(* Title: HOL/IMP/Denotation.thy
ID: $Id$
Author: Heiko Loetzbeyer & Robert Sandner, TUM
Copyright 1994 TUM
Denotational semantics of expressions & commands
*)
Denotation = Com +
consts
A :: "aexp => state => nat"
B :: "bexp => state => bool"
C :: "com => (state*state)set"
Gamma :: "[bexp,com,(state*state)set] => (state*state)set"
primrec A aexp
A_nat "A(N(n)) = (%s. n)"
A_loc "A(X(x)) = (%s. s(x))"
A_op1 "A(Op1(f,a)) = (%s. f(A(a,s)))"
A_op2 "A(Op2(f,a0,a1)) = (%s. f(A(a0,s),A(a1,s)))"
primrec B bexp
B_true "B(true) = (%s. True)"
B_false "B(false) = (%s. False)"
B_op "B(ROp(f,a0,a1)) = (%s. f(A(a0,s),A(a1,s)))"
B_not "B(noti(b)) = (%s. ~B(b,s))"
B_and "B(b0 andi b1) = (%s. B(b0,s) & B(b1,s))"
B_or "B(b0 ori b1) = (%s. B(b0,s) | B(b1,s))"
defs
Gamma_def "Gamma(b,c) == \
\ (%phi.{io. io : (phi O C(c)) & B(b,fst(io))} Un \
\ {io. io : id & ~B(b,fst(io))})"
primrec C com
C_skip "C(skip) = id"
C_assign "C(x := a) = {io . snd(io) = fst(io)[A(a,fst(io))/x]}"
C_comp "C(c0 ; c1) = C(c1) O C(c0)"
C_if "C(ifc b then c0 else c1) = \
\ {io. io:C(c0) & B(b,fst(io))} Un \
\ {io. io:C(c1) & ~B(b,fst(io))}"
C_while "C(while b do c) = lfp(Gamma(b,c))"
end