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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 (* This file was automatically generated: do not edit *********************)
19 (* $Id: Zsetoid.v,v 1.5 2004/04/07 15:08:08 lcf Exp $ *)
21 include "model/structures/Zsec.ma".
23 include "algebra/CSetoidFun.ma".
25 (*#* **Example of a setoid: [Z]
29 inline procedural "cic:/CoRN/model/setoids/Zsetoid/ap_Z_irreflexive.con" as lemma.
31 inline procedural "cic:/CoRN/model/setoids/Zsetoid/ap_Z_symmetric.con" as lemma.
33 inline procedural "cic:/CoRN/model/setoids/Zsetoid/ap_Z_cotransitive.con" as lemma.
35 inline procedural "cic:/CoRN/model/setoids/Zsetoid/ap_Z_tight.con" as lemma.
37 inline procedural "cic:/CoRN/model/setoids/Zsetoid/ap_Z_is_apartness.con" as definition.
39 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Z_as_CSetoid.con" as definition.
41 (*#* The term [Z_as_CSetoid] is of type [CSetoid]. Hence we have proven that [Z] is a constructive setoid.
43 We will prove now that the addition on the integers is a setoid function.
46 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zplus_wd.con" as lemma.
48 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zplus_strext.con" as lemma.
50 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_bin_fun.con" as definition.
52 (*#* What's more: the addition is also associative and commutative.
55 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_assoc.con" as lemma.
57 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_commut.con" as lemma.
60 Taking the opposite of an integer is a setoid function.
63 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zopp_wd.con" as lemma.
65 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zopp_strext.con" as lemma.
67 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zopp_is_fun.con" as definition.
69 (*#* ***Multiplication
70 Finally the multiplication is a setoid function and is associative and commutative.
73 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zmult_wd.con" as lemma.
75 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zmult_strext.con" as lemma.
77 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_bin_fun.con" as definition.
79 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_assoc.con" as lemma.
81 inline procedural "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_commut.con" as lemma.