : k = ⊥,
+    -- Any nonempty directed set certainly has sup above ⊥
+    { rintros _ _ ⟨x, hx⟩ _ _, use x, by simp only [hx, hbot, bot_le, and_self], },
+    { intros hk s hne hdir hsup,
+      obtain ⟨t, ht⟩ := hk s hsup,
+      -- If t were empty, its sup would be ⊥, which is not above k ≠ ⊥.
+      have tne : t.nonempty,
+      { by_contradiction n,
+        rw [finset.nonempty_iff_ne_empty, not_not] at n,
+        simp only [n, true_and, set.empty_subset, finset.coe_empty,
+          finset.sup_empty, le_bot_iff] at ht,
+        exact absurd ht hbot, },
+      -- certainly every element of t is below something in s, since ↑t ⊆ s.
+      have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y, from λ x hxt, ⟨x, ht.left hxt, by refl⟩,
+      obtain ⟨x, ⟨hxs, hsupx⟩⟩ := finset.sup_le_of_le_directed s hne hdir t t_below_s,
+      exact ⟨x, ⟨hxs, le_trans k (t.sup id) x ht.right hsupx⟩⟩, }, },
+  { intros hk s hsup,
+    -- Consider the set of finite joins of elements of the (plain) set s.
+    let S : set α := { x | ∃ t : finset α, ↑t ⊆ s ∧ x = t.sup id },
+    -- S is directed, nonempty, and still has sup above k.
+    have dir_US : directed_on (≤) S,
+    { rintros x ⟨c, hc⟩ y ⟨d, hd⟩,
+      use x ⊔ y,
+      split,
+      { use c ∪ d,
+        split,
+        { simp only [hc.left, hd.left, set.union_subset_iff, finset.coe_union, and_self], },
+        { simp only [hc.right, hd.right, finset.sup_union], }, },
+      simp only [and_self, le_sup_left, le_sup_right], },
+    have sup_S : Sup s ≤ Sup S,
+    { apply Sup_le_Sup,
+      intros x hx, use {x},
+      simpa only [and_true, id.def, finset.coe_singleton, eq_self_iff_true, finset.sup_singleton,
+        set.singleton_subset_iff], },
+    have Sne : S.nonempty,
+    { suffices : ⊥ ∈ S, from set.nonempty_of_mem this,
+      use ∅,
+      simp only [set.empty_subset, finset.coe_empty, finset.sup_empty,
+        eq_self_iff_true, and_self], },
+    -- Now apply the defn of compact and finish.
+    obtain ⟨j, ⟨hjS, hjk⟩⟩ := hk S Sne dir_US (le_trans k (Sup s) (Sup S) hsup sup_S),
+    obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS,
+    use t, exact ⟨htS, by rwa ←htsup⟩, },
+end
+
+lemma well_founded.is_Sup_finite_compact (h : well_founded ((>) : α → α → Prop)) :
+  is_Sup_finite_compact α :=
+begin
+  intros s,
+  let p : set α := { x | ∃ (t : finset α), ↑t ⊆ s ∧ t.sup id = x },
+  have hp : p.nonempty, { use [⊥, ∅], simp, },
+  obtain ⟨m, ⟨t, ⟨ht₁, ht₂⟩⟩, hm⟩ := well_founded.well_founded_iff_has_max'.mp h p hp,
+  use t, simp only [ht₁, ht₂, true_and], apply le_antisymm,
+  { apply Sup_le, intros y hy, classical,
+    have hy' : (insert y t).sup id ∈ p,
+    { use insert y t, simp, rw set.insert_subset, exact ⟨hy, ht₁⟩, },
+    have hm' : m ≤ (insert y t).sup id, { rw ← ht₂, exact finset.sup_mono (t.subset_insert y), },
+    rw ← hm _ hy' hm', simp, },
+  { rw [← ht₂, finset.sup_eq_Sup], exact Sup_le_Sup ht₁, },
+end
+
+lemma is_Sup_finite_compact.is_sup_closed_compact (h : is_Sup_finite_compact α) :
+  is_sup_closed_compact α :=
+begin
+  intros s hne hsc, obtain ⟨t, ht₁, ht₂⟩ := h s, clear h,
