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(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
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(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
Definition rho (h : TypeSpeed) : R := (g * (tan MaxBank / h))%R.
Definition r (h : TypeSpeed) : R := (h / rho h)%R.
Definition rho_ub (h : TypeSpeed) : R := (g * (tan_ub_MaxBank / h))%R.
Definition rho_lb (h : TypeSpeed) : R := (g * (tan_lb_MaxBank / h))%R.
intro; assert (H := v_cond1 h).
apply Rlt_le_trans with MinSpeed.
unfold MinSpeed in |- *; prove_sup.
assumption.
Qed.
unfold g in |- *; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 10%R.
prove_sup.
rewrite Rmult_0_r; rewrite <- Rmult_comm; rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ prove_sup | discrR ].
Qed.
intros; unfold rho, rho_ub in |- *; apply Rmult_lt_compat_l;
[ apply g_pos
| generalize TypeSpeed_pos; intros; generalize (H h); clear H; intro H;
generalize (Rinv_0_lt_compat h H); intro H1; generalize tan_MaxBank_ub;
intro H2;
generalize (Rmult_lt_compat_l (/ h) (tan MaxBank) tan_ub_MaxBank H1 H2);
rewrite Rmult_comm; rewrite <- (Rmult_comm tan_ub_MaxBank);
intro H3; assumption ].
Qed.

(**********)
Lemma rho_lb_0 : forall h : TypeSpeed, (rho_lb h < rho h)%R.