intro; unfold m_rho_ub, m in |- *; repeat rewrite Rmult_assoc;
apply Rmult_le_compat_l...
left; simpl in |- *; prove_sup0...
cut (0 <= r_lb V <= r_V)%R...
cut (0 <= sin_lb (rho_lb V * (t / 2)) <= sin (rho_V * (t / 2)))%R...
intros; elim H; intros H1 H2; elim H0; intros H3 H4...
apply Rmult_le_compat...
cut (0 < rho_lb V * (t / 2) <= rho_V * (t / 2))%R...
cut (rho_V * (t / 2) <= PI / 2)%R...
intros; elim H0; intros H1 H2; split...
left; apply sin_lb_gt_0...
apply Rle_trans with (rho_V * (t / 2))%R...
apply Rle_trans with (sin (rho_lb V * (t / 2)))...
generalize
(SIN (rho_lb V * (t / 2)) (Rlt_le 0 (rho_lb V * (t / 2)) H1)
(Rle_trans (rho_lb V * (t / 2)) (PI / 2) PI
(Rle_trans (rho_lb V * (t / 2)) (rho_V * (t / 2)) (PI / 2) H2 H)
(Rlt_le (PI / 2) PI PI2_Rlt_PI)))...
intro H3; elim H3; intros H4 H5...
apply sin_incr_1...
left; apply (Rlt_trans (- (PI / 2)) 0 (rho_lb V * (t / 2)) _PI2_RLT_0 H1)...
apply Rle_trans with (rho_V * (t / 2))%R...
left;
apply
(Rlt_trans (- (PI / 2)) 0 (rho_V * (t / 2)) _PI2_RLT_0
(Rlt_le_trans 0 (rho_lb V * (t / 2)) (rho_V * (t / 2)) H1 H2))...
left; apply Rlt_trans with (PI / 4)%R...
generalize (rho_t_PI2 t); intro H; unfold Rdiv in |- *;
replace (/ 4)%R with (/ 2 * / 2)%R...
repeat rewrite <- Rmult_assoc; apply Rmult_lt_compat_r...
apply Rinv_0_lt_compat; prove_sup0...
rewrite <- Rinv_mult_distr...
discrR...
discrR...
apply PI4_RLT_PI2...
split...
unfold Rdiv in |- *; repeat simple apply Rmult_lt_0_compat...
apply rho_lb_pos...
apply Rlt_le_trans with MinT...
apply MinT_is_pos...
apply (cond_1 t)...
apply Rinv_0_lt_compat; prove_sup0...
apply Rmult_le_compat_r...
left; unfold Rdiv in |- *; apply Rmult_lt_0_compat...
apply Rlt_le_trans with MinT...
apply MinT_is_pos...
apply (cond_1 t)...
apply Rinv_0_lt_compat; prove_sup0...
unfold rho_V in |- *...
left; apply rho_lb_0...
split...
unfold r_lb in |- *...
unfold Rdiv in |- *; left; apply Rmult_lt_0_compat...
apply (TypeSpeed_pos V)...
apply Rinv_0_lt_compat; apply rho_ub_pos...
unfold r_V in |- *; left; apply r_lb_0...
Qed.

(* Specifications: *)
(**********)
Lemma m_rho_T_pos : forall t : TimeT, (0 < m t)%R.
Proof with trivial.