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

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  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 include "CoRN.ma".
  18
  19 (* $Id: Npossetoid.v,v 1.3 2004/04/06 15:46:04 lcf Exp $ *)
  20
  21 include "model/setoids/Nsetoid.ma".
  22
  23 include "model/structures/Npossec.ma".
  24
  25 include "algebra/CSetoidFun.ma".
  26
  27 (*#* **Example of a setoid: [Npos]
  28
  29 *** Setoid
  30 The positive natural numbers [Npos] will be defined as a subsetoid of the
  31 natural numbers.
  32 *)
  33
  34 inline procedural "cic:/CoRN/model/setoids/Npossetoid/Npos.con" as definition.
  35
  36 inline procedural "cic:/CoRN/model/setoids/Npossetoid/NposP.con" as definition.
  37
  38 (*#* One and two are elements of it.
  39 *)
  40
  41 inline procedural "cic:/CoRN/model/setoids/Npossetoid/ONEpos.con" as definition.
  42
  43 inline procedural "cic:/CoRN/model/setoids/Npossetoid/TWOpos.con" as definition.
  44
  45 (*#* ***Addition and multiplication
  46 Because addition and multiplication preserve positivity, we can define
  47 them on this subsetoid.
  48 *)
  49
  50 inline procedural "cic:/CoRN/model/setoids/Npossetoid/plus_resp_Npos.con" as lemma.
  51
  52 inline procedural "cic:/CoRN/model/setoids/Npossetoid/Npos_plus.con" as definition.
  53
  54 inline procedural "cic:/CoRN/model/setoids/Npossetoid/mult_resp_Npos.con" as lemma.
  55
  56 inline procedural "cic:/CoRN/model/setoids/Npossetoid/Npos_mult.con" as definition.
  57
  58 (*#* The addition has no right unit on this set.
  59 *)
  60
  61 inline procedural "cic:/CoRN/model/setoids/Npossetoid/no_rht_unit_Npos1.con" as lemma.
  62
  63 (*#* And the multiplication doesn't have an inverse, because there can't be an
  64 inverse for 2.
  65 *)
  66
  67 inline procedural "cic:/CoRN/model/setoids/Npossetoid/no_inverse_Nposmult1.con" as lemma.
  68