[Haskell-cafe] Tutorial: Curry-Howard Correspondence
Dan Weston
westondan at imageworks.com
Wed Oct 17 21:49:24 EDT 2007
It would seem that this induction on SN requires strictness at every
stage. Or am I missing something?
Stefan O'Rear wrote:
> On Wed, Oct 17, 2007 at 03:06:33PM -0700, Dan Weston wrote:
>> 2) the function must halt for all defined arguments
>>
>> fix :: forall p . (p -> p) -> p
>> fix f = let x = f x in x
>
> consider:
>
> foo :: ((a -> a) -> a) -> a
> foo x = x id
>
> foo is a valid proof of a true theorem, but does not halt for the
> defined argument 'fix'.
>
> -
>
> An effective approach for handling this was found long ago in the
> context of prooving the strong normalization of the simply typed lambda
> calculus. Define a category of terms SN[a] for each type a by recursion
> on a.
>
> SN[a], for atomic a, consists of terms that halt.
> SN[a -> b] consists of functions that inhabit SN[b] when passed an
> argument in SN[a].
>
> With that definition, prooving that every term of the STLC of type a
> inhabits SN[a], and thus halts, becomess a trivial induction on the
> syntax of terms.
>
> Stefan
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