1 (* Title: HOL/wf.ML |
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2 ID: $Id$ |
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3 Author: Tobias Nipkow |
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4 Copyright 1992 University of Cambridge |
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5 |
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6 Well-founded Recursion |
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7 *) |
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8 |
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9 WF = Trancl + |
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10 consts |
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11 wf :: "('a * 'a)set => bool" |
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12 cut :: "['a => 'b, ('a * 'a)set, 'a] => 'a => 'b" |
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13 wftrec,wfrec :: "[('a * 'a)set, 'a, ['a,'a=>'b]=>'b] => 'b" |
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14 is_recfun :: "[('a * 'a)set, 'a, ['a,'a=>'b]=>'b, 'a=>'b] => bool" |
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15 the_recfun :: "[('a * 'a)set, 'a, ['a,'a=>'b]=>'b] => 'a=>'b" |
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16 |
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17 defs |
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18 wf_def "wf(r) == (!P. (!x. (!y. <y,x>:r --> P(y)) --> P(x)) --> (!x.P(x)))" |
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19 |
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20 cut_def "cut(f,r,x) == (%y. if(<y,x>:r, f(y), @z.True))" |
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21 |
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22 is_recfun_def "is_recfun(r,a,H,f) == (f = cut(%x.H(x, cut(f,r,x)), r, a))" |
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23 |
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24 the_recfun_def "the_recfun(r,a,H) == (@f.is_recfun(r,a,H,f))" |
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25 |
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26 wftrec_def "wftrec(r,a,H) == H(a, the_recfun(r,a,H))" |
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27 |
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28 (*version not requiring transitivity*) |
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29 wfrec_def "wfrec(r,a,H) == wftrec(trancl(r), a, %x f. H(x, cut(f,r,x)))" |
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30 end |
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