II. Notes on Data Structuring * - Cornell University

170 c.A.R. HOARE
(3) If X = T(xx, x2,..., x,, ki(x'i) ) for each i between 1 and m
X.S 1 ~. X 1 X.S 2 X 2 t~... & X.S n = X n
& x.k~ = x'i
Note: x.kz is undefined for l 4: i.
Abbreviations:
(4) Under the same condition as (3)
with x do {kl :$1, k2:$2 . . . . , kn:Sn} means
withx , i ao(S),,,~ i x~,x2 ..... ~. x, , .
and similarly with take instead of do.
(5) If n = O, k,(x',) stands for T(k,(x'i)).
12.4. ARRAYS
The general form of an array definition is:
type T = array D of R
(1) If r is an R then T(r) is a T
(2) IfxisaT, disaD, andrisanR
then T(x, d: r) is a T
(3) The only elements of T are as specified in (1) and (2).
(4) T(T(x, d:r), d':r') =
if d = d' then T(x, d':r')
(5) T(r)[d] = r.
else T( T(x, d' : r'), d : r ).
(6) T(x, d: r)[d'] = if d' = d then r else x [d'].
(7) (for i:D take E(i))[j] = E(j).
Abbreviations:
(8) x[d]: = r means x: = T(x, d: r).
(9) T(x, dx :rl, d2:r2, ..., d,:v,,) stands for
T(T(. . . T(T(x), d, :v,), d2 : v2). . . ), d,, : r,,).
(10) in (9), the x may be omitted, if dl, d2,..., d, exhaust the domain
type. Similarly, the T may be omitted in suitable contexts.
If the array type is ordered, the following axiom applies:
(11) x ~. y _= V d:D(y[d] < x[d] ~ ~ d':D(d' < d& x[d'] < y[d']))
Theorem:
x = y ~ V d:D(x[d] = y[d])