Section Permutation_properties.
Variable A:Type.
Implicit Types a b : A.
Implicit Types l m : list A.
(** Compatibility with others operations on lists *)
Global Instance Permutation_in' :
Proper (Logic.eq ==> @Permutation A ==> iff) (@In A) | 10.
Global Instance Permutation_app' :
Proper (@Permutation A ==> @Permutation A ==> @Permutation A) (@app A) | 10.
Local Hint Resolve Permutation_app_comm : core.
Local Hint Resolve Permutation_cons_app : core.
Local Hint Resolve Permutation_middle : core.
Global Instance Permutation_rev' :
Proper (@Permutation A ==> @Permutation A) (@rev A) | 10.
Global Instance Permutation_length' :
Proper (@Permutation A ==> Logic.eq) (@length A) | 10.
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'.