II. Notes on Data Structuring * - Cornell University

172 c.A.R. HOARE
(23) x down succ (n) = (x down n) down 1
(24) T( )downl = T( )
(25) (x v T(a))down 1 = (x down 1 v
if a -~ To. min then T(pred (a)) else T( )
(26)-(28) up is similarly defined, with succ for pred and max for min.
(29) b < a D range (a, b) = T( )
(30) a ~< b D range (a, b) = T(a)
(31) a < b = range (a, b) = range (a, pred (b)) v T(b)
(32) j in {i:D l B(i)} ~ B(j)
Abbreviations:
(33) T(al, a2,..., an)stands for T(al) v T(a2) v ... v T(a,,)
(34) {al, a2,..., an} stands for T(al, a2 ..... an)
(35) x: ^ y stands for x: = x ^ y
(36) x:vy .... x:= x v y
(37) a from x stands for a:= one of (x); x:- (a}
(38) if x = (al, a2, . . . , an} then
Theorems:
for a in x do S stands for
S a • S a • . S a
air a2 ~ " " • ~ an
where the ai are in natural order if the base type is ordered, and are in
arbitrary order otherwise; and they do not contain repetitions.
12.6 SEQUENCES
x =y ~Va:To(ainx~ainy)
ain(x v y) ~ (ainx v ainy)
ain(x ^y) ~ (a in x & a in y)
a in (x - y) ~. (a in x & -3 a in y)
The general form of a sequence definition is:
type T = sequence D;
(1) T( )isa T
(2) IfxisaTanddisaD
thenx T(d) is a T
(3) The only elements of T are as specified in (1) and (2)
(4) (x-'T(d)). last = d