(* Title: HOL/IMP/Denotation.thy
ID: $Id$
Author: Heiko Loetzbeyer & Robert Sandner, TUM
Copyright 1994 TUM
*)
header "Denotational Semantics of Commands"
theory Denotation imports Natural begin
types com_den = "(state\<times>state)set"
definition
Gamma :: "[bexp,com_den] => (com_den => com_den)" where
"Gamma b cd = (\<lambda>phi. {(s,t). (s,t) \<in> (cd O phi) \<and> b s} \<union>
{(s,t). s=t \<and> \<not>b s})"
primrec C :: "com => com_den"
where
C_skip: "C \<SKIP> = Id"
| C_assign: "C (x :== a) = {(s,t). t = s[x\<mapsto>a(s)]}"
| C_comp: "C (c0;c1) = C(c0) O C(c1)"
| C_if: "C (\<IF> b \<THEN> c1 \<ELSE> c2) = {(s,t). (s,t) \<in> C c1 \<and> b s} \<union>
{(s,t). (s,t) \<in> C c2 \<and> \<not>b s}"
| C_while: "C(\<WHILE> b \<DO> c) = lfp (Gamma b (C c))"
(**** mono (Gamma(b,c)) ****)
lemma Gamma_mono: "mono (Gamma b c)"
by (unfold Gamma_def mono_def) fast
lemma C_While_If: "C(\<WHILE> b \<DO> c) = C(\<IF> b \<THEN> c;\<WHILE> b \<DO> c \<ELSE> \<SKIP>)"
apply simp
apply (subst lfp_unfold [OF Gamma_mono]) --{*lhs only*}
apply (simp add: Gamma_def)
done
(* Operational Semantics implies Denotational Semantics *)
lemma com1: "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t \<Longrightarrow> (s,t) \<in> C(c)"
(* start with rule induction *)
apply (induct set: evalc)
apply auto
(* while *)
apply (unfold Gamma_def)
apply (subst lfp_unfold[OF Gamma_mono, simplified Gamma_def])
apply fast
apply (subst lfp_unfold[OF Gamma_mono, simplified Gamma_def])
apply auto
done
(* Denotational Semantics implies Operational Semantics *)
lemma com2: "(s,t) \<in> C(c) \<Longrightarrow> \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t"
apply (induct c arbitrary: s t)
apply auto
apply blast
(* while *)
apply (erule lfp_induct2 [OF _ Gamma_mono])
apply (unfold Gamma_def)
apply auto
done
(**** Proof of Equivalence ****)
lemma denotational_is_natural: "(s,t) \<in> C(c) = (\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t)"
by (fast elim: com2 dest: com1)
end