VDM-10 Language Manual

Chapter 3. Data Type Definitions
• Built-in operators for values belonging to the type 4 .
• Semantics of the built-in operators.
• Examples illustrating how the built-in operators can be used.
For each of the built-in operators the name, the symbol used and the type of the operator will
be given together with a description of its semantics (except that the semantics of Equality and
Inequality is not described, since it follows the usual semantics). In the semantics description
identifiers refer to those used in the corresponding definition of operator type, e.g. m, m1, s, s1
etc.
3.2.1 Set Types
A set is an unordered collection of values, all of the same type 5 , which is treated as a whole. All
sets in VDM languages are finite, i.e. they contain only a finite number of elements. The elements
of a set type can be arbitrarily complex, they could for example be sets themselves.
In the following this convention will be used: A is an arbitrary type, S is a set type, s, s1, s2
are set values, ss is a set of set values, e, e1, e2 and en are elements from the sets, bd1, bd2,
. . . , bdm are bindings of identifiers to sets or types, and P is a logical predicate.
Syntax:
type = set type
| . . . ;
set type = ‘set of’, type ;
Equation: S = set of A
Constructors:
Set enumeration: {e1, e2, ..., en} constructs a set of the enumerated elements.
The empty set is denoted by {}.
Set comprehension: {e | bd1, bd2, ..., bdm & P} constructs a set by evaluating
the expression e on all the bindings for which the predicate P evaluates to true. A
binding is either a set binding or a type binding 6 . A set bind bdn has the form pat1,
..., patp in set s, where pati is a pattern (normally simply an identifier),
and s is a set constructed by an expression. A type binding is similar, in the sense that
in set is replaced by a colon and s is replaced with a type expression.
4 These operators are used in either unary or binary expressions which are given with all the operators in section 6.3.
5 Note however that it is always possible to find a common type for two values by the use of a union type (see section
3.2.6.)
6 Notice that type bindings cannot be executed by the interpreter because in general they are not executable (see section 8
for further information about this).
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