matita/contribs/CoRN-Decl/algebra/CPoly_Degree.ma

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  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 set "baseuri" "cic:/matita/CoRN-Decl/algebra/CPoly_Degree".
  18
  19 include "CoRN.ma".
  20
  21 (* $Id: CPoly_Degree.v,v 1.5 2004/04/23 10:00:53 lcf Exp $ *)
  22
  23 include "algebra/CPoly_NthCoeff.ma".
  24
  25 include "algebra/CFields.ma".
  26
  27 (*#* *Degrees of Polynomials
  28 ** Degrees of polynomials over a ring
  29 %\begin{convention}%
  30 Let [R] be a ring and write [RX] for the ring of polynomials
  31 over [R].
  32 %\end{convention}%
  33 *)
  34
  35 (* UNEXPORTED
  36 Section Degree_def
  37 *)
  38
  39 alias id "R" = "cic:/CoRN/algebra/CPoly_Degree/Degree_def/R.var".
  40
  41 (* begin hide *)
  42
  43 (* NOTATION
  44 Notation RX := (cpoly_cring R).
  45 *)
  46
  47 (* end hide *)
  48
  49 (*#*
  50 The length of a polynomial is the number of its coefficients. This is
  51 a syntactical property, as the highest coefficient may be [0]. Note that
  52 the `zero' polynomial [cpoly_zero] has length [0],
  53 a constant polynomial has length [1] and so forth. So the length
  54 is always [1] higher than the `degree' (assuming that the highest
  55 coefficient is [[#]Zero])!
  56 *)
  57
  58 inline "cic:/CoRN/algebra/CPoly_Degree/lth_of_poly.con".
  59
  60 (*#*
  61 When dealing with constructive polynomials, notably over the reals or
  62 complex numbers, the degree may be unknown, as we can not decide
  63 whether the highest coefficient is [[#]Zero]. Hence,
  64 degree is a relation between polynomials and natural numbers; if the
  65 degree is unknown for polynomial [p], degree(n,p) doesn't hold for
  66 any [n].  If we don't know the degree of [p], we may still
  67 know it to be below or above a certain number. E.g. for the polynomial
  68 $p_0 +p_1 X +\cdots + p_{n-1} X^{n-1}$#p0 +p1 X + ... + p(n-1)
  69 X^(n-1)#, if $p_i \mathrel{\#}0$#pi apart from 0#, we can say that the
  70 `degree is at least [i]' and if $p_{j+1} = \ldots =p_n =0$#p(j+1)
  71 = ... =pn =0# (with [n] the length of the polynomial), we can say
  72 that the `degree is at most [j]'.
  73 *)
  74
  75 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le.con".
  76
  77 inline "cic:/CoRN/algebra/CPoly_Degree/degree.con".
  78
  79 inline "cic:/CoRN/algebra/CPoly_Degree/monic.con".
  80
  81 inline "cic:/CoRN/algebra/CPoly_Degree/odd_cpoly.con".
  82
  83 inline "cic:/CoRN/algebra/CPoly_Degree/even_cpoly.con".
  84
  85 inline "cic:/CoRN/algebra/CPoly_Degree/regular.con".
  86
  87 (* UNEXPORTED
  88 End Degree_def
  89 *)
  90
  91 (* UNEXPORTED
  92 Implicit Arguments degree_le [R].
  93 *)
  94
  95 (* UNEXPORTED
  96 Implicit Arguments degree [R].
  97 *)
  98
  99 (* UNEXPORTED
 100 Implicit Arguments monic [R].
 101 *)
 102
 103 (* UNEXPORTED
 104 Implicit Arguments lth_of_poly [R].
 105 *)
 106
 107 (* UNEXPORTED
 108 Section Degree_props
 109 *)
 110
 111 alias id "R" = "cic:/CoRN/algebra/CPoly_Degree/Degree_props/R.var".
 112
 113 (* begin hide *)
 114
 115 (* NOTATION
 116 Notation RX := (cpoly_cring R).
 117 *)
 118
 119 (* end hide *)
 120
 121 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_wd.con".
 122
 123 inline "cic:/CoRN/algebra/CPoly_Degree/degree_wd.con".
 124
 125 inline "cic:/CoRN/algebra/CPoly_Degree/monic_wd.con".
