intros; unfold m in |- *; repeat rewrite Rmult_assoc...
repeat apply Rmult_le_compat_l...
left; simpl in |- *; prove_sup...
left; apply r_V_is_pos...
generalize (rho_t_PI2 t1); intro H1...
generalize (rho_t_PI2 t2); intro H2...
assert (Hyp : (0 < 2)%R)...
prove_sup0...
generalize
(Rmult_lt_compat_l (/ 2) (rho_V * t1) (PI / 2) (Rinv_0_lt_compat 2 Hyp) H1)...
generalize
(Rmult_lt_compat_l (/ 2) (rho_V * t2) (PI / 2) (Rinv_0_lt_compat 2 Hyp) H2)...
replace (/ 2 * (PI / 2))%R with (PI / 4)%R...
clear H1 H2; intros H1 H2...
rewrite Rmult_comm in H1...
rewrite Rmult_comm in H2...
unfold Rdiv in |- *...
generalize (cond_1 t1)...
generalize (cond_1 t2)...
generalize MinT_is_pos; intro H7...
intros H3 H4...
generalize (Rlt_le_trans 0 MinT t1 H7 H4); clear H4; intro H4...
generalize (Rlt_le_trans 0 MinT t2 H7 H3); clear H3; intro H3...
apply sin_incr_1...
left; apply Rlt_trans with 0%R...
apply _PI2_RLT_0...
repeat simple apply Rmult_lt_0_compat...
apply rho_V_is_pos...
apply Rinv_0_lt_compat; simpl in |- *; prove_sup0...
left; apply Rlt_trans with (PI / 4)%R...
rewrite <- Rmult_assoc...
apply PI4_RLT_PI2...
left; apply Rlt_trans with 0%R...
apply _PI2_RLT_0...
repeat simple apply Rmult_lt_0_compat...
apply rho_V_is_pos...
apply Rinv_0_lt_compat; simpl in |- *; prove_sup0...
left; apply Rlt_trans with (PI / 4)%R...
rewrite <- Rmult_assoc...
apply PI4_RLT_PI2...
apply Rmult_le_compat_l...
left; apply rho_V_is_pos...
apply Rmult_le_compat_r...
left; apply Rinv_0_lt_compat; simpl in |- *; prove_sup0...
unfold Rdiv in |- *; rewrite (Rmult_comm (/ 2)); rewrite Rmult_assoc;
rewrite <- Rinv_mult_distr...
Qed.

(**********)
Lemma m_rho_ub_0 : forall t : TimeT, (m_rho_ub t <= m t)%R.
Proof with trivial.