intro t; unfold m in |- *; repeat simple apply Rmult_lt_0_compat...
simpl in |- *; prove_sup0...
apply r_V_is_pos...
generalize (cond_1 t); generalize MinT_is_pos; intros H H0;
generalize (Rlt_le_trans 0 MinT t H H0); clear H;
intro H; apply sin_gt_0...
repeat simple apply Rmult_lt_0_compat...
apply rho_V_is_pos...
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
try (apply Rinv_0_lt_compat; simpl in |- *; prove_sup0)...
generalize (rho_t_PI2 t); intro H1; assert (Hyp : (0 < 2)%R)...
prove_sup0...
generalize
(Rmult_lt_compat_r (/ 2) (rho_V * t) (PI / 2) (Rinv_0_lt_compat 2 Hyp) H1);
rewrite Rmult_assoc; replace (PI / 2 * / 2)%R with (PI / 4)%R...
intro H2; apply Rlt_trans with (PI / 4)%R...
apply (Rlt_trans (PI / 4) (PI / 2) PI PI4_RLT_PI2 PI2_Rlt_PI)...
unfold Rdiv in |- *; rewrite Rmult_assoc; rewrite <- Rinv_mult_distr...
Qed.
Variable T : TimeT.
Definition MaxDistance : R := (V * T + ConflictRange)%R.
Definition MinDistance : R := (m T - ConflictRange)%R.
Definition MaxDistance_ub : R := (V * MaxT + ConflictRange)%R.
Definition MinDistance_lb : R := (m_rho_ub MinT - ConflictRange)%R.

(**********)
Lemma MaxDistance_ub_0 : (MaxDistance <= MaxDistance_ub)%R.
unfold MaxDistance, MaxDistance_ub in |- *.