Proof with trivial.
intros; apply Rlt_le_trans with (r_vi * sin rho_vi)%R...
apply r_sin_rho...
apply Rle_trans with (r_vi * sin (rho_vi * t))%R...
apply Rmult_le_compat_l...
left; unfold r_vi in |- *; unfold r in |- *; unfold Rdiv in |- *;
apply Rmult_lt_0_compat...
apply TypeSpeed_pos...
apply Rinv_0_lt_compat; apply rho_pos...
case (Rle_dec (rho_vi * t) (PI / 2)); intro...
cut (rho_vi <= rho_vi * t)%R...
intro; apply sin_incr_1...
left; apply Rlt_trans with 0%R...
apply _PI2_RLT_0...
unfold rho_vi in |- *; apply rho_pos...
apply Rle_trans with (rho_vi * t)%R...
left; apply Rlt_le_trans with 0%R...
apply _PI2_RLT_0...
apply rho_t_pos...
left; apply Rlt_le_trans with 1%R; try apply Rlt_0_1...
pattern rho_vi at 1 in |- *; rewrite <- (Rmult_1_r rho_vi)...
apply Rmult_le_compat_l...
left; unfold rho_vi in |- *; apply rho_pos...
cut (PI / 2 < rho_vi * t)%R...
intro; cut (rho_vi <= PI - rho_vi * t)%R...
cut (PI - rho_vi * t <= PI / 2)%R...
intros; rewrite <- (sin_PI_x (rho_vi * t))...
apply sin_incr_1...
left; apply Rlt_trans with 0%R...
apply _PI2_RLT_0...
unfold rho_vi in |- *; apply rho_pos...
apply Rle_trans with (PI - rho_vi * t)%R...
apply Rle_trans with rho_vi...
left; apply Rlt_trans with 0%R...
apply _PI2_RLT_0...
unfold rho_vi in |- *; apply rho_pos...
left...
apply Rplus_lt_reg_l with (- (PI / 2) + rho_vi * t)%R...
replace (- (PI / 2) + rho_vi * t + (PI - rho_vi * t))%R with (PI / 2)%R...
replace (- (PI / 2) + rho_vi * t + PI / 2)%R with (rho_vi * t)%R...
ring...
unfold Rminus in |- *; unfold Rdiv in |- *;
replace (- (PI * / 2) + rho_vi * t + (PI + - (rho_vi * t)))%R with
(- (PI * / 2) + PI + (rho_vi * t + - (rho_vi * t)))%R...
rewrite Rplus_opp_r; rewrite Rplus_0_r...
pattern PI at 3 in |- *; rewrite double_var...
unfold Rdiv in |- *; ring...
ring...
apply Rplus_le_reg_l with (rho_vi * t - rho_vi)%R...
unfold Rminus in |- *...
repeat rewrite Rplus_assoc...
rewrite Rplus_opp_l; rewrite Rplus_0_r...
rewrite (Rplus_comm (rho_vi * t))...
repeat rewrite Rplus_assoc...
rewrite Rplus_opp_l; rewrite Rplus_0_r...
rewrite <- (Rplus_comm PI)...
auto with real...
cut (0 <= t)%R...
cut (rho_vi * t <= PI)%R...
intros...
generalize (xpt t H3 H2); intro...
generalize (xp0_pos H1); intro...
unfold Rminus in H4...
generalize (Rplus_le_compat_r (xp 0) (hx t) (xp t + - xp 0) H4)...
repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r...
intro...
generalize (plus_le_is_le (hx t) (xp 0) (xp t) H5 H6)...
apply Rle_trans with (PI - rho_vi)%R...
apply Rplus_le_reg_l with (- PI)%R...
unfold Rminus in |- *...
repeat rewrite <- Rplus_assoc...
repeat rewrite Rplus_opp_l...
rewrite Rplus_0_l...
rewrite <- Ropp_0...
apply Ropp_ge_le_contravar; apply Rle_ge; left; unfold rho_vi in |- *;
apply rho_pos...
left; apply Rlt_le_trans with 1%R; try apply Rlt_0_1...
Qed.

Lemma no_conflict_scenario_1_Omega :
forall t : R,
(1 <= t)%R ->
(rho_vi * t <= PI - rho_vi)%R ->
Omega (beta + thetat intr 0) = true ->
(Rsqr ConflictRange < Rsqr_evader_distance t)%R.