Theory Denotational

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theory Denotational
imports HOLCF Natural

(*  Title:      HOLCF/IMP/Denotational.thy
Author: Tobias Nipkow and Robert Sandner, TUM
Copyright 1996 TUM
*)


header "Denotational Semantics of Commands in HOLCF"

theory Denotational imports HOLCF "../../HOL/IMP/Natural" begin

text {* Disable conflicting syntax from HOL Map theory. *}

no_syntax
"_maplet" :: "['a, 'a] => maplet" ("_ /|->/ _")
"_maplets" :: "['a, 'a] => maplet" ("_ /[|->]/ _")
"" :: "maplet => maplets" ("_")
"_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
"_MapUpd" :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
"_Map" :: "maplets => 'a ~=> 'b" ("(1[_])")


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·(Discr y))"


primrec D :: "com => state discr -> state lift"
where
"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