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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 dnat = one ++ dnat
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The type is axiomatized as the least solution of the domain equation above.
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The functor term that specifies the domain equation is:
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FT = <++,K_{one},I>
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For details see chapter 5 of:
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[Franz Regensburger] HOLCF: Eine konservative Erweiterung von HOL um LCF,
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Dissertation, Technische Universit"at M"unchen, 1994
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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 functor term FT. *)
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(* domain equation: dnat = one ++ dnat *)
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(* functor term: FT = <++,K_{one},I> *)
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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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(sswhen`sinl`(sinr oo f)) oo dnat_rep )"
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dnat_reach "(fix`dnat_copy)`x=x"
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defs
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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.sswhen`(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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