Section multiset_defs.
Variable A : Type.
Variable eqA : A -> A -> Prop.
Hypothesis eqA_equiv : Equivalence eqA.
Hypothesis Aeq_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Inductive multiset : Type :=
Bag : (A -> nat) -> multiset.
Definition EmptyBag := Bag (fun a:A => 0).
Definition SingletonBag (a:A) :=
Bag (fun a':A => match Aeq_dec a a' with
| left _ => 1
| right _ => 0
end).
Definition multiplicity (m:multiset) (a:A) : nat := let (f) := m in f a.
(** multiset equality *)
Definition meq (m1 m2:multiset) :=
forall a:A, multiplicity m1 a = multiplicity m2 a.
(** multiset union *)
Definition munion (m1 m2:multiset) :=
Bag (fun a:A => multiplicity m1 a + multiplicity m2 a).
(** Here we should make multiset an abstract datatype, by hiding [Bag], [munion], [multiplicity]; all further properties are proved abstractly *)

Lemma munion_perm_left :
forall x y z:multiset, meq (munion x (munion y z)) (munion y (munion x z)).