End Permut.
Section Permut_map.
Variables A B : Type.
Variable eqA : relation A.
Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Hypothesis eqA_equiv : Equivalence eqA.
Variable eqB : B->B->Prop.
Hypothesis eqB_dec : forall x y:B, { eqB x y }+{ ~eqB x y }.
Hypothesis eqB_trans : Transitive eqB.
(** Permutation is compatible with map. *)

Lemma permut_map :
forall f,
(Proper (eqA==>eqB) f) ->
forall l1 l2, permutation _ eqA_dec l1 l2 ->
permutation _ eqB_dec (map f l1) (map f l2).