helm/software/matita/contribs/procedural/CoRN/algebra/CPoly_NthCoeff.mma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 include "CoRN.ma".
  18
  19 (* $Id: CPoly_NthCoeff.v,v 1.6 2004/04/23 10:00:53 lcf Exp $ *)
  20
  21 include "algebra/CPolynomials.ma".
  22
  23 (*#*
  24 * Polynomials: Nth Coefficient
  25 %\begin{convention}% Let [R] be a ring and write [RX] for the ring of
  26 polynomials over [R].
  27 %\end{convention}%
  28
  29 ** Definitions
  30 *)
  31
  32 (* UNEXPORTED
  33 Section NthCoeff_def
  34 *)
  35
  36 (* UNEXPORTED
  37 cic:/CoRN/algebra/CPoly_NthCoeff/NthCoeff_def/R.var
  38 *)
  39
  40 (* begin hide *)
  41
  42 (* NOTATION
  43 Notation RX := (cpoly_cring R).
  44 *)
  45
  46 (* end hide *)
  47
  48 (*#*
  49 The [n]-th coefficient of a polynomial. The default value is
  50 [Zero:CR] e.g. if the [n] is higher than the length. For the
  51 polynomial $a_0 +a_1 X +a_2 X^2 + \cdots + a_n X^n$ #a0 +a1 X +a2 X^2
  52 + ... + an X^n#, the [Zero]-th coefficient is $a_0$#a0#, the first
  53 is $a_1$#a1# etcetera.  *)
  54
  55 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff.con" as definition.
  56
  57 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_strext.con" as lemma.
  58
  59 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_wd.con" as lemma.
  60
  61 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_fun.con" as definition.
  62
  63 (*#*
  64 %\begin{shortcoming}%
  65 We would like to use [nth_coeff_fun n] all the time.
  66 However, Coq's coercion mechanism doesn't support this properly:
  67 the term
  68 [(nth_coeff_fun n p)] won't get parsed, and has to be written as
  69 [((nth_coeff_fun n) p)] instead.
  70
  71 So, in the names of lemmas, we write [(nth_coeff n p)],
  72 which always (e.g. in proofs) can be converted
  73 to [((nth_coeff_fun n) p)].
  74 %\end{shortcoming}%
  75 *)
  76
  77 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nonConst.con" as definition.
  78
  79 (*#*
  80 The following is probably NOT needed.  These functions are
  81 NOT extensional, that is, they are not CSetoid functions.
  82 *)
  83
  84 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_ok.con" as definition.
  85
  86 (* The in_coeff predicate*)
  87
  88 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/in_coeff.con" as definition.
  89
  90 (*#*
  91 The [cpoly_zero] case should be [c [=] Zero] in order to be extensional.
  92 *)
  93
  94 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_S.con" as lemma.
  95
  96 (* UNEXPORTED
  97 End NthCoeff_def
  98 *)
  99
 100 (* UNEXPORTED
 101 Implicit Arguments nth_coeff [R].
 102 *)
 103
 104 (* UNEXPORTED
 105 Implicit Arguments nth_coeff_fun [R].
 106 *)
 107
 108 (* UNEXPORTED
 109 Hint Resolve nth_coeff_wd: algebra_c.
 110 *)
 111
 112 (* UNEXPORTED
 113 Section NthCoeff_props
 114 *)
 115
 116 (*#* ** Properties of [nth_coeff] *)
 117
 118 (* UNEXPORTED
 119 cic:/CoRN/algebra/CPoly_NthCoeff/NthCoeff_props/R.var
 120 *)
 121
 122 (* begin hide *)
 123
 124 (* NOTATION
 125 Notation RX := (cpoly_cring R).
 126 *)
 127
 128 (* end hide *)
 129
 130 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_zero.con" as lemma.
 131
 132 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_O_lin.con" as lemma.
 133
 134 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_Sm_lin.con" as lemma.
 135
 136 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_O_c_.con" as lemma.
 137
 138 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_O_x_mult.con" as lemma.
 139
 140 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_Sm_x_mult.con" as lemma.
 141
 142 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_Sm_mult_x_.con" as lemma.
 143
 144 (* UNEXPORTED
 145 Hint Resolve nth_coeff_zero coeff_O_lin coeff_Sm_lin coeff_O_c_
 146   coeff_O_x_mult coeff_Sm_x_mult coeff_Sm_mult_x_: algebra.
 147 *)
 148
 149 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_ap_zero_imp.con" as lemma.
 150
 151 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_plus.con" as lemma.
 152
 153 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_inv.con" as lemma.
 154
 155 (* UNEXPORTED
 156 Hint Resolve nth_coeff_inv: algebra.
 157 *)
 158
 159 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_c_mult_p.con" as lemma.
 160
 161 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_p_mult_c_.con" as lemma.
 162
 163 (* UNEXPORTED
 164 Hint Resolve nth_coeff_c_mult_p nth_coeff_p_mult_c_ nth_coeff_plus: algebra.
 165 *)
 166
 167 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_complicated.con" as lemma.
 168
 169 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/all_nth_coeff_eq_imp.con" as lemma.
 170
 171 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/poly_at_zero.con" as lemma.
 172
 173 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_inv'.con" as lemma.
 174
 175 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_minus.con" as lemma.
 176
 177 (* UNEXPORTED
 178 Hint Resolve nth_coeff_minus: algebra.
 179 *)
 180
 181 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_sum0.con" as lemma.
 182
 183 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_sum.con" as lemma.
 184
 185 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_nexp_eq.con" as lemma.
 186
 187 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_nexp_neq.con" as lemma.
 188
 189 inline procedural "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_mult.con" as lemma.
 190
 191 (* UNEXPORTED
 192 End NthCoeff_props
 193 *)
 194
 195 (* UNEXPORTED
 196 Hint Resolve nth_coeff_wd: algebra_c.
 197 *)
 198
 199 (* UNEXPORTED
 200 Hint Resolve nth_coeff_complicated poly_at_zero nth_coeff_inv: algebra.
 201 *)
 202
 203 (* UNEXPORTED
 204 Hint Resolve nth_coeff_inv' nth_coeff_c_mult_p nth_coeff_mult: algebra.
 205 *)
 206
 207 (* UNEXPORTED
 208 Hint Resolve nth_coeff_zero nth_coeff_plus nth_coeff_minus: algebra.
 209 *)
 210
 211 (* UNEXPORTED
 212 Hint Resolve nth_coeff_nexp_eq nth_coeff_nexp_neq: algebra.
 213 *)
 214