intros.
cut (MinBeta <= 2 * PI - a)%R.
cut (2 * PI - a <= PI / 2)%R.
cut (l * sin (2 * PI - a) + r_V * (cos (rho_V * T) - 1) <= - y)%R.
intros; rewrite (Rsqr_neg y);
apply (Math_prop_no_conflict_1 (2 * PI - a) l x (- y) T H H0 H7 H6 H5).
replace (2 * PI - a)%R with (- a + 2 * INR 1 * PI)%R;
[ rewrite (cos_period (- a) 1); rewrite cos_neg; assumption | simpl; ring ].
replace (2 * PI - a)%R with (- a + 2 * INR 1 * PI)%R;
[ rewrite (sin_period (- a) 1); rewrite sin_neg;
replace (l * - sin a + r_V * (cos (rho_V * T) - 1))%R with
(- (l * sin a - r_V * (cos (rho_V * T) - 1)))%R;
[ apply Ropp_ge_le_contravar; apply Rle_ge; assumption | simpl; ring ]
| simpl; ring ].
generalize (Rplus_le_compat_l (PI / 2 - a) (3 * (PI / 2)) a H1);
replace (PI / 2 - a + 3 * (PI / 2))%R with (2 * PI - a)%R.
replace (PI / 2 - a + a)%R with (PI / 2)%R.
intro; assumption.
unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l;
rewrite Rplus_0_r; reflexivity.
rewrite double.
pattern PI at 1 2 in |- *; rewrite double_var.
ring.
generalize (Rplus_le_compat_l (MinBeta - a) a (2 * PI - MinBeta) H2);
replace (MinBeta - a + (2 * PI - MinBeta))%R with (2 * PI - a)%R;
[ replace (MinBeta - a + a)%R with MinBeta; [ intro; assumption | ring ]
| ring ].
Qed.

(**********)
Lemma Math_prop_alarm_2 :
forall (a l : R) (T : TimeT),
(MinDistance T <= l)%R ->
(l <= MaxDistance T)%R ->
(2 * PI - MinBeta <= a)%R ->
(a <= 2 * PI)%R ->
(Rsqr (l * cos a - v V * T) + Rsqr (l * sin a) <= Rsqr AlertRange)%R.