| author | wenzelm |
| Thu, 08 Jul 1999 18:26:24 +0200 | |
| changeset 6923 | 51c415f15007 |
| parent 243 | c22b85994e17 |
| permissions | -rw-r--r-- |
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(* Title: HOLCF/dnat.thy |
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ID: $Id$ |
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Author: Franz Regensburger |
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Copyright 1993 Technische Universitaet Muenchen |
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Theory for the domain of natural numbers |
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*) |
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Dnat = HOLCF + |
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types dnat 0 |
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(* ----------------------------------------------------------------------- *) |
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(* arrity axiom is valuated by semantical reasoning *) |
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arities dnat::pcpo |
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consts |
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(* ----------------------------------------------------------------------- *) |
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(* essential constants *) |
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dnat_rep :: " dnat -> (one ++ dnat)" |
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dnat_abs :: "(one ++ dnat) -> dnat" |
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(* ----------------------------------------------------------------------- *) |
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(* abstract constants and auxiliary constants *) |
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dnat_copy :: "(dnat -> dnat) -> dnat -> dnat" |
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dzero :: "dnat" |
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dsucc :: "dnat -> dnat" |
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dnat_when :: "'b -> (dnat -> 'b) -> dnat -> 'b" |
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is_dzero :: "dnat -> tr" |
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is_dsucc :: "dnat -> tr" |
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dpred :: "dnat -> dnat" |
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dnat_take :: "nat => dnat -> dnat" |
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dnat_bisim :: "(dnat => dnat => bool) => bool" |
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rules |
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(* ----------------------------------------------------------------------- *) |
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(* axiomatization of recursive type dnat *) |
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(* ----------------------------------------------------------------------- *) |
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(* (dnat,dnat_abs) is the initial F-algebra where *) |
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(* F is the locally continuous functor determined by domain equation *) |
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(* X = one ++ X *) |
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(* ----------------------------------------------------------------------- *) |
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(* dnat_abs is an isomorphism with inverse dnat_rep *) |
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(* identity is the least endomorphism on dnat *) |
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dnat_abs_iso "dnat_rep[dnat_abs[x]] = x" |
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dnat_rep_iso "dnat_abs[dnat_rep[x]] = x" |
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dnat_copy_def "dnat_copy == (LAM f. dnat_abs oo \ |
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\ (when[sinl][sinr oo f]) oo dnat_rep )" |
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dnat_reach "(fix[dnat_copy])[x]=x" |
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(* ----------------------------------------------------------------------- *) |
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(* properties of additional constants *) |
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(* ----------------------------------------------------------------------- *) |
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(* constructors *) |
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dzero_def "dzero == dnat_abs[sinl[one]]" |
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dsucc_def "dsucc == (LAM n. dnat_abs[sinr[n]])" |
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(* ----------------------------------------------------------------------- *) |
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(* discriminator functional *) |
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dnat_when_def "dnat_when == (LAM f1 f2 n.when[LAM x.f1][f2][dnat_rep[n]])" |
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(* ----------------------------------------------------------------------- *) |
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(* discriminators and selectors *) |
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is_dzero_def "is_dzero == dnat_when[TT][LAM x.FF]" |
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is_dsucc_def "is_dsucc == dnat_when[FF][LAM x.TT]" |
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dpred_def "dpred == dnat_when[UU][LAM x.x]" |
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(* ----------------------------------------------------------------------- *) |
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(* the taker for dnats *) |
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dnat_take_def "dnat_take == (%n.iterate(n,dnat_copy,UU))" |
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(* ----------------------------------------------------------------------- *) |
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(* definition of bisimulation is determined by domain equation *) |
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(* simplification and rewriting for abstract constants yields def below *) |
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dnat_bisim_def "dnat_bisim ==\ |
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\(%R.!s1 s2.\ |
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\ R(s1,s2) -->\ |
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\ ((s1=UU & s2=UU) |(s1=dzero & s2=dzero) |\ |
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\ (? s11 s21. s11~=UU & s21~=UU & s1=dsucc[s11] &\ |
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\ s2 = dsucc[s21] & R(s11,s21))))" |
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end |
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