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 *)
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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.
34 alias id "R" = "cic:/CoRN/reals/OddPolyRootIR/CPoly_Big/R.var".
38 inline procedural "cic:/CoRN/reals/OddPolyRootIR/CPoly_Big/RX.con" "CPoly_Big__" as definition.
42 inline procedural "cic:/CoRN/reals/OddPolyRootIR/Cbigger.con" as lemma.
44 inline procedural "cic:/CoRN/reals/OddPolyRootIR/Ccpoly_big.con" as lemma.
46 inline procedural "cic:/CoRN/reals/OddPolyRootIR/cpoly_pos.con" as lemma.
48 inline procedural "cic:/CoRN/reals/OddPolyRootIR/Ccpoly_pos'.con" as lemma.
58 (*#* **Flipping a polynomial
59 %\begin{convention}% Let [R] be a ring.
63 alias id "R" = "cic:/CoRN/reals/OddPolyRootIR/Flip_Poly/R.var".
67 inline procedural "cic:/CoRN/reals/OddPolyRootIR/Flip_Poly/RX.con" "Flip_Poly__" as definition.
71 inline procedural "cic:/CoRN/reals/OddPolyRootIR/flip.con" as definition.
73 inline procedural "cic:/CoRN/reals/OddPolyRootIR/flip_poly.con" as lemma.
75 inline procedural "cic:/CoRN/reals/OddPolyRootIR/flip_coefficient.con" as lemma.
78 Hint Resolve flip_coefficient: algebra.
81 inline procedural "cic:/CoRN/reals/OddPolyRootIR/flip_odd.con" as lemma.
88 Hint Resolve flip_poly: algebra.
95 (*#* ** Sign of a polynomial of odd degree
96 %\begin{convention}% Let [R] be an ordered field.
100 alias id "R" = "cic:/CoRN/reals/OddPolyRootIR/OddPoly_Signs/R.var".
104 inline procedural "cic:/CoRN/reals/OddPolyRootIR/OddPoly_Signs/RX.con" "OddPoly_Signs__" as definition.
108 inline procedural "cic:/CoRN/reals/OddPolyRootIR/oddpoly_pos.con" as lemma.
110 inline procedural "cic:/CoRN/reals/OddPolyRootIR/oddpoly_pos'.con" as lemma.
112 inline procedural "cic:/CoRN/reals/OddPolyRootIR/oddpoly_neg.con" as lemma.
122 (*#* ** The norm of a polynomial
123 %\begin{convention}% Let [R] be a field, and [RX] the polynomials over
128 alias id "R" = "cic:/CoRN/reals/OddPolyRootIR/Poly_Norm/R.var".
132 inline procedural "cic:/CoRN/reals/OddPolyRootIR/Poly_Norm/RX.con" "Poly_Norm__" as definition.
136 inline procedural "cic:/CoRN/reals/OddPolyRootIR/poly_norm_aux.con" as lemma.
138 inline procedural "cic:/CoRN/reals/OddPolyRootIR/poly_norm.con" as definition.
140 inline procedural "cic:/CoRN/reals/OddPolyRootIR/poly_norm_monic.con" as lemma.
142 inline procedural "cic:/CoRN/reals/OddPolyRootIR/poly_norm_apply.con" as lemma.
152 (*#* ** Roots of polynomials of odd degree
153 Polynomials of odd degree over the reals always have a root. *)
155 inline procedural "cic:/CoRN/reals/OddPolyRootIR/oddpoly_root'.con" as lemma.
157 inline procedural "cic:/CoRN/reals/OddPolyRootIR/oddpoly_root.con" as lemma.
159 inline procedural "cic:/CoRN/reals/OddPolyRootIR/realpolyn_oddhaszero.con" as lemma.