Proof with auto.
induction l...
simpl.
intros.
simpl.
rewrite IHl.
destruct (cmp_cmp (Ecmp ee (subscript a) (subscript d)) c)...
Qed.

Theorem qs_CM_U_map_cost_eq (l: list (Index ee ol)):
ne_tree.map (fst (A:=_:Set)(B:=_:Set)) (NDP.qs ee (map (@subscript ee ol) l))
= ne_tree.map (@length (_:Set) ∘ (@fst (_:Set) (_:Set))) (U.qs l).