unfold alt_perms.
induction l...
simpl.
unfold Basics.compose.
rewrite concatMap_insert_everywhere.
apply Permutation_app...
rewrite IHl. clear IHl.
destruct l...
rewrite map_concat.
rewrite map_map.
generalize (splits (t :: l)). intro.
setoid_rewrite map_map.
unfold compose.
rewrite concat_concat.
rewrite map_map.
simpl @fst. simpl @snd.
simpl.
unfold compose.
setoid_rewrite map_concat.
setoid_rewrite map_map...
Qed.

Lemma map_single A B (f: A -> B) x: map f [x] = [f x].