(* Title: HOLCF/IMP/Denotational.thy
ID: $Id$
Author: Tobias Nipkow and Robert Sandner, TUM
Copyright 1996 TUM
*)
header "Denotational Semantics of Commands in HOLCF"
theory Denotational imports HOLCF Natural begin
subsection "Definition"
definition
dlift :: "(('a::type) discr -> 'b::pcpo) => ('a lift -> 'b)" where
"dlift f = (LAM x. case x of UU => UU | Def y => f\<cdot>(Discr y))"
consts D :: "com => state discr -> state lift"
primrec
"D(\<SKIP>) = (LAM s. Def(undiscr s))"
"D(X :== a) = (LAM s. Def((undiscr s)[X \<mapsto> a(undiscr s)]))"
"D(c0 ; c1) = (dlift(D c1) oo (D c0))"
"D(\<IF> b \<THEN> c1 \<ELSE> c2) =
(LAM s. if b (undiscr s) then (D c1)\<cdot>s else (D c2)\<cdot>s)"
"D(\<WHILE> b \<DO> c) =
fix\<cdot>(LAM w s. if b (undiscr s) then (dlift w)\<cdot>((D c)\<cdot>s)
else Def(undiscr s))"
subsection
"Equivalence of Denotational Semantics in HOLCF and Evaluation Semantics in HOL"
lemma dlift_Def [simp]: "dlift f\<cdot>(Def x) = f\<cdot>(Discr x)"
by (simp add: dlift_def)
lemma cont_dlift [iff]: "cont (%f. dlift f)"
by (simp add: dlift_def)
lemma dlift_is_Def [simp]:
"(dlift f\<cdot>l = Def y) = (\<exists>x. l = Def x \<and> f\<cdot>(Discr x) = Def y)"
by (simp add: dlift_def split: lift.split)
lemma eval_implies_D: "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t ==> D c\<cdot>(Discr s) = (Def t)"
apply (induct set: evalc)
apply simp_all
apply (subst fix_eq)
apply simp
apply (subst fix_eq)
apply simp
done
lemma D_implies_eval: "!s t. D c\<cdot>(Discr s) = (Def t) --> \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t"
apply (induct c)
apply simp
apply simp
apply force
apply (simp (no_asm))
apply force
apply (simp (no_asm))
apply (rule fix_ind)
apply (fast intro!: adm_lemmas adm_chfindom ax_flat)
apply (simp (no_asm))
apply (simp (no_asm))
apply safe
apply fast
done
theorem D_is_eval: "(D c\<cdot>(Discr s) = (Def t)) = (\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t)"
by (fast elim!: D_implies_eval [rule_format] eval_implies_D)
end