intuition.
rewrite evalDist_seq_step.
rewrite (sumList_partition evt1).
eapply ratAdd_leRat_compat.
Transparent evalDist.
simpl in H.
eapply leRat_trans.
2:{
eapply H.
}
eapply sumList_le.
intuition.
rewrite ratMult_assoc.
eapply ratMult_leRat_compat.
intuition.
case_eq (evt1 a); intuition.
rewrite ratMult_1_r.
eapply leRat_trans.
eapply distro_irr_le.
intros.
rewrite orb_true_l.
simpl.
destruct (EqDec_dec bool_EqDec true true ).
eapply leRat_refl.
intuition.
destruct (EqDec_dec bool_EqDec true true); intuition.
destruct (EqDec_dec bool_EqDec false true); intuition.
discriminate.
rewrite ratMult_0_r; intuition.
assert ( sumList (getSupport c)
(fun a : A0 =>
evalDist c a * Pr  [y <-$ f a; ret evt1 a || evt2 y ] *
(if evt1 a then 0 else 1)) <=
sumList (getSupport c)
(fun a : A0 =>
evalDist c a *
(if evt1 a then 0 else 1) * k2)).
eapply sumList_le; intuition.
case_eq (evt1 a); intuition.
repeat rewrite ratMult_0_r.
eapply eqRat_impl_leRat.
symmetry.
apply ratMult_0_l.
repeat rewrite ratMult_1_r.
eapply ratMult_leRat_compat; intuition.
rewrite H1.
clear H1.
rewrite sumList_factor_constant_r.
eapply leRat_trans.
2:{
eapply eqRat_impl_leRat.
eapply ratMult_1_l.
}
eapply ratMult_leRat_compat; intuition.
assert ( sumList (getSupport c)
(fun a : A0 => evalDist c a * (if evt1 a then 0 else 1)) <=
sumList (getSupport c)
(fun a : A0 => evalDist c a)).
eapply sumList_le.
intuition.
destruct (evt1 a); intuition.
rewrite ratMult_0_r.
eapply rat0_le_all.
rewrite ratMult_1_r.
intuition.
rewrite H1. clear H1.
eapply evalDist_sum_le_1.
Qed.

Theorem RndInAdaptive_prob :
forall (c : OracleComp A B C)(q : nat),
queries_at_most c q ->
forall (S : Set)(evt : S -> bool)(eqd : EqDec S)(o : S -> A -> Comp (B * S))(k : Rat),
(forall a b,
evt a = false ->
Pr[ d <-$ o a b; ret (evt (snd d))] <= k)->
(forall a b x y,
In (a, b) (getSupport (o x y)) -> evt x = true -> evt b = true) ->
forall (s : S),
evt s = false ->
Pr[ d <-$ c _ _ o s; ret (evt (snd d))] <= q / 1 * k.