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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
19 (* $Id: Npossetoid.v,v 1.3 2004/04/06 15:46:04 lcf Exp $ *)
21 include "model/setoids/Nsetoid.ma".
23 include "model/structures/Npossec.ma".
25 include "algebra/CSetoidFun.ma".
27 (*#* **Example of a setoid: [Npos]
30 The positive natural numbers [Npos] will be defined as a subsetoid of the
34 inline procedural "cic:/CoRN/model/setoids/Npossetoid/Npos.con" as definition.
36 inline procedural "cic:/CoRN/model/setoids/Npossetoid/NposP.con" as definition.
38 (*#* One and two are elements of it.
41 inline procedural "cic:/CoRN/model/setoids/Npossetoid/ONEpos.con" as definition.
43 inline procedural "cic:/CoRN/model/setoids/Npossetoid/TWOpos.con" as definition.
45 (*#* ***Addition and multiplication
46 Because addition and multiplication preserve positivity, we can define
47 them on this subsetoid.
50 inline procedural "cic:/CoRN/model/setoids/Npossetoid/plus_resp_Npos.con" as lemma.
52 inline procedural "cic:/CoRN/model/setoids/Npossetoid/Npos_plus.con" as definition.
54 inline procedural "cic:/CoRN/model/setoids/Npossetoid/mult_resp_Npos.con" as lemma.
56 inline procedural "cic:/CoRN/model/setoids/Npossetoid/Npos_mult.con" as definition.
58 (*#* The addition has no right unit on this set.
61 inline procedural "cic:/CoRN/model/setoids/Npossetoid/no_rht_unit_Npos1.con" as lemma.
63 (*#* And the multiplication doesn't have an inverse, because there can't be an
67 inline procedural "cic:/CoRN/model/setoids/Npossetoid/no_inverse_Nposmult1.con" as lemma.