Proof with trivial.
intros; assert (H := cond_2 t)...
unfold MaxT in H; apply Rlt_trans with (PI_lb / 2)%R...
unfold rho_ub in |- *; cut (g < 33)%R...
cut (tan_ub_MaxBank < 4 / 5)%R...
cut (/ h < / 200)%R...
intros; generalize g_pos; intro H3; generalize tan_ub_MaxBank_pos; intro H4;
generalize (Rinv_0_lt_compat h (TypeSpeed_pos h));
intro H5...
generalize
(Rmult_le_0_lt_compat g 33 tan_ub_MaxBank (4 / 5)
(Rlt_le 0 g H3) (Rlt_le 0 tan_ub_MaxBank H4) H2 H1)...
intro H6; generalize (Rmult_lt_0_compat g tan_ub_MaxBank H3 H4); intro H7...
assert
(H8 :=
Rmult_le_0_lt_compat (g * tan_ub_MaxBank) (33 * (4 / 5))
(/ h) (/ 200) (Rlt_le 0 (g * tan_ub_MaxBank) H7)
(Rlt_le 0 (/ h) H5) H6 H0)...
apply Rlt_trans with (33 * / 200 * (4 / 5) * t)%R...
apply Rmult_lt_compat_r...
apply Rlt_le_trans with MinT...
apply MinT_is_pos...
apply (cond_1 t)...
rewrite Rmult_assoc; rewrite (Rmult_comm (/ 200)); unfold Rdiv in |- *;
do 2 rewrite <- Rmult_assoc...
apply Rle_lt_trans with (33 * / 200 * (4 / 5) * 10)%R...
apply Rmult_le_compat_l...
left; apply Rmult_lt_0_compat...
apply Rmult_lt_0_compat; [ prove_sup | apply Rinv_0_lt_compat; prove_sup ]...
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ prove_sup | apply Rinv_0_lt_compat; prove_sup ]...
apply Rmult_lt_reg_l with 2%R...
prove_sup...
apply Rmult_lt_reg_l with 200%R...
prove_sup...
apply Rmult_lt_reg_l with 5%R...
prove_sup...
set (x := 200%R); set (y := 33%R); set (z := 10%R); set (u := 5%R);
set (v := 4%R); set (w := 2%R)...
unfold Rdiv in |- *;
replace (u * (x * (w * (PI_lb * / w))))%R with (x * PI_lb * u * (w * / w))%R;
[ idtac | ring ]...
replace (u * (x * (w * (y * / x * (v * / u) * z))))%R with
(w * y * v * z * (u * / u) * (x * / x))%R; [ idtac | ring ]...
repeat rewrite <- Rinv_r_sym; unfold x, y, z, u, v, w in |- *; try discrR...
unfold PI_lb in |- *; prove_sup...
apply Rinv_lt_contravar...
apply Rmult_lt_0_compat...
prove_sup...
apply Rlt_le_trans with MinSpeed...
unfold MinSpeed in |- *; prove_sup...
apply (v_cond1 h)...
apply Rlt_le_trans with MinSpeed...
unfold MinSpeed in |- *; prove_sup...
apply (v_cond1 h)...
unfold tan_ub_MaxBank in |- *; unfold Rdiv in |- *...
apply Rmult_lt_reg_l with 10%R...
prove_sup...
rewrite (Rmult_comm 10); rewrite Rmult_assoc; rewrite <- Rinv_l_sym...
rewrite Rmult_1_r; apply Rmult_lt_reg_l with 5%R...
prove_sup...
pattern 5%R at 1 in |- *; rewrite (Rmult_comm 5); do 2 rewrite Rmult_assoc;
rewrite <- Rinv_l_sym...
prove_sup...
discrR...
discrR...
unfold g in |- *; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 10%R...
prove_sup...
rewrite <- Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym...
rewrite Rmult_1_r; prove_sup...
discrR...
unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2));
apply Rmult_lt_compat_l...
apply Rinv_0_lt_compat; prove_sup...
assert (H0 := PI_approx); elim H0...
Qed.

(**********)
Theorem rho_t_PI2 : forall t : TimeT, (rho_V * t < PI / 2)%R.
Proof with trivial.