[Agda] Re: [TYPES] strict unit type

Altenkirch Thorsten psztxa at exmail.nottingham.ac.uk
Thu May 10 17:41:46 CEST 2012

On 10/05/2012 00:04, "Vladimir Voevodsky" <vladimir at ias.edu> wrote:

>On May 1, 2012, at 5:24 AM, Altenkirch Thorsten wrote:
>> In general we shouldn't expect to decide eta equality by untyped
>> reduction. eta equality isa form of decidable extensionality and does
>> correspond well to computation to a normal form.
>> Clearly, the case of the unit type shows that we need to know the type
>> check equality.
>> Trying to decide eta equalities by reduction is very non-modular. This
>> basically the point of Vladimir's email.
>> However, if we implement it as a type directed decision procedure things
>> are much better. E.g. to terms f,g : Pi x:A.B x are convertible if f x
>> g x : B x are convertible (whewre x is a fresh variable) after beta
>> reduction. Similarily, two terms p,q : Sigma x:A.B x are convertible if
>> p.1 and q.1 : A are convertible after beta reduction and p.2 and q.2 : B
>> p.1 (or B q.1) are convertible after beta reduction. (The and here has
>> be read sequentially, and the choice B p.1 or B p.2 actually causes some
>> technical problems).
>Why just after beta reduction and not say iota-reduction or even some
>internal eta-reduction?

When I say beta I also mean beta for product and inductive types. The Coq
people seem to call this iota reduction.

What do you mean by internal eta-reduction.

Btw, I don't think you need to reduce under a lambda (which is also a form
of extensionality). So what I mean is to execute the purely computational
rules (these are the once used by any functional programming language).


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