 126
 127 inline "cic:/CoRN/algebra/CPoly_Degree/degree_imp_degree_le.con".
 128
 129 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_c_.con".
 130
 131 inline "cic:/CoRN/algebra/CPoly_Degree/degree_c_.con".
 132
 133 inline "cic:/CoRN/algebra/CPoly_Degree/monic_c_one.con".
 134
 135 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_x_.con".
 136
 137 inline "cic:/CoRN/algebra/CPoly_Degree/degree_x_.con".
 138
 139 inline "cic:/CoRN/algebra/CPoly_Degree/monic_x_.con".
 140
 141 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_mon.con".
 142
 143 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_inv.con".
 144
 145 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_plus.con".
 146
 147 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_minus.con".
 148
 149 inline "cic:/CoRN/algebra/CPoly_Degree/Sum_degree_le.con".
 150
 151 inline "cic:/CoRN/algebra/CPoly_Degree/degree_inv.con".
 152
 153 inline "cic:/CoRN/algebra/CPoly_Degree/degree_plus_rht.con".
 154
 155 inline "cic:/CoRN/algebra/CPoly_Degree/degree_minus_lft.con".
 156
 157 inline "cic:/CoRN/algebra/CPoly_Degree/monic_plus.con".
 158
 159 inline "cic:/CoRN/algebra/CPoly_Degree/monic_minus.con".
 160
 161 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_mult.con".
 162
 163 inline "cic:/CoRN/algebra/CPoly_Degree/degree_mult_aux.con".
 164
 165 (* UNEXPORTED
 166 Hint Resolve degree_mult_aux: algebra.
 167 *)
 168
 169 inline "cic:/CoRN/algebra/CPoly_Degree/monic_mult.con".
 170
 171 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_nexp.con".
 172
 173 inline "cic:/CoRN/algebra/CPoly_Degree/monic_nexp.con".
 174
 175 inline "cic:/CoRN/algebra/CPoly_Degree/lt_i_lth_of_poly.con".
 176
 177 inline "cic:/CoRN/algebra/CPoly_Degree/poly_degree_lth.con".
 178
 179 inline "cic:/CoRN/algebra/CPoly_Degree/Cpoly_ex_degree.con".
 180
 181 inline "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum''.con".
 182
 183 (* UNEXPORTED
 184 Hint Resolve poly_as_sum'': algebra.
 185 *)
 186
 187 inline "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum'.con".
 188
 189 inline "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum.con".
 190
 191 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_zero.con".
 192
 193 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_1_imp.con".
 194
 195 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_cpoly_linear.con".
 196
 197 inline "cic:/CoRN/algebra/CPoly_Degree/monic_cpoly_linear.con".
 198
 199 inline "cic:/CoRN/algebra/CPoly_Degree/monic_one.con".
 200
 201 inline "cic:/CoRN/algebra/CPoly_Degree/monic_apzero.con".
 202
 203 (* UNEXPORTED
 204 End Degree_props
 205 *)
 206
 207 (* UNEXPORTED
 208 Hint Resolve poly_as_sum'' poly_as_sum' poly_as_sum: algebra.
 209 *)
 210
 211 (* UNEXPORTED
 212 Hint Resolve degree_mult_aux: algebra.
 213 *)
 214
 215 (* UNEXPORTED
 216 Section degree_props_Field
 217 *)
 218
 219 (*#* ** Degrees of polynomials over a field
 220 %\begin{convention}% Let [F] be a field and write [FX] for the ring of
 221 polynomials over [F].
 222 %\end{convention}%
 223 *)
 224
 225 alias id "F" = "cic:/CoRN/algebra/CPoly_Degree/degree_props_Field/F.var".
 226
 227 (* begin hide *)
 228
 229 (* NOTATION
 230 Notation FX := (cpoly_cring F).
 231 *)
 232
 233 (* end hide *)
 234
 235 inline "cic:/CoRN/algebra/CPoly_Degree/degree_mult.con".
 236
 237 inline "cic:/CoRN/algebra/CPoly_Degree/degree_nexp.con".
 238
 239 inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_mult_imp.con".
 240
 241 inline "cic:/CoRN/algebra/CPoly_Degree/degree_mult_imp.con".
 242
 243 (* UNEXPORTED
 244 End degree_props_Field
 245 *)
 246