intros; case (Req_EM_var (div_tau intruder evader) 0); intro.
right; apply (R_equal_when_zero intruder evader t2 t1 e).
apply Rsqr_incr_0.
repeat rewrite R_Rmin.
apply Rplus_le_compat_r; apply Rmult_le_compat_l.
apply Ax2_R_pos.
case (Rtotal_order (tmin intruder evader) t1); intro.
generalize (Rle_lt_trans t2 (tmin intruder evader) t1 H H1); intro;
elim (Rlt_irrefl t1 (Rle_lt_trans t1 t2 t1 H0 H2)).
elim H1; intros.
rewrite H2 in H; generalize (Rle_antisym t1 t2 H0 H); intro; rewrite H3;
right; reflexivity.
rewrite Rsqr_neg; rewrite (Rsqr_neg (t1 - tmin intruder evader));
replace (- (t2 - tmin intruder evader))%R with (tmin intruder evader - t2)%R.
replace (- (t1 - tmin intruder evader))%R with (tmin intruder evader - t1)%R.
apply Rsqr_incr_1.
unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_ge_le_contravar;
apply Rle_ge; assumption.
apply Rplus_le_reg_l with t2; 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 t1; 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 t2;
assumption.
ring.
ring.
assumption.
assumption.
apply RR_pos.
apply RR_pos.
Qed.

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