unfold negligible.
intuition.
edestruct (H (1 + c)%nat).
exists (S x)%nat.
intuition.
eapply H0.
omega.
assert ((RatIntro (S O)
(@natToPosnat (expnat x0 (plus (S O) c))
(@expnat_nz (plus (S O) c) x0 pf_nz))) == (1 / x0) * (1 / (expnat x0 c))).
simpl.
rewrite <- ratMult_denom.
eapply eqRat_terms; intuition.
rewrite H3.
clear H3.
rewrite H2.
eapply leRat_trans.
2:{
eapply eqRat_impl_leRat.
symmetry.
eapply Def_equiv_2W.
}
eapply ratMult_leRat_compat; intuition.
unfold leRat.
eapply leRat_terms; intuition.
unfold natToPosnat, posnatToNat.
omega.
Qed.

Theorem TwoWorlds_equiv :
negligible (fun eta => (| Pr[f eta false] - Pr[f eta true] |)) <->
negligible (fun eta => | Pr[StandardDef_G eta] - 1 / 2 |).