intuition.
eapply (@rat_inf_limit_squeeze (fun n => (lowDistApprox_repeat c P a (Nat.sqrt (div2 n)))) (fun n => eqRat (evalDist (Repeat c P) a))).
eapply rat_inf_limit_mono.
eapply lowDistApprox_repeat_limit; trivial.
intuition.
eapply Nat.sqrt_le_mono.
eapply div2_le_mono.
trivial.
intuition.
exists (2 * (y * y))%nat.
rewrite div2_double.
rewrite Nat.sqrt_square.
trivial.
eapply rat_inf_limit_const.
intuition.
intuition.
rewrite <- H2.
rewrite H3.
intuition.
intuition.
eapply lowDistApprox_repeat_sqrt_le; eauto.
intuition.
eapply leRat_trans.
2:{
eapply eqRat_impl_leRat.
eapply H4.
}
eapply lowDistApprox_low.
destruct H0.
eapply well_formed_Repeat; eauto.
eapply comp_eq_dec; eauto.
eauto.
eapply lowDistApprox_repeat_sqrt_div2_left_total.
unfold left_total; intuition.
econstructor.
eapply eqRat_refl.
Qed.

Theorem evalDet_evalDist_equiv : forall (A : Set)(c : Comp A),
well_formed_comp c ->
forall a, rat_inf_limit (lowDistApprox c a) (evalDist c a).