Inductive perm (s1 s2 : seq A) : Prop :=
| permutation_nil of s1 = [::] & s2 = [::]
| permutation_skip x t1 t2 of s1 = x :: t1 & s2 = x :: t2 & perm t1 t2
| permutation_swap x y t of s1 = x :: y :: t & s2 = y :: x :: t
| permutation_trans t of perm s1 t & perm t s2.

(* Now the hard part; the opposite implications *)
Lemma perm_cat_consR s1 t1 s2 t2 x :
perm (s1 ++ x :: t1) (s2 ++ x :: t2) -> perm (s1 ++ t1) (s2 ++ t2).