Proof with trivial.
intros intr evad T hyp_intr hyp_evad; intros; unfold chktrack in |- *;
case (Rle_dec (tau (measure2state intr 0) (measure2state (tr evad) 0) 0) 0);
intro...
elim
(Rlt_irrefl 0
(Rlt_le_trans 0
(tau (measure2state intr 0) (measure2state (tr evad) 0) 0) 0 H3 r))...
rewrite Rplus_0_l;
case
(Rlt_dec AlertTime
(tau (measure2state intr 0) (measure2state (tr evad) 0) 0));
intro...
apply (alarm_NOT_Omega_AlertTime intr evad T)...
apply (alarm_NOT_Omega_tau intr evad T)...
Qed.
(**************************************************************)
(************************** THEOREM ***************************)
(**************************************************************)

Theorem ails_alarm_tau_gt0 :
forall (intr : Trajectory) (evad : EvaderTrajectory) (T : TimeT),
h intr = V ->
h (tr evad) = V ->
(MinDistance T <= l intr evad T)%R ->
(l intr evad T <= MaxDistance T)%R ->
Omega (beta intr evad T + thetat intr 0) = false ->
(0 < tau (measure2state intr 0) (measure2state (tr evad) 0) 0)%R ->
conflict intr evad T = true ->
ails_alert (measure2state intr 0) (measure2state (tr evad) 0).