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I think this is quite straightforward to show using the fact that
beta-eta-reduction is confluent on untyped terms.
Write |M| for the type-erasure of the Church-style simply-typed term M.
If M ~ N, then certainly |M| ~ |N|. Let M', N' be respectively the
normal forms of M and N. By confluence, |M'| = |N'|. The key property we
need is that given a Curry-style judgement, G |- P : A, where P is in
normal form, there is a unique Church-style term P* such that G |- P* :
A and |P*| = P. This means that M' = |M'|* = |N'|* = N'.
Sam
On 10/08/12 12:34, Derek Dreyer wrote:
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Hi, all. I'm having a hard time finding a (canonical) source for what
I thought would be a very simple question.
Consider the simply-typed lambda calculus (STLC), Church-style (with
type annotations on variable binders), and beta-eta reduction.
Beta-eta convertibility, which I'll write as M ~ N, means that M is
convertible to N by some sequence of beta/eta reductions/expansions,
which may very well go through ill-typed intermediate terms even if M
and N are themselves well-typed.
I want to show the following:
If G |- M : A and G |- N : A, and M ~ N, then M and N have the same
beta-eta (short) normal form.
It is well known that well-typed terms in STLC are strongly
normalizing wrt beta-eta reduction. But to prove the above, one also
needs confluence. The problem is that the intermediate terms in the
beta-eta conversion of M and N may be ill-typed, and
beta-eta-reduction is not confluent in this setting for ill-typed
terms. In particular,
(\x:A.(\y:B.y)x)
can eta-convert to
\y:B.y
or beta-convert to
\x:A.x
which are only equal if A = B.
One strategy would be to prove that beta-eta-convertibility --
*between terms of the same type* -- going through ill-typed terms is
equivalent to beta-eta convertibility restricted to going only through
well-typed terms. Intuitively, this seems like it should be true, but
it is not at all clear to me how to show it. In particular, the
*between terms of the same type* part seems critical in order to
outlaw the confluence counterexample given above.
I found some discussion of some related questions in a thread on the
Types list from 1989:
http://article.gmane.org/gmane.comp.science.types/266 (and subsequent posts)
But it doesn't seem to contain a direct answer to my question.
Thanks!
Derek
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