Theory Denotation

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theory Denotation
imports Com
begin

(*  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

Definitions

Misc lemmas

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 -> naty ∈ 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))