Proof with trivial.
intros evader intruder x H n; unfold arc_loop in |- *; rewrite mod_eq_0;
case (Rlt_le_dec 0 (trkrate (bank intruder)))...
cut (INR 0 = 0%R)...
intros; rewrite H0; repeat rewrite Rmult_0_l; repeat rewrite Rmult_0_r;
repeat rewrite Rplus_0_r;
cut
(mkState
(xt intruder +
Rsqr intruderSpeed / (g * tand (bank intruder)) *
(sind (heading intruder) - sind (heading intruder)))
(yt intruder +
Rsqr intruderSpeed / (g * tand (bank intruder)) *
(cosd (heading intruder) - cosd (heading intruder)))
(heading intruder) (bank intruder) = intruder)...
cut (mkState (xt evader) (yt evader) (heading evader) (bank evader) = evader)...
intros; rewrite H1; rewrite H2; unfold chktrack in |- *;
case (Rle_dec (tau intruder evader 0) 0); intro...
unfold chkrange in |- *; split...
unfold RR in |- *; unfold dx, dy in |- *; repeat rewrite Rmult_0_l;
repeat rewrite Rplus_0_r; unfold distance in H...
unfold AlertTime in |- *; left; prove_sup...
rewrite Rplus_0_l; case (Rlt_dec AlertTime (tau intruder evader 0)); intro...
cut (0 <= AlertTime)%R...
intro;
assert
(H4 :=
asymptotic_decrease_tau intruder evader 0 0 AlertTime (Rlt_le _ _ r0) H3);
repeat rewrite Rplus_0_l in H4; apply Rle_trans with (RR intruder evader 0)...
unfold RR in |- *; unfold dx, dy in |- *; repeat rewrite Rmult_0_l;
repeat rewrite Rplus_0_r...
unfold AlertTime in |- *; left; prove_sup...
assert (H3 := derivative_eq_zero_min intruder evader 0 0);
repeat rewrite Rplus_0_l in H3; apply Rle_trans with (RR intruder evader 0)...
unfold RR in |- *; unfold dx, dy in |- *; repeat rewrite Rmult_0_l;
repeat rewrite Rplus_0_r...
unfold xt, yt in |- *; case evader...
unfold xt, yt in |- *; case intruder; intros; unfold Rminus in |- *;
repeat rewrite Rplus_opp_r; repeat rewrite Rmult_0_r;
repeat rewrite Rplus_0_r...
cut (INR 0 = 0%R)...
intros; rewrite H0; repeat rewrite Rmult_0_l; repeat rewrite Rmult_0_r;
repeat rewrite Rplus_0_r;
cut
(mkState
(xt intruder +
Rsqr intruderSpeed / (g * tand (bank intruder)) *
(sind (heading intruder) - sind (heading intruder)))
(yt intruder +
Rsqr intruderSpeed / (g * tand (bank intruder)) *
(cosd (heading intruder) - cosd (heading intruder)))
(heading intruder) (bank intruder) = intruder)... cut
(mkState (xt evader) (yt evader) (heading evader) (bank evader) = evader)...
intros; rewrite H1; rewrite H2; unfold chktrack in |- *;
case (Rle_dec (tau intruder evader 0) 0); intro...
unfold chkrange in |- *; split...
unfold RR in |- *; unfold dx, dy in |- *; repeat rewrite Rmult_0_l;
repeat rewrite Rplus_0_r; unfold distance in H...
unfold AlertTime in |- *; left; prove_sup...
rewrite Rplus_0_l; case (Rlt_dec AlertTime (tau intruder evader 0)); intro...
cut (0 <= AlertTime)%R...
intro;
assert
(H4 :=
asymptotic_decrease_tau intruder evader 0 0 AlertTime (Rlt_le _ _ r0) H3);
repeat rewrite Rplus_0_l in H4; apply Rle_trans with (RR intruder evader 0)...
unfold RR in |- *; unfold dx, dy in |- *; repeat rewrite Rmult_0_l;
repeat rewrite Rplus_0_r...
unfold AlertTime in |- *; left; prove_sup...
assert (H3 := derivative_eq_zero_min intruder evader 0 0);
repeat rewrite Rplus_0_l in H3; apply Rle_trans with (RR intruder evader 0)...
unfold RR in |- *; unfold dx, dy in |- *; repeat rewrite Rmult_0_l;
repeat rewrite Rplus_0_r...
unfold xt, yt in |- *; case evader...
unfold xt, yt in |- *; case intruder; intros; unfold Rminus in |- *;
repeat rewrite Rplus_opp_r; repeat rewrite Rmult_0_r;
repeat rewrite Rplus_0_r...
Qed.

Theorem alarm_at_alerting_distance :
forall evader intruder : State,
alerting_distance evader intruder -> ails_alert intruder evader.