helm/software/matita/contribs/procedural/CoRN/model/setoids/Zsetoid.mma

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