(* Title: ZF/IMP/Denotation.thy ID: $Id: Denotation.thy,v 1.15 2006/11/17 01:20:04 wenzelm Exp $ Author: Heiko Loetzbeyer and Robert Sandner, TU München *) header {* Denotational semantics of expressions and commands *} theory Denotation imports Com begin subsection {* Definitions *} consts A :: "i => i => i" B :: "i => i => i" C :: "i => i" definition Gamma :: "[i,i,i] => i" ("Γ") where "Γ(b,cden) == (λphi. {io ∈ (phi O cden). B(b,fst(io))=1} ∪ {io ∈ id(loc->nat). B(b,fst(io))=0})" primrec "A(N(n), sigma) = n" "A(X(x), sigma) = sigma`x" "A(Op1(f,a), sigma) = f`A(a,sigma)" "A(Op2(f,a0,a1), sigma) = f`<A(a0,sigma),A(a1,sigma)>" primrec "B(true, sigma) = 1" "B(false, sigma) = 0" "B(ROp(f,a0,a1), sigma) = f`<A(a0,sigma),A(a1,sigma)>" "B(noti(b), sigma) = not(B(b,sigma))" "B(b0 andi b1, sigma) = B(b0,sigma) and B(b1,sigma)" "B(b0 ori b1, sigma) = B(b0,sigma) or B(b1,sigma)" primrec "C(\<SKIP>) = id(loc->nat)" "C(x \<ASSN> a) = {io ∈ (loc->nat) × (loc->nat). snd(io) = fst(io)(x := A(a,fst(io)))}" "C(c0\<SEQ> c1) = C(c1) O C(c0)" "C(\<IF> b \<THEN> c0 \<ELSE> c1) = {io ∈ C(c0). B(b,fst(io)) = 1} ∪ {io ∈ C(c1). B(b,fst(io)) = 0}" "C(\<WHILE> b \<DO> c) = lfp((loc->nat) × (loc->nat), Γ(b,C(c)))" subsection {* Misc lemmas *} lemma A_type [TC]: "[|a ∈ aexp; sigma ∈ loc->nat|] ==> A(a,sigma) ∈ nat" by (erule aexp.induct) simp_all lemma B_type [TC]: "[|b ∈ bexp; sigma ∈ loc->nat|] ==> B(b,sigma) ∈ bool" by (erule bexp.induct, simp_all) lemma C_subset: "c ∈ com ==> C(c) ⊆ (loc->nat) × (loc->nat)" apply (erule com.induct) apply simp_all apply (blast dest: lfp_subset [THEN subsetD])+ done lemma C_type_D [dest]: "[| <x,y> ∈ C(c); c ∈ com |] ==> x ∈ loc->nat & y ∈ loc->nat" by (blast dest: C_subset [THEN subsetD]) lemma C_type_fst [dest]: "[| x ∈ C(c); c ∈ com |] ==> fst(x) ∈ loc->nat" by (auto dest!: C_subset [THEN subsetD]) lemma Gamma_bnd_mono: "cden ⊆ (loc->nat) × (loc->nat) ==> bnd_mono ((loc->nat) × (loc->nat), Γ(b,cden))" by (unfold bnd_mono_def Gamma_def) blast end
lemma A_type:
[| a ∈ aexp; sigma ∈ loc -> nat |] ==> A(a, sigma) ∈ nat
lemma B_type:
[| b ∈ bexp; sigma ∈ loc -> nat |] ==> B(b, sigma) ∈ bool
lemma C_subset:
c ∈ com ==> C(c) ⊆ (loc -> nat) × (loc -> nat)
lemma C_type_D:
[| 〈x, y〉 ∈ C(c); c ∈ com |] ==> x ∈ loc -> nat ∧ y ∈ loc -> nat
lemma C_type_fst:
[| x ∈ C(c); c ∈ com |] ==> fst(x) ∈ loc -> nat
lemma Gamma_bnd_mono:
cden ⊆ (loc -> nat) × (loc -> nat)
==> bnd_mono((loc -> nat) × (loc -> nat), Γ(b, cden))