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1 (* Title: ZF/Order.thy |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1994 University of Cambridge |
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5 |
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6 Orders in Zermelo-Fraenkel Set Theory |
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7 *) |
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8 |
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9 Order = WF + Perm + |
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10 consts |
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11 part_ord :: "[i,i]=>o" (*Strict partial ordering*) |
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12 linear, tot_ord :: "[i,i]=>o" (*Strict total ordering*) |
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13 well_ord :: "[i,i]=>o" (*Well-ordering*) |
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14 ord_iso :: "[i,i,i,i]=>i" (*Order isomorphisms*) |
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15 pred :: "[i,i,i]=>i" (*Set of predecessors*) |
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16 |
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17 rules |
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18 part_ord_def "part_ord(A,r) == irrefl(A,r) & trans[A](r)" |
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19 |
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20 linear_def "linear(A,r) == (ALL x:A. ALL y:A. <x,y>:r | x=y | <y,x>:r)" |
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21 |
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22 tot_ord_def "tot_ord(A,r) == part_ord(A,r) & linear(A,r)" |
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23 |
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24 well_ord_def "well_ord(A,r) == tot_ord(A,r) & wf[A](r)" |
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25 |
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26 ord_iso_def "ord_iso(A,r,B,s) == \ |
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27 \ {f: bij(A,B). ALL x:A. ALL y:A. <x,y>:r <-> <f`x,f`y>:s}" |
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28 |
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29 pred_def "pred(A,x,r) == {y:A. <y,x>:r}" |
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30 |
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31 end |