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).

Theorem Permutation_cons_app_inv l l1 l2 a :
Permutation (a :: l) (l1 ++ a :: l2) -> Permutation l (l1 ++ l2).