Section Permut.
(** * From lists to multisets *)
Variable A : Type.
Variable eqA : relation A.
Hypothesis eqA_equiv : Equivalence eqA.
Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Let emptyBag := EmptyBag A.
Let singletonBag := SingletonBag _ eqA_dec.
(** contents of a list *)
Fixpoint list_contents (l:list A) : multiset A :=
match l with
| [] => emptyBag
| a :: l => munion (singletonBag a) (list_contents l)
end.
(** * [permutation]: definition and basic properties *)
Definition permutation (l m:list A) := meq (list_contents l) (list_contents m).

Lemma permut_add_inside_eq :
forall a a' l1 l2 l3 l4, eqA a a' ->
permutation (l1 ++ l2) (l3 ++ l4) ->
permutation (l1 ++ a :: l2) (l3 ++ a' :: l4).