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_Add_inv a l1 l2 : Permutation l1 l2 -> forall l1' l2', Add a l1' l1 -> Add a l2' l2 -> Permutation l1' l2'.