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(* Copyright 2006 Milad Niqui                                           *)
(* This file is distributed under the terms of the                      *)
(* GNU Lesser General Public License Version 2.1                        *)
(* A copy of the license can be found at                                *)
(*                  <http://www.gnu.org/licenses/lgpl-2.1.html>         *)
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Require Export Qsyntax.
Require Export Field_Theory_Q.
Require Export Q_ordered_field_properties.
Definition Qabs (q:Q):Q:=
match q with
| Zero => Zero
| Qpos qp => Qpos qp
| Qneg qp => Qpos qp
end.
intros q p H; apply (proj1 (Qabs_5 _ _ H)).
Qed.

Lemma lower_bound_affine_base_interval_twice:forall a b c d x, -Qone<=x -> x<=Qone ->
-Qabs (a*c) + -Qabs (a*d+c*b)+b*d <= (a*x+b)*(c*x+d).