(** 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.

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