intros; unfold Rminus in |- *; apply Rplus_le_compat.
apply Rmult_le_compat.
left; unfold r_lb in |- *; unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply TypeSpeed_pos | apply Rinv_0_lt_compat; apply rho_ub_pos ].
apply sin_lb_ge_0.
left; apply Rmult_lt_0_compat; [ apply rho_lb_pos | apply MinT_is_pos ].
apply Rle_trans with (rho_V * MinT)%R.
apply Rmult_le_compat_r.
left; apply MinT_is_pos.
unfold rho_V in |- *; left; apply rho_lb_0.
cut (MinT <= MinT)%R.
intro; left; apply (rho_t_PI2 (mkTimeT MinT H3 MinT_MaxT)).
right; reflexivity.
unfold r_V in |- *; left; apply r_lb_0.
apply Rle_trans with (sin (rho_lb V * MinT)).
cut (0 <= rho_lb V * MinT <= PI)%R.
intro; elim H3; intros.
generalize (SIN (rho_lb V * MinT) H4 H5); intro; elim H6; intros; assumption.
split.
left; apply Rmult_lt_0_compat.
apply rho_lb_pos.
apply MinT_is_pos.
left; apply Rle_lt_trans with (PI / 2)%R.
apply Rle_trans with (rho_V * MinT)%R.
apply Rmult_le_compat_r.
left; apply MinT_is_pos.
unfold rho_V in |- *; left; apply rho_lb_0.
cut (MinT <= MinT)%R.
intro; left; apply (rho_t_PI2 (mkTimeT MinT H3 MinT_MaxT)).
right; reflexivity.
apply PI2_Rlt_PI.
apply sin_incr_1.
left; apply Rlt_trans with 0%R.
apply _PI2_RLT_0.
apply Rmult_lt_0_compat.
apply rho_lb_pos.
apply MinT_is_pos.
apply Rle_trans with (rho_V * MinT)%R.
apply Rmult_le_compat_r.
left; apply MinT_is_pos.
unfold rho_V in |- *; left; apply rho_lb_0.
cut (MinT <= MinT)%R.
intro; left; apply (rho_t_PI2 (mkTimeT MinT H3 MinT_MaxT)).
right; reflexivity.
left; apply Rlt_trans with 0%R.
apply _PI2_RLT_0.
apply Rmult_lt_0_compat.
apply rho_V_is_pos.
apply Rlt_le_trans with MinT.
apply MinT_is_pos.
apply (cond_1 T).
left; apply (rho_t_PI2 T).
apply Rmult_le_compat.
left; apply rho_lb_pos.
left; apply MinT_is_pos.
left; unfold rho_V in |- *; apply rho_lb_0.
apply (cond_1 T).
apply Ropp_ge_le_contravar; apply Rle_ge.
apply Rmult_le_compat.
assumption.
apply cos_ge_0.
left; apply Rlt_le_trans with 0%R.
apply _PI2_RLT_0.
assumption.
assumption.
assumption.
cut (- PI / 2 <= a)%R.
intro; generalize (COS a H3 H0); intro; elim H4; intros; assumption.
left; apply Rlt_le_trans with 0%R.
replace (- PI / 2)%R with (- (PI / 2))%R.
apply _PI2_RLT_0.
unfold Rdiv in |- *; symmetry  in |- *; apply Ropp_mult_distr_l_reverse.
assumption.
Qed.

(**********)
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.