intros; case (Req_EM_var (div_tau intruder evader) 0); intro.
right; apply (R_equal_when_zero intruder evader t1 t2 e).
apply Rsqr_incr_0.
repeat rewrite R_Rmin.
apply Rplus_le_compat_r; apply Rmult_le_compat_l.
apply Ax2_R_pos.
apply Rsqr_incr_1.
unfold Rminus in |- *; apply Rplus_le_compat_r; assumption.
apply Rplus_le_reg_l with (tmin intruder evader); rewrite Rplus_0_r;
rewrite Rplus_comm; unfold Rminus in |- *; repeat rewrite Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_r; assumption.
apply Rplus_le_reg_l with (tmin intruder evader); rewrite Rplus_0_r;
rewrite Rplus_comm; unfold Rminus in |- *; repeat rewrite Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_r; apply Rle_trans with t1;
assumption.
assumption.
assumption.
apply RR_pos.
apply RR_pos.
Qed.

Lemma asymptotic_increase_tau :
forall (intruder evader : State) (t t1 t2 : R),
(tau intruder evader t <= t1)%R ->
(t1 <= t2)%R ->
(RR intruder evader (t + t1) <= RR intruder evader (t + t2))%R.