II. Notes on Data Structuring * - Cornell University

106 c.A.R. HOARE
variable of type raster, r.x and r.y are commonly known as the projections
of r onto the x and y axes respectively; however, we shall refer to the functions
x and y as selectors rather than projections.
The cardinality of a Cartesian product type is obtained by multiplying
together the cardinalities of the constituent types. This is fairly obvious
from the visualisation of a Cartesian product as a rectangle or box with
sides equal in length to the cardinalities of the types which form the axes.
thus the cardinaiity of the card type is thirteen times four, i.e., fifty-two, which
is, as you might expect, the number of cards in a standard pack. The number
of dates is 26 040, which slightly overestimates the actual number of days in
the interval, since as explained above, it includes a small number of invalid
dates.
4.1 MANIPULATION
Apart from assignment and test of equality, which are common to all types,
the main operations defined for a product type are just those of constructing
a value in terms of component values, and of selecting the components.
When constructing a value of a Cartesian product type, it is in principle
necessary to quote the name of the type as a transfer function. However,
it is often more convenient to follow the traditional mathematical practice,
and leave the transfer function implicit in cases where no confusion would
arise. This is in any case necessary when a type is not even given an explicit
name. For example, one may write (heart, Jack) instead of cardface (heart,
Jack).
For selection of a component, a dot-notation has been used, e.g.,
n. imagpart. This is more convenient than the normal functional notation
imagpart (n), since it avoids unnecessarily deep nesting of brackets.
Another most important operation is the selective updating of the com-
ponents of a variable. This may be denoted by placing the component name
on the left of an assignment
u. imagpart: = 0;
r.x:= axr.x + bxr.y.
If a Cartesian product is declared as ordered, it is necessary that all the
constituent types be ordered, and it is natural to define the ordering in a
lexicographic manner, taking the earlier components as the more significant.
Thus if suit and rank are ordered, the cardface type could be declared as
ordered in the traditional ranking whereby all clubs precede all diamonds,
and these are followed by all hearts and all spades; whereas within each suit,
the cards are ordered in accordance with their rank.