diff --git a/theories/collections.v b/theories/collections.v index bf72aa636e92da0b27c21d48633bcee648cace04..e0d70b8d77f6b29f66d41c01970d77bb82ef74a8 100644 --- a/theories/collections.v +++ b/theories/collections.v @@ -324,6 +324,10 @@ Section list_unfold. intros ??; constructor. by rewrite elem_of_app, (set_unfold (x ∈ l) P), (set_unfold (x ∈ k) Q). Qed. + Global Instance set_unfold_included l k (P Q : A → Prop) : + (∀ x, SetUnfold (x ∈ l) (P x)) → (∀ x, SetUnfold (x ∈ k) (Q x)) → + SetUnfold (l `included` k) (∀ x, P x → Q x). + Proof. by constructor; unfold included; set_unfold. Qed. End list_unfold. (** * Guard *) diff --git a/theories/list.v b/theories/list.v index 7dcf11c6cc76f21940c5e047746ba274f7398c73..36be6e46faa42b96cced5acb2134c0b7c8828e9c 100644 --- a/theories/list.v +++ b/theories/list.v @@ -282,6 +282,9 @@ Inductive Forall3 {A B C} (P : A → B → C → Prop) : P x y z → Forall3 P l k k' → Forall3 P (x :: l) (y :: k) (z :: k'). (** Set operations on lists *) +Definition included {A} (l1 l2 : list A) := ∀ x, x ∈ l1 → x ∈ l2. +Infix "`included`" := included (at level 70) : C_scope. + Section list_set. Context {A} {dec : ∀ x y : A, Decision (x = y)}. Global Instance elem_of_list_dec {dec : ∀ x y : A, Decision (x = y)} @@ -2017,6 +2020,12 @@ Section contains_dec. abstract (rewrite Permutation_alt; tauto). Defined. End contains_dec. + +(** ** Properties of [included] *) +Global Instance included_preorder : PreOrder (@included A). +Proof. split; firstorder. Qed. +Lemma included_nil l : [] `included` l. +Proof. intros x. by rewrite elem_of_nil. Qed. End more_general_properties. (** ** Properties of the [Forall] and [Exists] predicate *)