Proof with trivial.
intros intr evad T hyp_intr hyp_evad; intros; apply Rsqr_incr_0_var...
rewrite R_T...
generalize (conflict_beta_theta intr evad T hyp_intr H H0 H1 H2); intro;
elim H3; intro...
cut (0 <= beta intr evad T + thetat intr 0)%R...
intro;
apply
(Math_prop_alarm_1 (beta intr evad T + thetat intr 0)
(l intr evad T) T H H0 H5 H4)...
generalize (beta_def intr evad T); intro; decompose [and] H5; intros...
cut (beta intr evad T + thetat intr 0 <= 2 * PI)%R...
intro;
apply
(Math_prop_alarm_2 (beta intr evad T + thetat intr 0)
(l intr evad T) T H H0
(Rlt_le (2 * PI - MinBeta) (beta intr evad T + thetat intr 0) H4))...
generalize (beta_def intr evad T); intro; decompose [and] H5; intros; left...
(*Rewrite hyp_evad.*)
left; apply AlertRange_pos...
Qed.

Lemma alarm_NOT_Omega_tau :
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 ->
conflict intr evad T = true ->
(0 < tau (measure2state intr 0) (measure2state (tr evad) 0) 0)%R ->
(RR (measure2state intr 0) (measure2state (tr evad) 0)
(tau (measure2state intr 0) (measure2state (tr evad) 0) 0) <= AlertRange)%R.