1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
19 (* $Id: OddPolyRootIR.v,v 1.5 2004/04/23 10:01:05 lcf Exp $ *)
21 include "reals/IVT.ma".
23 (*#* * Roots of polynomials of odd degree *)
29 (*#* ** Monic polynomials are positive near infinity
30 %\begin{convention}% Let [R] be an ordered field.
35 cic:/CoRN/reals/OddPolyRootIR/CPoly_Big/R.var
40 inline procedural "cic:/CoRN/reals/OddPolyRootIR/CPoly_Big/RX.con" "CPoly_Big__" as definition.
44 inline procedural "cic:/CoRN/reals/OddPolyRootIR/Cbigger.con" as lemma.
46 inline procedural "cic:/CoRN/reals/OddPolyRootIR/Ccpoly_big.con" as lemma.
48 inline procedural "cic:/CoRN/reals/OddPolyRootIR/cpoly_pos.con" as lemma.
50 inline procedural "cic:/CoRN/reals/OddPolyRootIR/Ccpoly_pos'.con" as lemma.
60 (*#* **Flipping a polynomial
61 %\begin{convention}% Let [R] be a ring.
66 cic:/CoRN/reals/OddPolyRootIR/Flip_Poly/R.var
71 inline procedural "cic:/CoRN/reals/OddPolyRootIR/Flip_Poly/RX.con" "Flip_Poly__" as definition.
75 inline procedural "cic:/CoRN/reals/OddPolyRootIR/flip.con" as definition.
77 inline procedural "cic:/CoRN/reals/OddPolyRootIR/flip_poly.con" as lemma.
79 inline procedural "cic:/CoRN/reals/OddPolyRootIR/flip_coefficient.con" as lemma.
82 Hint Resolve flip_coefficient: algebra.
85 inline procedural "cic:/CoRN/reals/OddPolyRootIR/flip_odd.con" as lemma.
92 Hint Resolve flip_poly: algebra.
99 (*#* ** Sign of a polynomial of odd degree
100 %\begin{convention}% Let [R] be an ordered field.
105 cic:/CoRN/reals/OddPolyRootIR/OddPoly_Signs/R.var
110 inline procedural "cic:/CoRN/reals/OddPolyRootIR/OddPoly_Signs/RX.con" "OddPoly_Signs__" as definition.
114 inline procedural "cic:/CoRN/reals/OddPolyRootIR/oddpoly_pos.con" as lemma.
116 inline procedural "cic:/CoRN/reals/OddPolyRootIR/oddpoly_pos'.con" as lemma.
118 inline procedural "cic:/CoRN/reals/OddPolyRootIR/oddpoly_neg.con" as lemma.
128 (*#* ** The norm of a polynomial
129 %\begin{convention}% Let [R] be a field, and [RX] the polynomials over
135 cic:/CoRN/reals/OddPolyRootIR/Poly_Norm/R.var
140 inline procedural "cic:/CoRN/reals/OddPolyRootIR/Poly_Norm/RX.con" "Poly_Norm__" as definition.
144 inline procedural "cic:/CoRN/reals/OddPolyRootIR/poly_norm_aux.con" as lemma.
146 inline procedural "cic:/CoRN/reals/OddPolyRootIR/poly_norm.con" as definition.
148 inline procedural "cic:/CoRN/reals/OddPolyRootIR/poly_norm_monic.con" as lemma.
150 inline procedural "cic:/CoRN/reals/OddPolyRootIR/poly_norm_apply.con" as lemma.
160 (*#* ** Roots of polynomials of odd degree
161 Polynomials of odd degree over the reals always have a root. *)
163 inline procedural "cic:/CoRN/reals/OddPolyRootIR/oddpoly_root'.con" as lemma.
165 inline procedural "cic:/CoRN/reals/OddPolyRootIR/oddpoly_root.con" as lemma.
167 inline procedural "cic:/CoRN/reals/OddPolyRootIR/realpolyn_oddhaszero.con" as lemma.