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

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