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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 *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/model/setoids/Qpossetoid".
19 (* $Id: Qpossetoid.v,v 1.4 2004/04/06 15:46:05 lcf Exp $ *)
33 (*#* **Example of a setoid: [Qpos]
35 We will examine the subsetoid of positive rationals of the setoid of
39 inline cic:/CoRN/model/setoids/Qpossetoid/Qpos.con.
41 inline cic:/CoRN/model/setoids/Qpossetoid/QposP.con.
43 (*#* One, two and four are elements of it.
46 inline cic:/CoRN/model/setoids/Qpossetoid/QONEpos.con.
48 inline cic:/CoRN/model/setoids/Qpossetoid/QTWOpos.con.
50 inline cic:/CoRN/model/setoids/Qpossetoid/QFOURpos.con.
52 (*#* ***Multiplication
53 As we have seen, multiplication preserves positivity, so we can restrict it
54 to the positive rationals. We see that this restricted multiplication has some
58 inline cic:/CoRN/model/setoids/Qpossetoid/Qmult_pres_pos1.con.
60 inline cic:/CoRN/model/setoids/Qpossetoid/Qpos_mult.con.
62 inline cic:/CoRN/model/setoids/Qpossetoid/associative_Qpos_mult.con.
65 We restrict the domain of the inverse to the set of positive rationals.
68 inline cic:/CoRN/model/setoids/Qpossetoid/Qpos_inv.con.
70 (*#* The restricted inverse preserves positivity.
73 inline cic:/CoRN/model/setoids/Qpossetoid/inv_pres_pos1.con.
75 (*#* Now, we can also restrict the co-domain.
78 inline cic:/CoRN/model/setoids/Qpossetoid/Qpos_Qpos_inv.con.
80 (*#* This restricted inverse map appears a setoid function.
83 inline cic:/CoRN/model/setoids/Qpossetoid/Qpos_Qpos_inv_strong_ext.con.
85 inline cic:/CoRN/model/setoids/Qpossetoid/Qpos_Qpos_inv_op.con.
87 (*#* ***Special multiplication and inverse
88 We define [multdiv2]: $(x,y) \mapsto xy/2$ #(x,y) ↦ xy/2#.
91 inline cic:/CoRN/model/setoids/Qpossetoid/Qpos_div2.con.
93 inline cic:/CoRN/model/setoids/Qpossetoid/multdiv2.con.
95 inline cic:/CoRN/model/setoids/Qpossetoid/associative_multdiv2.con.
97 (*#* And its inverse [multdiv4]: $x \mapsto 4/x$ #x ↦ 4/x#.
100 inline cic:/CoRN/model/setoids/Qpossetoid/mult4.con.
102 inline cic:/CoRN/model/setoids/Qpossetoid/divmult4.con.