(* Verifiable in MuPAD *)
Axiom
  Math_prop_alarm_1 :
    forall (a l : R) (T : TimeT),
    (MinDistance T <= l)%R ->
    (l <= MaxDistance T)%R ->
    (0 <= a)%R ->
    (a <= MinBeta)%R ->
    (Rsqr (l * cos a - v V * T) + Rsqr (l * sin a) <= Rsqr AlertRange)%R.(* Verifiable in MuPAD *)
Axiom
  Math_prop_no_conflict_1 :
    forall (a l x y : R) (T : TimeT),
    (MinDistance T <= l)%R ->
    (l <= MaxDistance T)%R ->
    (MinBeta <= a)%R ->
    (a <= PI / 2)%R ->
    (l * sin a + r_V * (cos (rho_V * T) - 1) <= y)%R ->
    (r_V * sin (rho_V * T) - l * cos a <= x)%R ->
    (Rsqr ConflictRange < Rsqr x + Rsqr y)%R.

(* Specifications: *)
Lemma MinBeta_pos : (0 < MinBeta)%R.
Lemma Math_prop_y_a_PI2 : forall (a l : R) (T : TimeT), (MinBeta <= a)%R -> (a <= PI / 2)%R -> (0 <= l)%R -> (l * sin MinBeta + r_V * (cos (rho_V * T) - 1) <= l * sin a + r_V * (cos (rho_V * T) - 1))%R.
Lemma Math_prop_approx_y_a_PI2 : forall (a l1 l2 : R) (T : TimeT), (0 <= a)%R -> (a <= PI / 2)%R -> (0 <= l1)%R -> (l1 <= l2)%R -> (l1 * sin_lb a + r_ub V * (cos_lb (rho_ub V * MaxT) - 1) <= l2 * sin a + r_V * (cos (rho_V * T) - 1))%R.
Lemma Math_prop_x_a_PI2 : forall (a l : R) (T : TimeT), (MinBeta <= a)%R -> (a <= PI / 2)%R -> (0 <= l)%R -> (r_V * sin (rho_V * T) - l * cos MinBeta <= r_V * sin (rho_V * T) - l * cos a)%R.
Lemma Math_prop_approx_x_a_PI2 : forall (a l1 l2 : R) (T : TimeT), (0 <= a)%R -> (a <= PI / 2)%R -> (0 <= l1)%R -> (l1 <= l2)%R -> (r_lb V * sin_lb (rho_lb V * MinT) - l2 * cos_ub a <= r_V * sin (rho_V * T) - l1 * cos a)%R.
Lemma Math_prop_no_conflict_2 : forall (a l x y : R) (T : TimeT), (MinDistance T <= l)%R -> (l <= MaxDistance T)%R -> (3 * (PI / 2) <= a)%R -> (a <= 2 * PI - MinBeta)%R -> (y <= l * sin a - r_V * (cos (rho_V * T) - 1))%R -> (r_V * sin (rho_V * T) - l * cos a <= x)%R -> (Rsqr ConflictRange < Rsqr x + Rsqr y)%R.
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.