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

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   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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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 set "baseuri" "cic:/matita/CoRN-Decl/algebra/CPoly_Degree".
  18
  19 (* $Id: CPoly_Degree.v,v 1.5 2004/04/23 10:00:53 lcf Exp $ *)
  20
  21 (* INCLUDE
  22 CPoly_NthCoeff
  23 *)
  24
  25 (* INCLUDE
  26 CFields
  27 *)
  28
  29 (*#* *Degrees of Polynomials
  30 ** Degrees of polynomials over a ring
  31 %\begin{convention}%
  32 Let [R] be a ring and write [RX] for the ring of polynomials
  33 over [R].
  34 %\end{convention}%
  35 *)
  36
  37 (* UNEXPORTED
  38 Section Degree_def.
  39 *)
  40
  41 inline cic:/CoRN/algebra/CPoly_Degree/R.var.
  42
  43 (* begin hide *)
  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 cic:/CoRN/algebra/CPoly_Degree/lth_of_poly.con.
  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 cic:/CoRN/algebra/CPoly_Degree/degree_le.con.
  74
  75 inline cic:/CoRN/algebra/CPoly_Degree/degree.con.
  76
  77 inline cic:/CoRN/algebra/CPoly_Degree/monic.con.
  78
  79 inline cic:/CoRN/algebra/CPoly_Degree/odd_cpoly.con.
  80
  81 inline cic:/CoRN/algebra/CPoly_Degree/even_cpoly.con.
  82
  83 inline cic:/CoRN/algebra/CPoly_Degree/regular.con.
  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 inline cic:/CoRN/algebra/CPoly_Degree/R.var.
 110
 111 (* begin hide *)
 112
 113 (* end hide *)
 114
 115 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_wd.con.
 116
 117 inline cic:/CoRN/algebra/CPoly_Degree/degree_wd.con.
 118
 119 inline cic:/CoRN/algebra/CPoly_Degree/monic_wd.con.
 120
 121 inline cic:/CoRN/algebra/CPoly_Degree/degree_imp_degree_le.con.
 122
 123 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_c_.con.
 124
 125 inline cic:/CoRN/algebra/CPoly_Degree/degree_c_.con.
 126
 127 inline cic:/CoRN/algebra/CPoly_Degree/monic_c_one.con.
 128
 129 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_x_.con.
 130
 131 inline cic:/CoRN/algebra/CPoly_Degree/degree_x_.con.
 132
 133 inline cic:/CoRN/algebra/CPoly_Degree/monic_x_.con.
 134
 135 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_mon.con.
 136
 137 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_inv.con.
 138
 139 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_plus.con.
 140
 141 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_minus.con.
 142
 143 inline cic:/CoRN/algebra/CPoly_Degree/Sum_degree_le.con.
 144
 145 inline cic:/CoRN/algebra/CPoly_Degree/degree_inv.con.
 146
 147 inline cic:/CoRN/algebra/CPoly_Degree/degree_plus_rht.con.
 148
 149 inline cic:/CoRN/algebra/CPoly_Degree/degree_minus_lft.con.
 150
 151 inline cic:/CoRN/algebra/CPoly_Degree/monic_plus.con.
 152
 153 inline cic:/CoRN/algebra/CPoly_Degree/monic_minus.con.
 154
 155 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_mult.con.
 156
 157 inline cic:/CoRN/algebra/CPoly_Degree/degree_mult_aux.con.
 158
 159 (* UNEXPORTED
 160 Hint Resolve degree_mult_aux: algebra.
 161 *)
 162
 163 inline cic:/CoRN/algebra/CPoly_Degree/monic_mult.con.
 164
 165 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_nexp.con.
 166
 167 inline cic:/CoRN/algebra/CPoly_Degree/monic_nexp.con.
 168
 169 inline cic:/CoRN/algebra/CPoly_Degree/lt_i_lth_of_poly.con.
 170
 171 inline cic:/CoRN/algebra/CPoly_Degree/poly_degree_lth.con.
 172
 173 inline cic:/CoRN/algebra/CPoly_Degree/Cpoly_ex_degree.con.
 174
 175 inline cic:/CoRN/algebra/CPoly_Degree/poly_as_sum''.con.
 176
 177 (* UNEXPORTED
 178 Hint Resolve poly_as_sum'': algebra.
 179 *)
 180
 181 inline cic:/CoRN/algebra/CPoly_Degree/poly_as_sum'.con.
 182
 183 inline cic:/CoRN/algebra/CPoly_Degree/poly_as_sum.con.
 184
 185 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_zero.con.
 186
 187 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_1_imp.con.
 188
 189 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_cpoly_linear.con.
 190
 191 inline cic:/CoRN/algebra/CPoly_Degree/monic_cpoly_linear.con.
 192
 193 inline cic:/CoRN/algebra/CPoly_Degree/monic_one.con.
 194
 195 inline cic:/CoRN/algebra/CPoly_Degree/monic_apzero.con.
 196
 197 (* UNEXPORTED
 198 End Degree_props.
 199 *)
 200
 201 (* UNEXPORTED
 202 Hint Resolve poly_as_sum'' poly_as_sum' poly_as_sum: algebra.
 203 *)
 204
 205 (* UNEXPORTED
 206 Hint Resolve degree_mult_aux: algebra.
 207 *)
 208
 209 (* UNEXPORTED
 210 Section degree_props_Field.
 211 *)
 212
 213 (*#* ** Degrees of polynomials over a field
 214 %\begin{convention}% Let [F] be a field and write [FX] for the ring of
 215 polynomials over [F].
 216 %\end{convention}%
 217 *)
 218
 219 inline cic:/CoRN/algebra/CPoly_Degree/F.var.
 220
 221 (* begin hide *)
 222
 223 (* end hide *)
 224
 225 inline cic:/CoRN/algebra/CPoly_Degree/degree_mult.con.
 226
 227 inline cic:/CoRN/algebra/CPoly_Degree/degree_nexp.con.
 228
 229 inline cic:/CoRN/algebra/CPoly_Degree/degree_le_mult_imp.con.
 230
 231 inline cic:/CoRN/algebra/CPoly_Degree/degree_mult_imp.con.
 232
 233 (* UNEXPORTED
 234 End degree_props_Field.
 235 *)
 236