src/ZF/IMP/Denotation.thy

better representation of Sigma

(*  Title:      ZF/IMP/Denotation.thy
    ID:         $Id$
    Author:     Heiko Loetzbeyer & Robert Sandner, TUM
    Copyright   1994 TUM

Denotational semantics of expressions & commands
*)

Denotation = Com + 

consts
  A     :: i => i => i
  B     :: i => i => i
  C     :: i => i
  Gamma :: [i,i,i] => i

rules   (*NOT definitional*)
  A_nat_def     "A(N(n)) == (%sigma. n)"
  A_loc_def     "A(X(x)) == (%sigma. sigma`x)" 
  A_op1_def     "A(Op1(f,a)) == (%sigma. f`A(a,sigma))"
  A_op2_def     "A(Op2(f,a0,a1)) == (%sigma. f`<A(a0,sigma),A(a1,sigma)>)"


  B_true_def    "B(true) == (%sigma. 1)"
  B_false_def   "B(false) == (%sigma. 0)"
  B_op_def      "B(ROp(f,a0,a1)) == (%sigma. f`<A(a0,sigma),A(a1,sigma)>)" 


  B_not_def     "B(noti(b)) == (%sigma. not(B(b,sigma)))"
  B_and_def     "B(b0 andi b1) == (%sigma. B(b0,sigma) and B(b1,sigma))"
  B_or_def      "B(b0 ori b1) == (%sigma. B(b0,sigma) or B(b1,sigma))"

  C_skip_def    "C(skip) == id(loc->nat)"
  C_assign_def  "C(x := a) == {io:(loc->nat)*(loc->nat). 
                               snd(io) = fst(io)[A(a,fst(io))/x]}"

  C_comp_def    "C(c0 ; c1) == C(c1) O C(c0)"
  C_if_def      "C(ifc b then c0 else c1) == {io:C(c0). B(b,fst(io))=1} Un 
                                             {io:C(c1). B(b,fst(io))=0}"

  Gamma_def     "Gamma(b,c) == (%phi.{io : (phi O C(c)). B(b,fst(io))=1} Un 
                                     {io : id(loc->nat). B(b,fst(io))=0})"

  C_while_def   "C(while b do c) == lfp((loc->nat)*(loc->nat), Gamma(b,c))"

end