Commit b483ce86 by Jean-Christophe Filliâtre

### Bags added to the standard library

parent 479621eb
 theory Bag use import int.Int type bag 'a (* the most basic operation is the number of occurrences *) function nb_occ (x: 'a) (b: bag 'a): int axiom occ_non_negative : forall b: bag 'a, x: 'a. nb_occ x b >= 0 (* equality of bags *) predicate eq_bag (a b : bag 'a) = forall x:'a. nb_occ x a = nb_occ x b axiom bag_extensionality: forall a b : bag 'a. eq_bag a b -> a = b (* basic constructors of bags *) function empty_bag : bag 'a axiom occ_empty : forall x: 'a. nb_occ x empty_bag = 0 lemma is_empty : forall b: bag 'a. (forall x: 'a. nb_occ x b = 0) -> b = empty_bag function singleton (x: 'a) : bag 'a axiom occ_singleton: forall x y: 'a. (x = y /\ (nb_occ y (singleton x)) = 1) \/ (x <> y /\ (nb_occ y (singleton x)) = 0) lemma occ_singleton_eq : forall x y: 'a. x = y -> (nb_occ y (singleton x)) = 1 lemma occ_singleton_neq : forall x y: 'a. x <> y -> (nb_occ y (singleton x)) = 0 function union (bag 'a) (bag 'a) : bag 'a axiom occ_union : forall x: 'a, a b : bag 'a. nb_occ x (union a b) = (nb_occ x a) + (nb_occ x b) (* union commutative, associative with identity empty_bag *) lemma Union_comm : forall a b:bag 'a. (union a b) = (union b a) lemma Union_identity : forall a:bag 'a. (union a empty_bag) = a lemma Union_assoc : forall a b c:bag 'a. (union a (union b c)) = (union (union a b) c) lemma bag_simpl : forall a b c: bag 'a. (union a b) = (union c b) -> a = c lemma bag_simpl_left : forall a b c: bag 'a. (union a b) = (union a c) -> b = c (* add operation *) function add (x : 'a) (b: bag 'a) : bag 'a = union (singleton x) b lemma occ_add_eq : forall b: bag 'a, x y: 'a. x = y -> nb_occ x (add x b) = (nb_occ x b) + 1 lemma occ_add_neq : forall b: bag 'a, x y: 'a. x <> y -> nb_occ y (add x b) = (nb_occ y b) (* cardinality of bags *) function card (bag 'a) : int axiom Card_empty : card (empty_bag: bag 'a) = 0 axiom Card_singleton : forall x:'a. card (singleton x) = 1 axiom Card_union: forall x y: bag 'a. card (union x y) = (card x) + (card y) axiom Card_zero_empty : forall x: bag 'a. card (x) = 0 -> x = empty_bag (* bag difference *) use import int.MinMax function diff (bag 'a) (bag 'a) : bag 'a axiom Diff_occ: forall b1 b2:bag 'a, x:'a. nb_occ x (diff b1 b2) = max 0 (nb_occ x b1 - nb_occ x b2) lemma Diff_empty_right: forall b:bag 'a. diff b empty_bag = b lemma Diff_empty_left: forall b:bag 'a. diff empty_bag b = empty_bag lemma Diff_add: forall b:bag 'a, x:'a. diff (add x b) (singleton x) = b lemma Diff_comm: forall b b1 b2:bag 'a. diff (diff b b1) b2 = diff (diff b b2) b1 lemma Add_diff: forall b:bag 'a, x:'a. nb_occ x b > 0 -> add x (diff b (singleton x)) = b end (* Local Variables: compile-command: "why3ide -I . proofs" End: *)
theories/bag.why 0 → 100644