apply Rlt_le_trans with (PI_lb / 2)%R.
unfold PI_lb in |- *; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2%R.
prove_sup.
rewrite Rmult_1_r; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m;
[ prove_sup | discrR ].
left; unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2));
apply Rmult_lt_compat_l; [ apply Rinv_0_lt_compat; prove_sup | assumption ].
Qed.

(**********)
Lemma D_G_pos :
forall t : R,
(0 <= t)%R ->
(t < 2 / rho (vi intr))%R ->
(G t <= 0)%R -> (0 <= derive_pt G t (G_derivable t))%R.