# Theory Denotational

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

section "Denotational Semantics of Commands in HOLCF"

theory Denotational imports HOLCF "HOL-IMP.Big_Step" 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⋅(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 := aval a (undiscr s))))"
| "D(c0 ;; c1) = (dlift(D c1) oo (D c0))"
| "D(IF b THEN c1 ELSE c2) =
(LAM s. if bval b (undiscr s) then (D c1)⋅s else (D c2)⋅s)"
| "D(WHILE b DO c) =
fix⋅(LAM w s. if bval 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)"

lemma cont_dlift [iff]: "cont (%f. dlift f)"

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) ⇒ t ⟹ D c⋅(Discr s) = (Def t)"
apply (induct rule: big_step_induct)
apply (auto)
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) ⇒ t"
apply (induct c)
apply fastforce
apply fastforce
apply force
apply (simp (no_asm))
apply force
apply (simp (no_asm))
apply (rule fix_ind)
apply (simp (no_asm))
apply (simp (no_asm))
apply force
done

theorem D_is_eval: "(D c⋅(Discr s) = (Def t)) = ((c,s) ⇒ t)"
by (fast elim!: D_implies_eval [rule_format] eval_implies_D)

end
```