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

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   4 (*      ||A||       A project by Andrea Asperti                           *)
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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 (* $Id: Zsetoid.v,v 1.5 2004/04/07 15:08:08 lcf Exp $ *)
  20
  21 (* INCLUDE
  22 Zsec
  23 *)
  24
  25 (* INCLUDE
  26 CSetoidFun
  27 *)
  28
  29 (*#* **Example of a setoid: [Z]
  30 *** [Z]
  31 *)
  32
  33 inline cic:/CoRN/model/setoids/Zsetoid/ap_Z_irreflexive.con.
  34
  35 inline cic:/CoRN/model/setoids/Zsetoid/ap_Z_symmetric.con.
  36
  37 inline cic:/CoRN/model/setoids/Zsetoid/ap_Z_cotransitive.con.
  38
  39 inline cic:/CoRN/model/setoids/Zsetoid/ap_Z_tight.con.
  40
  41 inline cic:/CoRN/model/setoids/Zsetoid/ap_Z_is_apartness.con.
  42
  43 inline cic:/CoRN/model/setoids/Zsetoid/Z_as_CSetoid.con.
  44
  45 (*#* The term [Z_as_CSetoid] is of type [CSetoid]. Hence we have proven that [Z] is a constructive setoid.
  46 ***Addition
  47 We will prove now that the addition on the integers is a setoid function.
  48 *)
  49
  50 inline cic:/CoRN/model/setoids/Zsetoid/Zplus_wd.con.
  51
  52 inline cic:/CoRN/model/setoids/Zsetoid/Zplus_strext.con.
  53
  54 inline cic:/CoRN/model/setoids/Zsetoid/Zplus_is_bin_fun.con.
  55
  56 (*#* What's more: the addition is also associative and commutative.
  57 *)
  58
  59 inline cic:/CoRN/model/setoids/Zsetoid/Zplus_is_assoc.con.
  60
  61 inline cic:/CoRN/model/setoids/Zsetoid/Zplus_is_commut.con.
  62
  63 (*#* ***Opposite
  64 Taking the opposite of an integer is a setoid function.
  65 *)
  66
  67 inline cic:/CoRN/model/setoids/Zsetoid/Zopp_wd.con.
  68
  69 inline cic:/CoRN/model/setoids/Zsetoid/Zopp_strext.con.
  70
  71 inline cic:/CoRN/model/setoids/Zsetoid/Zopp_is_fun.con.
  72
  73 (*#* ***Multiplication
  74 Finally the multiplication is a setoid function and is associative and commutative.
  75 *)
  76
  77 inline cic:/CoRN/model/setoids/Zsetoid/Zmult_wd.con.
  78
  79 inline cic:/CoRN/model/setoids/Zsetoid/Zmult_strext.con.
  80
  81 inline cic:/CoRN/model/setoids/Zsetoid/Zmult_is_bin_fun.con.
  82
  83 inline cic:/CoRN/model/setoids/Zsetoid/Zmult_is_assoc.con.
  84
  85 inline cic:/CoRN/model/setoids/Zsetoid/Zmult_is_commut.con.
  86