(** 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.

(* Specifications: *)
Lemma permut_app_inv2 :
forall l l1 l2, permutation (l ++ l1) (l ++ l2) -> permutation l1 l2.
Lemma permut_remove_hd_eq :
forall l l1 l2 a b, eqA a b ->
permutation (a :: l) (l1 ++ b :: l2) -> permutation l (l1 ++ l2).
Lemma permut_remove_hd :
forall l l1 l2 a,
permutation (a :: l) (l1 ++ a :: l2) -> permutation l (l1 ++ l2).