matita/contribs/CoRN-Decl/model/setoids/Zsetoid.ma

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  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 set "baseuri" "cic:/matita/CoRN-Decl/model/setoids/Zsetoid".
  18
  19 include "CoRN.ma".
  20
  21 (* $Id: Zsetoid.v,v 1.5 2004/04/07 15:08:08 lcf Exp $ *)
  22
  23 include "model/structures/Zsec.ma".
  24
  25 include "algebra/CSetoidFun.ma".
  26
  27 (*#* **Example of a setoid: [Z]
  28 *** [Z]
  29 *)
  30
  31 inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_irreflexive.con".
  32
  33 inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_symmetric.con".
  34
  35 inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_cotransitive.con".
  36
  37 inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_tight.con".
  38
  39 inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_is_apartness.con".
  40
  41 inline "cic:/CoRN/model/setoids/Zsetoid/Z_as_CSetoid.con".
  42
  43 (*#* The term [Z_as_CSetoid] is of type [CSetoid]. Hence we have proven that [Z] is a constructive setoid.
  44 ***Addition
  45 We will prove now that the addition on the integers is a setoid function.
  46 *)
  47
  48 inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_wd.con".
  49
  50 inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_strext.con".
  51
  52 inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_bin_fun.con".
  53
  54 (*#* What's more: the addition is also associative and commutative.
  55 *)
  56
  57 inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_assoc.con".
  58
  59 inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_commut.con".
  60
  61 (*#* ***Opposite
  62 Taking the opposite of an integer is a setoid function.
  63 *)
  64
  65 inline "cic:/CoRN/model/setoids/Zsetoid/Zopp_wd.con".
  66
  67 inline "cic:/CoRN/model/setoids/Zsetoid/Zopp_strext.con".
  68
  69 inline "cic:/CoRN/model/setoids/Zsetoid/Zopp_is_fun.con".
  70
  71 (*#* ***Multiplication
  72 Finally the multiplication is a setoid function and is associative and commutative.
  73 *)
  74
  75 inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_wd.con".
  76
  77 inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_strext.con".
  78
  79 inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_bin_fun.con".
  80
  81 inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_assoc.con".
  82
  83 inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_commut.con".
  84