Equivalence between two definitions of permutations of lists

Among many ways to define permutations between lists (please have a look at Standard Library !), consider the following ones: Show that both definitions are logically equivalent.
Use this equivalence to prove easily the following fact:
Lemma counterexample : ~ permI (2::3::5::5::8::7::4::nil) 
                               (3::2::8::7::4::3::5::nil).

Solution

This file
Look also at the following exercises on permutations : this one and this one .

This exercise is proposed by Houda Anoun