Commit 588b9559 by Robbert Krebbers

### Minimal elements in a set.

parent 3ecb4b2e
 ... ... @@ -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.
 ... ... @@ -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. ... ...
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