VDM-10 Language Manual

Chapter 5. Function Definitions
This proof obligation will ensure that the recursive function will terminate and thus sooner or later
reach the base case.
5.1 Polymorphic Functions
Functions can also be polymorphic. This means that we can create generic functions that can
be used on values of several different types. For this purpose type parameters (or type variables
which are written like normal identifiers prefixed with a @ sign) are used. Consider the polymorphic
function to create an empty bag: 1
✞
empty_bag[@elem] : () +> (map @elem to nat1)
empty_bag() ==
{ |-> }
✡✝
Before we can use the above function, we have to instantiate the function empty bag with a type,
for example integers (see also section 6.12):
✞
emptyInt = empty_bag[int]
✡✝
Now we can use the function emptyInt to create a new bag to store integers. More examples of
polymorphic functions are:
✞
num_bag[@elem] : @elem * (map @elem to nat1) +> nat
num_bag(e, m) ==
if e in set dom m
then m(e)
else 0;
✆
✆
plus_bag[@elem] : @elem * (map @elem to nat1) +>
(map @elem to nat1)
plus_bag(e, m) ==
m ++ { e |-> num_bag[@elem](e, m) + 1 }
✡✝
✆
If pre- and or post-conditions are defined for polymorphic functions, the corresponding predicate
functions are also polymorphic. For instance if num bag was defined as
✞
num_bag[@elem] : @elem * (map @elem to nat1) +> nat
1 The examples for polymorphic functions are taken from [Dawes91]. Bags are modelled as maps from the elements to
their multiplicity in the bag. The multiplicity is at least 1, i.e. a non-element is not part of the map, rather than being
mapped to 0.
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