Section expected_poly_time_theory.
Context `{ept : expected_poly_time_predicate}.
intuition.
eapply (@expected_poly_time_compose _ _ _ _ _ _ (fun n a b => (f2 n b))); eauto.
eapply expected_poly_time_const; eauto.
Qed.
intuition.
eapply (@expected_poly_time_compose _ _ _ _ _ _ (fun n a b => f3 n (f1 n a) b)); eauto.
eapply (@expected_poly_time_compose _ _ _ _ _ _ (fun n a b => f3 n b)); eauto.
eapply expected_poly_time_const.
eauto.
Qed.
intuition.
eapply (@expected_poly_time_compose_binary _ _ _ _ _ _ (fun n b c => pair b c)); eauto.
eapply expected_poly_time_pair.
Qed.
intuition.
eapply (@expected_poly_time_compose_binary _ _ _ _ _ _ BVxor); eauto.
eapply expected_poly_time_bvxor.
Qed.
intuition.
eapply (@expected_poly_time_compose_binary _ _ _ _ _ _ (fun n a b => eqb a b)); eauto.
eapply expected_poly_time_eqb_bool.
Qed.
intuition.
eapply (@expected_poly_time_compose_simp _ _ _ _ (fun n a => fst a)); eauto.
eapply expected_poly_time_fst.
Qed.
intuition.
eapply (@expected_poly_time_compose_simp _ _ _ _ (fun n a => snd a)); eauto.
eapply expected_poly_time_snd.
Qed.
intuition.
eapply (@expected_poly_time_compose_0 _ _ _ _ _ (fun n b => f3 n (f1 n) b)); eauto.
eapply (@expected_poly_time_compose_0 _ _ _ _ _ (fun n b => f3 n b)); eauto.
Qed.
intuition.
eapply (@expected_poly_time_compose_0_binary _ _ _ a b (fun n a b => Bind a b)); eauto.
eapply expected_poly_time_bind.
Qed.
intuition.
eapply (@expected_poly_time_compose_binary _ _ _ _ b c (fun n a b => Bind a b)); eauto.
eapply expected_poly_time_bind.
Qed.
intuition.
eapply (@expected_poly_time_compose_simp _ _ _ _ (fun n b => Ret (eqd n) b)); eauto.
eapply expected_poly_time_ret.
Qed.
eapply expected_poly_time_rnd.
exists 1%nat.
exists 1%nat.
exists 0%nat.
intuition.
simpl.
omega.
Qed.
eapply expected_poly_time_bind_0_f.
eapply expected_poly_time_rnd.
exists 0%nat.
exists 1%nat.
exists 0%nat.
intuition.
eapply expected_poly_time_ret_f.
eapply expected_poly_time_vec_hd.
Qed.

Theorem expected_poly_time_if_f :
forall (A B: nat -> Type)(f_b : forall n, (A n) -> bool)(f_so f_not : forall n, A n -> B n),
efficient _ f_b ->
efficient _ f_so ->
efficient _ f_not ->
efficient _ (fun n (a : A n) => if (f_b n a) then (f_so n a) else (f_not n a)).