Global Instance Permutation_map' :
  Proper (@Permutation A ==> @Permutation B) (map f).

Lemma Permutation_elt_map_inv: forall l1 l2 l3 l4 a, Permutation (l1 ++ a :: l2) (l3 ++ map f l4) -> (forall b, a <> f b) -> exists l1' l2', l3 = l1' ++ a :: l2'.