[Haskell-cafe] Learn You a Haskell for Great Good - a few doubts

[Daniel Fischer]

On Friday 04 March 2011 22:33:20, Alexander Solla wrote:
> > Unfortunately, Haskell's tuples aren't quite products.[1]
> 
> I'm not seeing this either.  (A,B) is certainly the Cartesian product of
> A and B.

Not quite in Haskell, there

(A,B) = A×B \union {_|_}

_|_ and (_|_,b) are distinguishable.

(A,()) contains

- (a,()) for a in A
- (a, _|_) for a in A
- _|_

the three classes are distinguishable

case x of
   (a,b) -> do
    putStrLn "Bona fide tuple"
    case b of
      () -> putStrLn "With defined second component"

will produce different output for them.

In Haskell, |(A,B)| = |A|×|B| + 1 (and |()| = 2, () = { (), _|_ }),
and |(A,B,C)| = |A|×|B|×|C| + 1 etc.
So one would expect |(A)| = |A| + 1 by consistency for 1-tuples.

> In what sense are you using "product" here?

Set theoretic or more general, category theoretic, I presume.

> Is your complaint
> a continuation of your previous (implicit) line of thought regarding
> distinct bottoms?

I don't think distinguishing bottoms is the issue, but distinuishing bottom 
from partially defined values.