--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Dnat.thy Wed Jan 19 17:35:01 1994 +0100
@@ -0,0 +1,101 @@
+(* Title: HOLCF/dnat.thy
+ ID: $Id$
+ Author: Franz Regensburger
+ Copyright 1993 Technische Universitaet Muenchen
+
+Theory for the domain of natural numbers
+
+*)
+
+Dnat = HOLCF +
+
+types dnat 0
+
+(* ----------------------------------------------------------------------- *)
+(* arrity axiom is valuated by semantical reasoning *)
+
+arities dnat::pcpo
+
+consts
+
+(* ----------------------------------------------------------------------- *)
+(* essential constants *)
+
+dnat_rep :: " dnat -> (one ++ dnat)"
+dnat_abs :: "(one ++ dnat) -> dnat"
+
+(* ----------------------------------------------------------------------- *)
+(* abstract constants and auxiliary constants *)
+
+dnat_copy :: "(dnat -> dnat) -> dnat -> dnat"
+
+dzero :: "dnat"
+dsucc :: "dnat -> dnat"
+dnat_when :: "'b -> (dnat -> 'b) -> dnat -> 'b"
+is_dzero :: "dnat -> tr"
+is_dsucc :: "dnat -> tr"
+dpred :: "dnat -> dnat"
+dnat_take :: "nat => dnat -> dnat"
+dnat_bisim :: "(dnat => dnat => bool) => bool"
+
+rules
+
+(* ----------------------------------------------------------------------- *)
+(* axiomatization of recursive type dnat *)
+(* ----------------------------------------------------------------------- *)
+(* (dnat,dnat_abs) is the initial F-algebra where *)
+(* F is the locally continuous functor determined by domain equation *)
+(* X = one ++ X *)
+(* ----------------------------------------------------------------------- *)
+(* dnat_abs is an isomorphism with inverse dnat_rep *)
+(* identity is the least endomorphism on dnat *)
+
+dnat_abs_iso "dnat_rep[dnat_abs[x]] = x"
+dnat_rep_iso "dnat_abs[dnat_rep[x]] = x"
+dnat_copy_def "dnat_copy == (LAM f. dnat_abs oo \
+\ (when[sinl][sinr oo f]) oo dnat_rep )"
+dnat_reach "(fix[dnat_copy])[x]=x"
+
+(* ----------------------------------------------------------------------- *)
+(* properties of additional constants *)
+(* ----------------------------------------------------------------------- *)
+(* constructors *)
+
+dzero_def "dzero == dnat_abs[sinl[one]]"
+dsucc_def "dsucc == (LAM n. dnat_abs[sinr[n]])"
+
+(* ----------------------------------------------------------------------- *)
+(* discriminator functional *)
+
+dnat_when_def "dnat_when == (LAM f1 f2 n.when[LAM x.f1][f2][dnat_rep[n]])"
+
+
+(* ----------------------------------------------------------------------- *)
+(* discriminators and selectors *)
+
+is_dzero_def "is_dzero == dnat_when[TT][LAM x.FF]"
+is_dsucc_def "is_dsucc == dnat_when[FF][LAM x.TT]"
+dpred_def "dpred == dnat_when[UU][LAM x.x]"
+
+
+(* ----------------------------------------------------------------------- *)
+(* the taker for dnats *)
+
+dnat_take_def "dnat_take == (%n.iterate(n,dnat_copy,UU))"
+
+(* ----------------------------------------------------------------------- *)
+(* definition of bisimulation is determined by domain equation *)
+(* simplification and rewriting for abstract constants yields def below *)
+
+dnat_bisim_def "dnat_bisim ==\
+\(%R.!s1 s2.\
+\ R(s1,s2) -->\
+\ ((s1=UU & s2=UU) |(s1=dzero & s2=dzero) |\
+\ (? s11 s21. s11~=UU & s21~=UU & s1=dsucc[s11] &\
+\ s2 = dsucc[s21] & R(s11,s21))))"
+
+end
+
+
+
+