intuition.
case_eq (getSupport c);
intuition.
unfold sumList.
simpl.
assert (Pr[c] == 0).
eapply getSupport_not_In_evalDist.
rewrite H.
simpl; intuition.
rewrite H0.
clear H0.
assert (evalDist c false == 0).
eapply getSupport_not_In_evalDist.
rewrite H.
simpl; intuition.
rewrite H0.
clear H0.
repeat rewrite ratMult_0_l.
rewrite <- ratAdd_0_r.
reflexivity.
destruct b.
rewrite sumList_cons.
eapply ratAdd_eqRat_compat; intuition.
destruct l.
unfold sumList.
simpl.
assert (evalDist c false == 0).
eapply getSupport_not_In_evalDist.
rewrite H.
simpl; intuition.
rewrite H0.
clear H0.
rewrite ratMult_0_l.
intuition.
destruct b.
specialize (getSupport_NoDup c); intuition.
rewrite H in H0.
inversion H0; clear H0; subst.
simpl in *; intuition.
rewrite sumList_cons.
symmetry.
rewrite ratAdd_0_r at 1.
eapply ratAdd_eqRat_compat.
intuition.
destruct l.
unfold sumList; simpl; intuition.
specialize (getSupport_NoDup c); intuition.
rewrite H in H0.
destruct b.
inversion H0; clear H0; subst; simpl in *; intuition.
inversion H0; clear H0; subst.
inversion H4; clear H4; subst; simpl in *; intuition.
rewrite sumList_cons.
symmetry.
rewrite ratAdd_comm.
eapply ratAdd_eqRat_compat; intuition.
destruct l.
unfold sumList.
simpl.
assert (Pr[c] == 0).
eapply getSupport_not_In_evalDist.
rewrite H.
simpl; intuition.
rewrite H0.
clear H0.
eapply ratMult_0_l.
destruct b.
rewrite sumList_cons.
rewrite ratAdd_0_r at 1.
eapply ratAdd_eqRat_compat.
intuition.
destruct l.
unfold sumList.
simpl.
intuition.
specialize (getSupport_NoDup c); intuition.
rewrite H in H0.
destruct b.
inversion H0; clear H0; subst.
inversion H4; clear H4; subst; simpl in *; intuition.
inversion H0; clear H0; subst; simpl in *; intuition.
specialize (getSupport_NoDup c); intuition.
rewrite H in H0.
inversion H0; clear H0; subst; simpl in *; intuition.
Qed.

(* Specifications: *)
Theorem prob_fold_and_eq_exp_h :
forall (A B : Set)(c : Comp A)(ls : list B)(P : A -> bool),
Pr[compFold _ (fun b _ => x <-$ c; ret b && (P x)) true ls] ==
expRat (Pr[x <-$c; ret (P x)]) (length ls).