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theory Denotational(* Title: HOLCF/IMP/Denotational.thy ID: $Id: Denotational.thy,v 1.12 2005/06/17 14:15:09 haftmann Exp $ 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" constdefs dlift :: "(('a::type) discr -> 'b::pcpo) => ('a lift -> 'b)" "dlift f == (LAM x. case x of UU => UU | Def y => f·(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)·s else (D c2)·s)" "D(\<WHILE> b \<DO> c) = fix·(LAM w s. if b (undiscr s) then (dlift w)·((D c)·s) else Def(undiscr s))" subsection "Equivalence of Denotational Semantics in HOLCF and Evaluation Semantics in HOL" lemma dlift_Def [simp]: "dlift f·(Def x) = f·(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·l = Def y) = (∃x. l = Def x ∧ f·(Discr x) = Def y)" by (simp add: dlift_def split: lift.split) lemma eval_implies_D: "〈c,s〉 -->c t ==> D c·(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·(Discr s) = (Def t) --> 〈c,s〉 -->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·(Discr s) = (Def t)) = (〈c,s〉 -->c t)" by (fast elim!: D_implies_eval [rule_format] eval_implies_D) end
lemma dlift_Def:
dlift f·(Def x) = f·(Discr x)
lemma cont_dlift:
cont dlift
lemma dlift_is_Def:
(dlift f·l = Def y) = (∃x. l = Def x ∧ f·(Discr x) = Def y)
lemma eval_implies_D:
〈c,s〉 -->c t ==> D c·(Discr s) = Def t
lemma D_implies_eval:
∀s t. D c·(Discr s) = Def t --> 〈c,s〉 -->c t
theorem D_is_eval:
(D c·(Discr s) = Def t) = 〈c,s〉 -->c t