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    <p>There's generally two approaches for this: The inductive definition you gave, or an abstract data type using something else for internal representation.</p> <p>Note that the inductive representation is not terribly efficient for large numbers; however, it can be lazy, which lets you do things like see which of two nats is larger without evaluating further than the size of the smaller one.</p> <p>An abstract data type is one which is defined in a separate module and does not export its constructors, examples being <code>IO</code> or <code>Data.Set.Set</code>. You can define something like this:</p> <pre><code>module Nat (Nat() {- etc. -} ) where newtype Nat = Nat { unNat :: Integer } </code></pre> <p>...where you export various operations on <code>Nat</code> such that, even though the <em>internal representation</em> is just <code>Integer</code>, you ensure that no value of type <code>Nat</code> is constructed holding a negative value.</p> <p>In both cases, if you want to use numeric literals, you'll need a definition of <code>fromInteger</code>, which is attached to the <code>Num</code> type class, which is completely wrong for natural numbers but oh well.</p> <p>If you don't mind making a broken instance just to get syntactic niceties, you can do something like this:</p> <pre><code>instance Num Nat where Zero + n = n n + Zero = n (Succ n1) + (Succ n2) = Succ . Succ $ n1 + n2 fromInteger 0 = Zero fromInteger i | i &gt; 0 = Succ . fromInteger $ i - 1 </code></pre> <p>...and so on, for the other functions. The same can be done for the abstract data type approach, just be careful to <em>not use <code>deriving</code> to get an automatic <code>Num</code> instance</em>, because it will happily break your non-negative constraint.</p>
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