End Permut_map.
Section Permut_permut.
Variable A : Type.
Variable eqA : relation A.
Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Hypothesis eqA_equiv : Equivalence eqA.

(* Specifications: *)
Lemma Permutation_impl_permutation : forall l l',
Permutation l l' -> permutation _ eqA_dec l l'.
Lemma permut_eqA : forall l l', Forall2 eqA l l' -> permutation _ eqA_dec l l'.
Lemma permutation_Permutation : forall l l',
permutation _ eqA_dec l l' <->
exists l'', Permutation l l'' /\ Forall2 eqA l'' l'.