Under what notion of equality are typeclass laws written?

Question

Haskell typeclasses often come with laws; for instance, instances of Monoid are expected to observe that x <> mempty = mempty <> x = x.

Typeclass laws are often written with single-equals (=) rather than double-equals (==). This suggests that the notion of equality used in typeclass laws is something other than that of Eq (which makes sense, since Eq is not a superclass of Monoid)

Searching around, I was unable to find any authoritative statement on the meaning of = in typeclass laws. For instance:

• The Haskell 2010 report does not even contain the word "law" in it
• Speaking with other Haskell users, most people seem to believe that = usually means extensional equality or substitution but is fundamentally context-dependent. Nobody provided any authoritative source for this claim.
• The Haskell wiki article on monad laws states that = is extensional, but, again, fails to provide a source, and I wasn't able to track down any way to contact the author of the relevant edit.

The question, then: Is there any authoritative source on or standard for the semantics for = in typeclass laws? If so, what is it? Additionally, are there examples where the intended meaning of = is particularly exotic?

---

(As a side note, treating = extensionally can get tricky. For instance, there is a Monoid (IO a) instance, but it's not really clear what extensional equality of IO values looks like.)

Answer 1

I suspect most folks use = to mean "moral equality" as from Fast and Loose Reasoning is Morally Correct, which you can think of as extensional equality up to defined-ness.

But there's no hard-and-fast rule here. There's a lot of libraries, and a lot of authors, and if you take any two authors they probably have some minor detail about = on which they disagree.

Answer 2

Typeclass laws are not part of the Haskell language, so they are not subject to the same kind of language-theoretic semantic analysis as the language itself.

Instead, these laws are typically presented as an informal mathematical notation. Most presentations do not need a more detailed mathematical exposition, so they do not provide one.