Ltac InvAdd := repeat (match goal with
| H: Add ?x _ (_ :: _) |- _ => inversion H; clear H; subst
end).
Ltac finish_basic_perms H :=
try constructor; try rewrite perm_swap; try constructor; trivial;
(rewrite <- H; now apply Permutation_Add) ||
(rewrite H; symmetry; now apply Permutation_Add).

Lemma Permutation_vs_elt_inv : forall l l1 l2 a,
Permutation l (l1 ++ a :: l2) -> exists l' l'', l = l' ++ a :: l''.