Program Semantics and Classical Logic



for each ` N {ω}, each m N and each k (1 k n) the relation
R. by setting R0,k = 0 and Rmk+1 = V(Ak, α[Rm√Xι,..., Rmn∕Xn, R'∕ξ])•
Then R
0k R'k ... Rnk ... for each k and '. Moreover, it
,,     ,

is easily proved by induction on m, using the monotonicity of the Ak and
the definition of R
mk that Rmk Rm+1,k if ' N, so that we also have
chains R
mk Rmk ... Rmk .... We prove by induction on m that
S' Rmk = Rmk. This is trivial for m = 0. Suppose the statement holds for
m. Then, using the continuity of the
Ak and the induction hypothesis, we
find that

R'm,k+1  =

'

[ V(Ak, a[Rm1∕X1,..., Rm/Xn, R'∕ξ]) =
'

V(Ak,a


U Rm1∕X1,... , Rmn/Xn, U R'∕ξ ) =

'   ''

V(Ak, a Rωm,1∕X1,..., Rωm,n∕Xn, Rω ∕ξ ) =

Rωm,+k 1

Since Rωω,k = m  ' R'm,k = '  m R'm,k it follows that (*) Rωω,k =  'R'ω,k. Now

we reason as follows, using Lemma 8 twice and using (*) and the continuity
of
B in ξ

[V(λr.X1 .. .Xn(^n Xi = Ai → B(r)), a[R'∕ξ]) =

'                        i=1

[ V(B, a[R'ω,1∕X1,..., R'ω,n∕Xn, R'∕ξ]) =
'

V(B, a


R'ω,1∕X1,...,   R'ω,n∕Xn,   R'∕ξ

)=


'   ''

V(B,a Rωω,1∕X1,..., Rωω,n∕Xn, Rω∕ξ ) =

V(λr.X1...Xn(^n Xi =Ai → B(r)), a[Rω ∕ξ])

i=1

This concludes the proof

That translations of programs are continuous is now easily proved. The
proposition will allow us to apply the Knaster-Tarski theorem to those trans-
lations.

21



More intriguing information

1. A THEORETICAL FRAMEWORK FOR EVALUATING SOCIAL WELFARE EFFECTS OF NEW AGRICULTURAL TECHNOLOGY
2. Giant intra-abdominal hydatid cysts with multivisceral locations
3. The name is absent
4. Markets for Influence
5. Detecting Multiple Breaks in Financial Market Volatility Dynamics
6. Should Local Public Employment Services be Merged with the Local Social Benefit Administrations?
7. Land Police in Mozambique: Future Perspectives
8. THE CHANGING RELATIONSHIP BETWEEN FEDERAL, STATE AND LOCAL GOVERNMENTS
9. The name is absent
10. Qualifying Recital: Lisa Carol Hardaway, flute