Proof with auto.
intro.
set (fun x y: T * list T => fst x = fst y /\ Permutation (snd x) (snd y)).
intros.
induction H.
simpl...
simpl.
apply permuted_skip...
split...
apply Permuted_map.
unfold on.
subst P.
simpl.
apply (Permuted_sub IHPermutation).
intuition.
simpl.
subst P.
apply alt_permuted_swap...
repeat rewrite map_map.
simpl.
apply elemsR_map_map.
simpl...
eauto.
Qed.
Inductive merges_spec: list T -> list T -> list (list T) -> Prop :=
| merges_left_nil x: merges_spec nil x (x :: nil)
| merges_right_nil x: merges_spec x nil (x :: nil)
| merges_cons x y h t r r':
merges_spec y (h :: t) r ->
merges_spec (x :: y) t r' ->
merges_spec (x :: y) (h :: t) (map (cons x) r ++ map (cons h) r').
Hint Constructors merges_spec.

Lemma merges_uniq a b r:
merges_spec a b r ->
forall r', merges_spec a b r' -> r = r'.