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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 *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/algebra/CPoly_ApZero".
21 (* $Id: CPoly_ApZero.v,v 1.3 2004/04/23 10:00:53 lcf Exp $ *)
23 include "algebra/CPoly_Degree.ma".
25 include "algebra/COrdFields2.ma".
27 (*#* * Polynomials apart from zero *)
29 inline "cic:/CoRN/algebra/CPoly_ApZero/distinct1.con".
32 Implicit Arguments distinct1 [A].
36 Section Poly_Representation.
40 ** Representation of polynomials
41 %\begin{convention}% Let [R] be a field, [RX] the ring of polynomials
42 over [R], [a_ : nat->R] with [(distinct1 a_)] and let [f] be a
43 polynomial over [R], [n] a natural with [(degree_le n f)], i.e. [f]
44 has degree at most [n].
48 inline "cic:/CoRN/algebra/CPoly_ApZero/R.var".
50 inline "cic:/CoRN/algebra/CPoly_ApZero/a_.var".
52 inline "cic:/CoRN/algebra/CPoly_ApZero/distinct_a_.var".
54 inline "cic:/CoRN/algebra/CPoly_ApZero/f.var".
56 inline "cic:/CoRN/algebra/CPoly_ApZero/n.var".
58 inline "cic:/CoRN/algebra/CPoly_ApZero/degree_f.var".
64 include "tactics/Transparent_algebra.ma".
66 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_linear_shifted.con".
68 include "tactics/Opaque_algebra.ma".
70 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_linear_factor.con".
72 inline "cic:/CoRN/algebra/CPoly_ApZero/zero_poly.con".
74 inline "cic:/CoRN/algebra/CPoly_ApZero/identical_poly.con".
76 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor'.con".
78 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor'_degree.con".
80 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor'_zero.con".
82 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor'_apzero.con".
85 Hint Resolve poly_01_factor'_zero.
88 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor.con".
90 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor_degree.con".
92 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor_zero.con".
94 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor_one.con".
97 Hint Resolve poly_01_factor_zero poly_01_factor_one: algebra.
100 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01.con".
102 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_degree'.con".
104 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_degree.con".
106 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_zero.con".
108 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_one.con".
111 Hint Resolve poly_01_zero poly_01_one: algebra.
114 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_representation''.con".
116 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_representation'.con".
118 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_representation.con".
121 Hint Resolve poly_representation: algebra.
124 inline "cic:/CoRN/algebra/CPoly_ApZero/Cpoly_choose_apzero.con".
127 End Poly_Representation.
131 Section Characteristic_zero.
135 If we are in a field of characteristic zero, the previous result can be
139 inline "cic:/CoRN/algebra/CPoly_ApZero/R.var".
143 inline "cic:/CoRN/algebra/CPoly_ApZero/H.var".
151 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_apzero.con".
154 Also, in this situation polynomials are extensional functions.
157 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_extensional.con".
160 End Characteristic_zero.
164 ** Polynomials are nonzero on any interval
168 Section Poly_ApZero_Interval.
171 inline "cic:/CoRN/algebra/CPoly_ApZero/R.var".
177 inline "cic:/CoRN/algebra/CPoly_ApZero/Cpoly_apzero_interval.con".
180 End Poly_ApZero_Interval.