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    <p><a href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems" rel="nofollow">Gödel's incompleteness theorems</a> imply that any <code>Eq</code> instance for functions must either give inaccurate results or sometimes return ⊥. That's not what we expect from <code>Eq</code> instances (at least for finite data).</p> <p><code>show</code> is supposed to provide Haskell source code that evaluates to its input. That is awkward when compiling a Haskell program, because now you must keep a copy of the source code for every function, bloating the executable, <em>even if the <code>Show</code> instance for functions is never used</em>.</p> <p>It is possible to provide a <code>Show</code> instance for functions that breaks this rule, e.g. by always returning <code>"{-function-}"</code>, or (for some types) returning the type of the function. Early versions of Haskell did. But it was felt that breaking this rule was not a good idea.</p>
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    1. COI disagree with both these answers. The first one because it is _very_ common to have `Eq` instances for infinite data that work exactly as you describe, and since we can represent many functions simply as infinite mappings of input -> output the problem is in fact much the same. Dragging Godel into this seems rather far-fetched (and even dragging in the halting problem is besides the point). Your second point feels similarly silly.
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    2. COThe real issue in both cases is intensional vs. extensional equality (or i guess syntactic vs semantic). The semantics of Haskell dictate the latter but all we can capture for your proposed `Show` is the former. `Eq`, on the other hand, is possible over finite domain things, but that's a narrow subset of what we might want, and inefficient to boot.
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    3. CONo, @sclv, Godel's incompleteness Theorem _is_ relevant. Consider a function that returns all the even numbers and another one that returns the even numbers that are the sum of two smaller primes. Are they equal? I'm impressed if you can write a Haskell program that can solve a problem unsolved by mathematicians for centuries, and which we don't know is solvable. Godel shows that there are unprovable truths in any mathematical system sophisticated enough to include arithmetic - so any Eq instance for functions must be incorrect or incomplete, and incomplete in the sense of does not terminate.
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