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 R0k ⊆ 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 Rmk that Rmk ⊆ Rm+1,k if ' ∈ N, so that we also have
chains Rmk ⊆ 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. Ongoing Emergence: A Core Concept in Epigenetic Robotics2. Transfer from primary school to secondary school
3. Qualifying Recital: Lisa Carol Hardaway, flute
4. The name is absent
5. The name is absent
6. Activation of s28-dependent transcription in Escherichia coli by the cyclic AMP receptor protein requires an unusual promoter organization
7. Revisiting The Bell Curve Debate Regarding the Effects of Cognitive Ability on Wages
8. Death as a Fateful Moment? The Reflexive Individual and Scottish Funeral Practices
9. Strategic Policy Options to Improve Irrigation Water Allocation Efficiency: Analysis on Egypt and Morocco
10. How do investors' expectations drive asset prices?