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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 alias id "R" = "cic:/CoRN/algebra/CPoly_ApZero/Poly_Representation/R.var".
50 alias id "a_" = "cic:/CoRN/algebra/CPoly_ApZero/Poly_Representation/a_.var".
52 alias id "distinct_a_" = "cic:/CoRN/algebra/CPoly_ApZero/Poly_Representation/distinct_a_.var".
54 alias id "f" = "cic:/CoRN/algebra/CPoly_ApZero/Poly_Representation/f.var".
56 alias id "n" = "cic:/CoRN/algebra/CPoly_ApZero/Poly_Representation/n.var".
58 alias id "degree_f" = "cic:/CoRN/algebra/CPoly_ApZero/Poly_Representation/degree_f.var".
63 Notation RX := (cpoly_cring R).
68 include "tactics/Transparent_algebra.ma".
70 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_linear_shifted.con".
72 include "tactics/Opaque_algebra.ma".
74 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_linear_factor.con".
76 inline "cic:/CoRN/algebra/CPoly_ApZero/zero_poly.con".
78 inline "cic:/CoRN/algebra/CPoly_ApZero/identical_poly.con".
80 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor'.con".
82 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor'_degree.con".
84 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor'_zero.con".
86 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor'_apzero.con".
89 Hint Resolve poly_01_factor'_zero.
92 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor.con".
94 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor_degree.con".
96 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor_zero.con".
98 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_factor_one.con".
101 Hint Resolve poly_01_factor_zero poly_01_factor_one: algebra.
104 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01.con".
106 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_degree'.con".
108 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_degree.con".
110 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_zero.con".
112 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_01_one.con".
115 Hint Resolve poly_01_zero poly_01_one: algebra.
118 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_representation''.con".
120 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_representation'.con".
122 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_representation.con".
125 Hint Resolve poly_representation: algebra.
128 inline "cic:/CoRN/algebra/CPoly_ApZero/Cpoly_choose_apzero.con".
131 End Poly_Representation
135 Section Characteristic_zero
139 If we are in a field of characteristic zero, the previous result can be
143 alias id "R" = "cic:/CoRN/algebra/CPoly_ApZero/Characteristic_zero/R.var".
147 alias id "H" = "cic:/CoRN/algebra/CPoly_ApZero/Characteristic_zero/H.var".
154 Notation RX := (cpoly_cring R).
159 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_apzero.con".
162 Also, in this situation polynomials are extensional functions.
165 inline "cic:/CoRN/algebra/CPoly_ApZero/poly_extensional.con".
168 End Characteristic_zero
172 ** Polynomials are nonzero on any interval
176 Section Poly_ApZero_Interval
179 alias id "R" = "cic:/CoRN/algebra/CPoly_ApZero/Poly_ApZero_Interval/R.var".
184 Notation RX := (cpoly_cring R).
189 inline "cic:/CoRN/algebra/CPoly_ApZero/Cpoly_apzero_interval.con".
192 End Poly_ApZero_Interval