Minimal elements in a set.

theories/collections.v

@@ -993,3 +993,34 @@ Section seq_set.
     seq_set start (S len) = {[ start + len ]} ∪ seq_set start len.
   Proof. unfold_leibniz. apply seq_set_S_union. Qed.
 End seq_set.
+
+(** Mimimal elements *)
+Definition minimal `{ElemOf A C} (R : relation A) (x : A) (X : C) : Prop :=
+  ∀ y, y ∈ X → R y x → y = x.
+Instance: Params (@minimal) 5.
+
+Section minimal.
+  Context `{SimpleCollection A C} {R : relation A}.
+
+  Global Instance minimal_proper x : Proper (@equiv C _ ==> iff) (minimal R x).
+  Proof. intros X X' y; unfold minimal; set_solver. Qed.
+  Lemma empty_minimal x : minimal R x ∅.
+  Proof. unfold minimal; set_solver. Qed.
+  Lemma singleton_minimal x : minimal R x {[ x ]}.
+  Proof. unfold minimal; set_solver. Qed.
+  Lemma singleton_minimal_not_above y x : ¬R y x → minimal R x {[ y ]}.
+  Proof. unfold minimal; set_solver. Qed.
+  Lemma union_minimal X Y x :
+    minimal R x X → minimal R x Y → minimal R x (X ∪ Y).
+  Proof. unfold minimal; set_solver. Qed.
+  Lemma minimal_subseteq X Y x : minimal R x X → Y ⊆ X → minimal R x Y.
+  Proof. unfold minimal; set_solver. Qed.
+
+  Lemma minimal_weaken `{!StrictOrder R} X x x' :
+    minimal R x X → R x' x → minimal R x' X.
+  Proof.
+    intros Hmin ? y ??.
+    assert (y = x) as -> by (apply (Hmin y); [done|by trans x']).
+    destruct (irreflexivity R x). by trans x'.
+  Qed.
+End minimal.

theories/fin_collections.v

@@ -184,6 +184,27 @@ Lemma collection_fold_proper {B} (R : relation B) `{!Equivalence R}
   Proper ((≡) ==> R) (collection_fold f b : C → B).
 Proof. intros ?? E. apply (foldr_permutation R f b); auto. by rewrite E. Qed.
 
+Lemma minimal_exists `{!StrictOrder R, ∀ x y, Decision (R x y)} (X : C) :
+  X ≢ ∅ → ∃ x, x ∈ X ∧ minimal R x X.
+Proof.
+  pattern X; apply collection_ind; clear X.
+  { by intros X X' HX; setoid_rewrite HX. }
+  { done. }
+  intros x X ? IH Hemp. destruct (collection_choose_or_empty X) as [[z ?]|HX].
+  { destruct IH as (x' & Hx' & Hmin); [set_solver|].
+    destruct (decide (R x x')).
+    - exists x; split; [set_solver|].
+      eauto using union_minimal, singleton_minimal, minimal_weaken.
+    - exists x'; split; [set_solver|].
+      auto using union_minimal, singleton_minimal_not_above. }
+  exists x; split; [set_solver|].
+  rewrite HX, (right_id _ (∪)). apply singleton_minimal.
+Qed.
+Lemma minimal_exists_L
+    `{!LeibnizEquiv C, !StrictOrder R, ∀ x y, Decision (R x y)} (X : C) :
+  X ≠ ∅ → ∃ x, x ∈ X ∧ minimal R x X.
+Proof. unfold_leibniz. apply minimal_exists. Qed.
+
 (** * Decision procedures *)
 Global Instance set_Forall_dec `(P : A → Prop)
   `{∀ x, Decision (P x)} X : Decision (set_Forall P X) | 100.