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).
(** we can use [multiplicity] to define [InA] and [NoDupA]. *)
Fact if_eqA_then : forall a a' (B:Type)(b b':B),
eqA a a' -> (if eqA_dec a a' then b else b') = b.
Fact if_eqA_else : forall a a' (B:Type)(b b':B),
~eqA a a' -> (if eqA_dec a a' then b else b') = b'.
Fact if_eqA_refl : forall a (B:Type)(b b':B),
(if eqA_dec a a then b else b') = b.
(** PL: Inutilisable dans un rewrite sans un change prealable. *)
Global Instance if_eqA (B:Type)(b b':B) :
Proper (eqA==>eqA==>@eq _) (fun x y => if eqA_dec x y then b else b').
Fact if_eqA_rewrite_l : forall a1 a1' a2 (B:Type)(b b':B),
eqA a1 a1' -> (if eqA_dec a1 a2 then b else b') =
(if eqA_dec a1' a2 then b else b').
Fact if_eqA_rewrite_r : forall a1 a2 a2' (B:Type)(b b':B),
eqA a2 a2' -> (if eqA_dec a1 a2 then b else b') =
(if eqA_dec a1 a2' then b else b').
Global Instance multiplicity_eqA (l:list A) :
Proper (eqA==>@eq _) (multiplicity (list_contents l)).

Lemma multiplicity_NoDupA : forall l,
NoDupA eqA l <-> (forall a, multiplicity (list_contents l) a <= 1).