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<a class = mltona href="Home">MLton MLTONWIKIVERSION</a>
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EqualityTypeVariable
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An equality type variable is a type variable that starts with two or more primes, as in <tt>''a</tt> or <tt>''b</tt>. The canonical use of equality type variables is in specifying the type of the <a href="PolymorphicEquality">PolymorphicEquality</a> function, which is <tt>''a * ''a -> bool</tt>. Equality type variables ensure that polymorphic equality is only used on <a href="EqualityType">equality types</a>, by requiring that at every use of a polymorphic value, equality type variables are instantiated by equality types. <p>
For example, the following program is type correct because polymorphic equality is applied to variables of type <tt>''a</tt>.
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">fun</FONT></B> f (x: ''a, y: ''a): bool = x = y
</PRE>
<p>
</p>
<p>
On the other hand, the following program is not type correct, because polymorphic equality is applied to variables of type <tt>'a</tt>, which is not an equality type.
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">fun</FONT></B> f (x: 'a, y: 'a): bool = x = y
</PRE>
<p>
</p>
<p>
MLton reports the following error, indicating that polymorphic equality expects equality types, but didn't get them.
</p>
<pre>Error: z.sml 1.32.
Function applied to incorrect argument.
expects: [<equality>] * [<equality>]
but got: [<non-equality>] * [<non-equality>]
in: = (x, y)
</pre><p>
As an example of using such a function that requires equality types, suppose that <tt>f</tt> has polymorphic type <tt>''a -> unit</tt>. Then, <tt>f 13</tt> is type correct because <tt>int</tt> is an equality type. On the other hand, <tt>f 13.0</tt> and <tt>f (fn x => x)</tt> are not type correct, because <tt>real</tt> and arrow types are not equality types. We can test these facts with the following short programs. First, we verify that such an <tt>f</tt> can be applied to integers.
</p>
<pre class=code>
<B><FONT COLOR="#0000FF">functor</FONT></B> Ok (<B><FONT COLOR="#A020F0">val</FONT></B> f: ''a -> unit): <B><FONT COLOR="#0000FF">sig</FONT></B> <B><FONT COLOR="#0000FF">end</FONT></B> =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> () = f <B><FONT COLOR="#5F9EA0">13</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> () = f <B><FONT COLOR="#5F9EA0">14</FONT></B>
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
<p>
We can do better, and verify that such an <tt>f</tt> can be applied to any integer.
</p>
<pre class=code>
<B><FONT COLOR="#0000FF">functor</FONT></B> Ok (<B><FONT COLOR="#A020F0">val</FONT></B> f: ''a -> unit): <B><FONT COLOR="#0000FF">sig</FONT></B> <B><FONT COLOR="#0000FF">end</FONT></B> =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#A020F0">fun</FONT></B> g (x: int) = f x
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
<p>
Even better, we don't need to introduce a dummy function name; we can use a type constraint.
</p>
<pre class=code>
<B><FONT COLOR="#0000FF">functor</FONT></B> Ok (<B><FONT COLOR="#A020F0">val</FONT></B> f: ''a -> unit): <B><FONT COLOR="#0000FF">sig</FONT></B> <B><FONT COLOR="#0000FF">end</FONT></B> =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> _ = f: int -> unit
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
<p>
Even better, we can use a signature constraint.
</p>
<pre class=code>
<B><FONT COLOR="#0000FF">functor</FONT></B> Ok (S: <B><FONT COLOR="#0000FF">sig</FONT></B> <B><FONT COLOR="#A020F0">val</FONT></B> f: ''a -> unit <B><FONT COLOR="#0000FF">end</FONT></B>):
<B><FONT COLOR="#0000FF">sig</FONT></B> <B><FONT COLOR="#A020F0">val</FONT></B> f: int -> unit <B><FONT COLOR="#0000FF">end</FONT></B> = S
</PRE>
<p>
</p>
<p>
This functor concisely verifies that a function of polymorphic type <tt>''a -> unit</tt> can be safely used as a function of type <tt>int -> unit</tt>.
</p>
<p>
As above, we can verify that such an <tt>f</tt> can not be used at non equality types.
</p>
<pre class=code>
<B><FONT COLOR="#0000FF">functor</FONT></B> Bad (S: <B><FONT COLOR="#0000FF">sig</FONT></B> <B><FONT COLOR="#A020F0">val</FONT></B> f: ''a -> unit <B><FONT COLOR="#0000FF">end</FONT></B>):
<B><FONT COLOR="#0000FF">sig</FONT></B> <B><FONT COLOR="#A020F0">val</FONT></B> f: real -> unit <B><FONT COLOR="#0000FF">end</FONT></B> = S
</PRE>
<p>
</p>
<pre class=code>
<B><FONT COLOR="#0000FF">functor</FONT></B> Bad (S: <B><FONT COLOR="#0000FF">sig</FONT></B> <B><FONT COLOR="#A020F0">val</FONT></B> f: ''a -> unit <B><FONT COLOR="#0000FF">end</FONT></B>):
<B><FONT COLOR="#0000FF">sig</FONT></B> <B><FONT COLOR="#A020F0">val</FONT></B> f: ('a -> 'a) -> unit <B><FONT COLOR="#0000FF">end</FONT></B> = S
</PRE>
<p>
</p>
<p>
For each of these programs, MLton reports the following error.
</p>
<pre>Error: z.sml 2.4.
Variable type in structure disagrees with signature.
variable: f
structure: [<equality>] -> _
signature: [<non-equality>] -> _
</pre><h2 id="head-298e9694028673faa9fcbabf774bc23e4d2fbbe0">Equality type variables in type and datatype declarations</h2>
<p>
Equality type variables can be used in type and datatype declarations; however they play no special role. For example,
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> 'a t </FONT></B>=<B><FONT COLOR="#228B22"> 'a * int
</FONT></B></PRE>
<p>
</p>
<p>
is completely identical to
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> ''a t </FONT></B>=<B><FONT COLOR="#228B22"> ''a * int
</FONT></B></PRE>
<p>
</p>
<p>
In particular, such a definition does <em>not</em> require that <tt>t</tt> only be applied to equality types.
</p>
<p>
Similarly,
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">datatype</FONT></B><B><FONT COLOR="#228B22"> 'a t </FONT></B>=<B><FONT COLOR="#228B22"> <FONT COLOR="#B8860B">A</FONT> </FONT></B>|<B><FONT COLOR="#228B22"> <FONT COLOR="#B8860B">B</FONT> <B><FONT COLOR="#A020F0">of</FONT></B> 'a
</FONT></B></PRE>
<p>
</p>
<p>
is completely identical to
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">datatype</FONT></B><B><FONT COLOR="#228B22"> ''a t </FONT></B>=<B><FONT COLOR="#228B22"> <FONT COLOR="#B8860B">A</FONT> </FONT></B>|<B><FONT COLOR="#228B22"> <FONT COLOR="#B8860B">B</FONT> <B><FONT COLOR="#A020F0">of</FONT></B> ''a
</FONT></B></PRE>
<p>
</p>
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<p>
<hr>
Last edited on 2005-12-01 04:00:38 by <span title="ppp-71-139-183-221.dsl.snfc21.pacbell.net"><a href="StephenWeeks">StephenWeeks</a></span>.
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