Local Opaque evalDist.
induction 1; intuition; simpl in *.
inline_first.
assert (
Pr
[a <-$ c S eqds o s;
p <-$ ([z, s']<-2 a; (f z) S eqds o s'); ret evt (snd p) ] <=
q1 / 1 * k (i * q1 + count s)%nat +
q2 / 1 * k (i * q2 + (i * q1 + count s))%nat
).
eapply evalDist_bind_event_le.
eapply (@IHqueries_at_most _ _ _ count evt); intuition.
intros.
comp_simp.
eapply leRat_trans.
eapply (@H1 _ _ _ _ _ count evt).
eapply H2.
trivial.
intuition.
intuition.
eapply ratMult_leRat_compat; intuition.
eapply H2.
eapply plus_le_compat; intuition.
eapply le_trans.
eapply (qam_count_gen _ _ _ _ _ i).
intuition.
eapply H5.
eauto.
eauto.
rewrite mult_comm.
intuition.
rewrite H6.
clear H6.
eapply leRat_trans.
2:{
eapply eqRat_impl_leRat.
symmetry.
rewrite ratAdd_num.
eapply ratMult_distrib_r.
}
eapply ratAdd_leRat_compat;
eapply ratMult_leRat_compat; intuition.
eapply H2.
rewrite mult_plus_distr_l.
rewrite plus_comm.
repeat rewrite <- plus_assoc.
eapply plus_le_compat; intuition.
rewrite H1; intuition.
rewrite <- ratMult_1_l.
eapply ratMult_leRat_compat; intuition.
eapply leRat_terms; intuition.
inline_first.
eapply distro_irr_le.
intuition.
comp_simp.
simpl.
rewrite H0.
rewrite evalDist_ret_0.
eapply rat0_le_all.
congruence.
inline_first.
assert (EqDec C).
eapply oc_EqDec; intros.
eapply c.
assert (B * S).
eapply oc_base_exists.
eapply oc.
trivial.
trivial.
intuition.
assert (B' * S0).
eapply comp_base_exists.
eapply o.
trivial.
trivial.
intuition.
intuition.
trivial.
assert
(
(evalDist
(Bind
(c (prod S S0) (pair_EqDec eqds eqds0)
(fun (x : prod S S0) (y : A) =>
Bind ((oc (fst x) y) S0 eqds0 o (snd x))
(fun p : prod (prod B S) S0 =>
Ret (EqDec_dec (pair_EqDec eqdb (pair_EqDec eqds eqds0)))
(pair (fst (fst p)) (pair (snd (fst p)) (snd p)))))
(pair s s0))
(fun a : prod C (prod S S0) =>
Bind
(Ret
(EqDec_dec
(pair_EqDec
(pair_EqDec
(oc_EqDec c
(fun x : A =>
fst
(oc_base_exists (oc s x)
(fun y : A' =>
fst (comp_base_exists (o s0 y)))))
(fun x : A =>
EqDec_pair_l
(oc_EqDec (oc s x)
(fun y : A' =>
fst (comp_base_exists (o s0 y)))
(fun y : A' =>
EqDec_pair_l (comp_EqDec (o s0 y)) s0)) s))
eqds) eqds0))
(pair (pair (fst a) (fst (snd a))) (snd (snd a))))
(fun p : prod (prod C S) S0 =>
Ret (EqDec_dec bool_EqDec) (evt (snd p))))) true)
==
Pr
[a <-$
c (S * S0)%type (pair_EqDec eqds eqds0)
(fun (x : S * S0) (y : A) =>
p <-$ (oc (fst x) y) S0 eqds0 o (snd x);
ret (fst (fst p), (snd (fst p), snd p))) (s, s0);
ret (evt (snd (snd a))) ]
).
comp_skip.
simpl.
intuition.
rewrite H7.
clear H7.
eapply leRat_trans.
eapply (@IHqueries_at_most _ _
((fun (x : S * S0) (y : A) =>
p <-$ (oc (fst x) y) S0 eqds0 o (snd x);
ret (fst (fst p), (snd (fst p), snd p))))
(fun p => count (snd p)) (fun p => evt (snd p))
_ (fun x => q2 / 1 * k (x)%nat) (i * q2)%nat).
intuition.
eapply ratMult_leRat_compat; intuition.
trivial.
intuition.
comp_inline_l.
assert (
Pr
[a1 <-$ (oc (fst (a, b)) a0) S0 eqds0 o (snd (a, b));
p <-$ ret (fst (fst a1), (snd (fst a1), snd a1)); ret evt (snd (snd p)) ]
==
Pr
[a1 <-$ (oc (fst (a, b)) a0) S0 eqds0 o (snd (a, b));
ret evt (snd a1) ]
).
comp_skip.
simpl.
intuition.
rewrite H8.
clear H8.
simpl.
eapply leRat_trans.
eapply (@H1 ); intuition.
reflexivity.
intuition.
repeat simp_in_support.
destruct x.
simpl in *.
rewrite mult_comm.
eapply qam_count_gen.
eapply H0.
intuition.
eapply H5.
eapply H7.
eapply H8.
simpl.
rewrite <- ratMult_assoc.
eapply ratMult_leRat_compat; intuition.
eapply eqRat_impl_leRat.
rewrite <- ratMult_num_den.
eapply eqRat_terms; intuition.
eapply H2.
eapply plus_le_compat; intuition.
rewrite <- mult_assoc.
eapply mult_le_compat; intuition.
rewrite mult_comm.
intuition.
rewrite IHqueries_at_most.
eapply ratMult_leRat_compat; intuition.
eapply leRat_terms; intuition.
intuition.
trivial.
intuition.
intuition.
Grab Existential Variables.
trivial.
repeat econstructor; eauto.
Qed.

(* Specifications: *)
Theorem oc_eventProb_0_1 :
forall (S : Set)(count : S -> nat)(evt : S -> bool)(k : nat -> Rat)
(A B C : Set)(c : OracleComp A B C) n,
queries_at_most c n ->
forall
(eqds : EqDec S)(o : S -> A -> Comp (B * S))
(s : S),
(forall (n1 n2 : nat), (n1 <= n2)%nat -> (k n1 <= k n2)) ->
evt s = false ->
(forall s a, evt s = false -> Pr[p <-$ o s a; ret (evt (snd p))] <= (k (1 + (count s))%nat)) ->
(forall s s' a b, In (b, s') (getSupport (o s a)) ->
count s' <= 1 + (count s))%nat ->
count s = 0%nat ->
Pr[p <-$ c _ _ o s; ret (evt (snd p))] <= (n / 1) * (k n).