induction ls; intuition.
simpl.
rewrite evalDist_ret_0; intuition.
simpl.
inline_first.
eapply leRat_impl_eqRat.
eapply distro_irr_le.
intuition.
comp_simp.
rewrite IHls.
intuition.
eapply rat0_le_all.
Qed.

(* Specifications: *)
Theorem sumList_support_bool':
forall (c : Comp bool) (f : bool -> Rat),
sumList (getSupport c) (fun x => evalDist c x * f x) ==
Pr  [c ] * (f true) + (evalDist c false * (f false)).