Ord: Reflexivity law is wrong?
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研究方向
Read the Ord law documentation and the linked issue #174 and pull request #298 for context; pull request #301 is already open for this issue. The proposed change is to state reflexivity as “if a == b then a <= b”; determine whether that law is appropriate for Ord and how the example of unreduced rationals affects it before making a change.
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描述
A while back, after some discussion, we updated the antisymetry law of Ord from:
-- | - Antisymmetry: if `a <= b` and `b <= a` then `a = b`
to
-- | - Antisymmetry: if `a <= b` and `b <= a` then `a == b`
so as to connect Eq and Ord. See issue #174 and PR #298.
Ever since, the reflexivity law has been bugging me and I think we need to update that too. Currently, it reads
-- | - Reflexivity: `a <= a`
and I think it should read
-- | - Reflexivity: if `a == b` then `a <= b`
To explain why, I return to my old example of unreduced rational numbers. Take a rational number represented as a record with two fields (n and d) where d is not zero but without the requirement that n and d are coprime. We can define equality as
eq (Rat x) (Rat y) = x.n * y.d == y.n * x.d
But what if we then define Ord using dictionary order
compare (Rat x) (Rat y) = case compare x.n y.n of
EQ -> compare x.d y.d
neq -> neq
These two definitions satisfy the current laws, specifically Ord is reflexive, but:
> x = Rat {n: 2, d: 4}
> y = Rat {n: 1, d: 2}
> x == y
true
> x <= y
false
which is surely not what we want.
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