/*  File Nat.java

    Part of Natural Number Example
    H. Conrad Cunningham, Professor

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2004-01-17: Original
2016-10-26: Formatting changes, compilation check
    
The cluster of classes consisting of base class Nat and its subclasses
Zero, Err, and Succ defines a representation for the "natural numbers"
with a hierarchical class structure inspired by Peano's Postulates.
It does not use any builtin integer type.

We consider 0 as being a natural number as is customary for many
computer scientists.

Mathematically, we can define the set Nat (of natural numbers) as
follows:

    x in Nat if and only if 
            (1) x = 0                            -- zero
         or (2) (Exists y in Nat :: x = Succ(y)) -- successor function

This design uses the Composite pattern to represent natural numbers.
The abstract base class for the set is Nat.

Subclass Zero is the leaf case of the Composite pattern; it
represents 0.

Subclass Succ is the composite case of the Composite pattern; it
represents a successor of the single Nat it contains.

Subclass Err is another leaf case of the Composit pattern; it
represents errors.

The design and implementations of Zero and Err also reflect the
Singleton design pattern, which provides for exactly one instance to
be created.  

Err also reflects the Null Object design pattern, in that its
instances are null or inert objects that can be safely returned by
operations in error situations.

This implementation essentially gives a a unary representation for the
natural numbers -- one Succ object in a linear structure for each
natural number value greater than 0.

*/

/* 
    Class Nat is the base class of our Natural numbers.  It specifies 
    the operations that Nats must support.  Additional operations
    should be included.  
 */

abstract public class Nat 
{ 
    abstract public Nat add(Nat n); 
    abstract public Nat sub(Nat n); 
    abstract public boolean isLess(Nat n); 
}