# [isabelle] sets in HOL/CF

```Dear Isabelle community,

I am exploring HOLCF at the moment. As a starting point, I tried to
define this function with fixrec:
f :: nat → nat set
f b = {b} ∪ f b
I’d expect to find that "f b = {b}".

This approach worked:
______________________________________________________
fixrec f :: "int lift → (int ⇒ one)" where
"f⋅b = (case b of ⊥ ⇒ ⊥ | Def a ⇒ (f⋅b) (a := ONE))"
declare f.simps[simp del]

lemma "f⋅(Def b) = (λ c. if b = c then ONE else ⊥)"
apply (rule ext)
apply auto
apply (subst f.unfold)
apply simp
apply(induct rule:f.induct)
apply auto
done
______________________________________________________

But I’m wondering: Why can I not write:
______________________________________________________
fixrec f :: "int lift → int set" where
"f⋅b = (case b of ⊥ ⇒ ⊥ | Def a ⇒ {a} ∪ (f⋅b)"
declare f.simps[simp del]
______________________________________________________
The error message
*** Type unification failed: No type arity bool :: cpo
seems to indicate that fixrec has a problem with me using "int set" as a
pcpo. I assume the reason is that "int set" is just an abbreviation for
"int ⇒ bool", therefore my above work-around. But I’m missing the
convenient syntax of sets in Isabelle.

Is this really the correct way to work with sets in HOL/CF or am I just
overlooking something?

Thank you,
Joachim

--
Joachim Breitner
e-Mail: mail at joachim-breitner.de
Homepage: http://www.joachim-breitner.de
ICQ#: 74513189
Jabber-ID: nomeata at joachim-breitner.de
```

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