# Ord instance

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What is the meaning of the `Ord` instance? Certainly most people agree that an `Ord` instance shall provide an total ordering. However opinions differ whether there shall be more in it:

• An `Ord` instance may also suggest a notion of magnitude
• An `Ord` instance may be free of any other association

Depending on these opinions we come to different conclusions whether there should be `Ord` instances for `Bool` and `Complex` numbers. In most circumstances expressions like `a < b` are certainly a bug, when `a` and `b` are `Bool` or `Complex` numbers. Consider someone rewrites an algorithm for real numbers to complex numbers and he relies on the type system to catch all inconsistencies. The field operations can remain the same, but `(<)` has to be applied to results of `abs`, `realPart` or other functions that yield a real. The truth of `False < True` relies on the encoding of `False` by 0 and `True` by 1. However there are also programming languages that represent "true" by -1, because this has bit pattern 1....1. The CPU has an instruction to fill a byte with the content of a flag and you can use this bit pattern for bitwise AND and OR operations. This makes that representation very efficient. In principle we could also provide machine dependent efficient representations of boolean values in Haskell. If "true" is associated with -1 then it holds `False > True`. If you use the numeric value of boolean values for arithmetics like in `2 * fromEnum bool - 1` in order to map `False` to -1 and `True` to 1 without an `if-then-else`, then porting a program between different representations of boolean values becomes error-prone.

However you like to work with `Set`s of boolean values and complex numbers, and `Set` requires an `Ord` instance. You may consider using the `Ord` instance by `Set` operations an abuse, since they do not require a particular ordering. Ordering is only needed for implementation of efficient `Set` operations and the operations would be as efficient, if the order is reversed or otherwise modified. But we would certainly not like to have `1 < 2` for `Integer`s and `1 > 2` for, say, `Rational`s.

The solution might be a type class especially for `Set` and `Map`. However it would miss automatic instance deriving.