helm/software/matita/contribs/CoRN-Procedural/algebra/CPoly_Degree.mma

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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 include "CoRN.ma".
  18
  19 (* $Id: CPoly_Degree.v,v 1.5 2004/04/23 10:00:53 lcf Exp $ *)
  20
  21 include "algebra/CPoly_NthCoeff.ma".
  22
  23 include "algebra/CFields.ma".
  24
  25 (*#* *Degrees of Polynomials
  26 ** Degrees of polynomials over a ring
  27 %\begin{convention}%
  28 Let [R] be a ring and write [RX] for the ring of polynomials
  29 over [R].
  30 %\end{convention}%
  31 *)
  32
  33 (* UNEXPORTED
  34 Section Degree_def
  35 *)
  36
  37 alias id "R" = "cic:/CoRN/algebra/CPoly_Degree/Degree_def/R.var".
  38
  39 (* begin hide *)
  40
  41 (* NOTATION
  42 Notation RX := (cpoly_cring R).
  43 *)
  44
  45 (* end hide *)
  46
  47 (*#*
  48 The length of a polynomial is the number of its coefficients. This is
  49 a syntactical property, as the highest coefficient may be [0]. Note that
  50 the `zero' polynomial [cpoly_zero] has length [0],
  51 a constant polynomial has length [1] and so forth. So the length
  52 is always [1] higher than the `degree' (assuming that the highest
  53 coefficient is [[#]Zero])!
  54 *)
  55
  56 inline procedural "cic:/CoRN/algebra/CPoly_Degree/lth_of_poly.con" as definition.
  57
  58 (*#*
  59 When dealing with constructive polynomials, notably over the reals or
  60 complex numbers, the degree may be unknown, as we can not decide
  61 whether the highest coefficient is [[#]Zero]. Hence,
  62 degree is a relation between polynomials and natural numbers; if the
  63 degree is unknown for polynomial [p], degree(n,p) doesn't hold for
  64 any [n].  If we don't know the degree of [p], we may still
  65 know it to be below or above a certain number. E.g. for the polynomial
  66 $p_0 +p_1 X +\cdots + p_{n-1} X^{n-1}$#p0 +p1 X + ... + p(n-1)
  67 X^(n-1)#, if $p_i \mathrel{\#}0$#pi apart from 0#, we can say that the
  68 `degree is at least [i]' and if $p_{j+1} = \ldots =p_n =0$#p(j+1)
  69 = ... =pn =0# (with [n] the length of the polynomial), we can say
  70 that the `degree is at most [j]'.
  71 *)
  72
  73 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le.con" as definition.
  74
  75 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree.con" as definition.
  76
  77 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic.con" as definition.
  78
  79 inline procedural "cic:/CoRN/algebra/CPoly_Degree/odd_cpoly.con" as definition.
  80
  81 inline procedural "cic:/CoRN/algebra/CPoly_Degree/even_cpoly.con" as definition.
  82
  83 inline procedural "cic:/CoRN/algebra/CPoly_Degree/regular.con" as definition.
  84
  85 (* UNEXPORTED
  86 End Degree_def
  87 *)
  88
  89 (* UNEXPORTED
  90 Implicit Arguments degree_le [R].
  91 *)
  92
  93 (* UNEXPORTED
  94 Implicit Arguments degree [R].
  95 *)
  96
  97 (* UNEXPORTED
  98 Implicit Arguments monic [R].
  99 *)
 100
 101 (* UNEXPORTED
 102 Implicit Arguments lth_of_poly [R].
 103 *)
 104
 105 (* UNEXPORTED
 106 Section Degree_props
 107 *)
 108
 109 alias id "R" = "cic:/CoRN/algebra/CPoly_Degree/Degree_props/R.var".
 110
 111 (* begin hide *)
 112
 113 (* NOTATION
 114 Notation RX := (cpoly_cring R).
 115 *)
 116
 117 (* end hide *)
 118
 119 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_wd.con" as lemma.
 120
 121 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_wd.con" as lemma.
 122
 123 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_wd.con" as lemma.
 124
 125 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_imp_degree_le.con" as lemma.
 126
 127 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_c_.con" as lemma.
 128
 129 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_c_.con" as lemma.
 130
 131 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_c_one.con" as lemma.
 132
 133 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_x_.con" as lemma.
 134
 135 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_x_.con" as lemma.
 136
 137 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_x_.con" as lemma.
 138
 139 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_mon.con" as lemma.
 140
 141 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_inv.con" as lemma.
 142
 143 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_plus.con" as lemma.
 144
 145 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_minus.con" as lemma.
 146
 147 inline procedural "cic:/CoRN/algebra/CPoly_Degree/Sum_degree_le.con" as lemma.
 148
 149 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_inv.con" as lemma.
 150
 151 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_plus_rht.con" as lemma.
 152
 153 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_minus_lft.con" as lemma.
 154
 155 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_plus.con" as lemma.
 156
 157 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_minus.con" as lemma.
 158
 159 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_mult.con" as lemma.
 160
 161 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_mult_aux.con" as lemma.
 162
 163 (* UNEXPORTED
 164 Hint Resolve degree_mult_aux: algebra.
 165 *)
 166
 167 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_mult.con" as lemma.
 168
 169 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_nexp.con" as lemma.
 170
 171 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_nexp.con" as lemma.
 172
 173 inline procedural "cic:/CoRN/algebra/CPoly_Degree/lt_i_lth_of_poly.con" as lemma.
 174
 175 inline procedural "cic:/CoRN/algebra/CPoly_Degree/poly_degree_lth.con" as lemma.
 176
 177 inline procedural "cic:/CoRN/algebra/CPoly_Degree/Cpoly_ex_degree.con" as lemma.
 178
 179 inline procedural "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum''.con" as lemma.
 180
 181 (* UNEXPORTED
 182 Hint Resolve poly_as_sum'': algebra.
 183 *)
 184
 185 inline procedural "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum'.con" as lemma.
 186
 187 inline procedural "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum.con" as lemma.
 188
 189 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_zero.con" as lemma.
 190
 191 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_1_imp.con" as lemma.
 192
 193 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_cpoly_linear.con" as lemma.
 194
 195 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_cpoly_linear.con" as lemma.
 196
 197 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_one.con" as lemma.
 198
 199 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_apzero.con" as lemma.
 200
 201 (* UNEXPORTED
 202 End Degree_props
 203 *)
 204
 205 (* UNEXPORTED
 206 Hint Resolve poly_as_sum'' poly_as_sum' poly_as_sum: algebra.
 207 *)
 208
 209 (* UNEXPORTED
 210 Hint Resolve degree_mult_aux: algebra.
 211 *)
 212
 213 (* UNEXPORTED
 214 Section degree_props_Field
 215 *)
 216
 217 (*#* ** Degrees of polynomials over a field
 218 %\begin{convention}% Let [F] be a field and write [FX] for the ring of
 219 polynomials over [F].
 220 %\end{convention}%
 221 *)
 222
 223 alias id "F" = "cic:/CoRN/algebra/CPoly_Degree/degree_props_Field/F.var".
 224
 225 (* begin hide *)
 226
 227 (* NOTATION
 228 Notation FX := (cpoly_cring F).
 229 *)
 230
 231 (* end hide *)
 232
 233 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_mult.con" as lemma.
 234
 235 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_nexp.con" as lemma.
 236
 237 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_mult_imp.con" as lemma.
 238
 239 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_mult_imp.con" as lemma.
 240
 241 (* UNEXPORTED
 242 End degree_props_Field
 243 *)
 244