+  cases t.eq_empty_or_nonempty with h h,
+  { subst h, rw finset.sup_empty at ht₂, rw ht₂,
+    simp [eq_singleton_bot_of_Sup_eq_bot_of_nonempty ht₂ hne], },
+  { rw ht₂, exact t.sup_closed_of_sup_closed h ht₁ hsc, },
+end
+
+lemma is_sup_closed_compact.well_founded (h : is_sup_closed_compact α) :
+  well_founded ((>) : α → α → Prop) :=
+begin
+  rw rel_embedding.well_founded_iff_no_descending_seq, rintros ⟨a⟩,
+  suffices : Sup (set.range a) ∈ set.range a,
+  { obtain ⟨n, hn⟩ := set.mem_range.mp this,
+    have h' : Sup (set.range a) < a (n+1), { change _ > _, simp [← hn, a.map_rel_iff], },
+    apply lt_irrefl (a (n+1)), apply lt_of_le_of_lt _ h', apply le_Sup, apply set.mem_range_self, },
+  apply h (set.range a),
+  { use a 37, apply set.mem_range_self, },
+  { rintros x y ⟨m, hm⟩ ⟨n, hn⟩, use m ⊔ n, rw [← hm, ← hn], apply a.to_rel_hom.map_sup, },
+end
+
+lemma is_Sup_finite_compact_iff_all_elements_compact :
+  is_Sup_finite_compact α ↔ (∀ k : α, is_compact_element k) :=
+begin
+  split,
+  { intros h k s hs,
+    obtain ⟨t, ⟨hts, htsup⟩⟩ := h s,
+    use [t, hts],
+    rwa ←htsup, },
+  { intros h s,
+    obtain ⟨t, ⟨hts, htsup⟩⟩ := h (Sup s) s (by refl),
+    have : Sup s = t.sup id,
+    { suffices : t.sup id ≤ Sup s, by { apply le_antisymm; assumption },
+      simp only [id.def, finset.sup_le_iff],
+      intros x hx,
+      apply le_Sup, exact hts hx, },
+    use [t, hts], assumption, },
+end
+
+lemma well_founded_characterisations :
+  tfae [well_founded ((>) : α → α → Prop),
+        is_Sup_finite_compact α,
+        is_sup_closed_compact α,
+        ∀ k : α, is_compact_element k] :=
+begin
+  tfae_have : 1 → 2, by { exact well_founded.is_Sup_finite_compact α, },
+  tfae_have : 2 → 3, by { exact is_Sup_finite_compact.is_sup_closed_compact α, },
+  tfae_have : 3 → 1, by { exact is_sup_closed_compact.well_founded α, },
+  tfae_have : 2 ↔ 4, by { exact is_Sup_finite_compact_iff_all_elements_compact α },
+  tfae_finish,
+end
+
+lemma well_founded_iff_is_Sup_finite_compact :
+  well_founded ((>) : α → α → Prop) ↔ is_Sup_finite_compact α :=
+(well_founded_characterisations α).out 0 1
+
+lemma is_Sup_finite_compact_iff_is_sup_closed_compact :
+  is_Sup_finite_compact α ↔ is_sup_closed_compact α :=
+(well_founded_characterisations α).out 1 2
+
+lemma is_sup_closed_compact_iff_well_founded :
+  is_sup_closed_compact α ↔ well_founded ((>) : α → α → Prop) :=
+(well_founded_characterisations α).out 2 0
+
+alias well_founded_iff_is_Sup_finite_compact ↔ _ is_Sup_finite_compact.well_founded
+alias is_Sup_finite_compact_iff_is_sup_closed_compact ↔
+      _ is_sup_closed_compact.is_Sup_finite_compact
+alias is_sup_closed_compact_iff_well_founded ↔ _ well_founded.is_sup_closed_compact
+
+lemma compactly_generated_of_well_founded (h : well_founded ((>) : α → α → Prop)) :
+  is_compactly_generated α :=
+begin
+  rw [well_founded_iff_is_Sup_finite_compact, is_Sup_finite_compact_iff_all_elements_compact] at h,
+  -- x is the join of the set of compact elements {x}
+  exact λ x, ⟨{x}, ⟨λ x _, h x, Sup_singleton⟩⟩,
+end
+
+end complete_lattice