That's an interesting question.  I was thinking of operations and
equalities, and annotations are neither of those, but I can see that
they're somewhat similar in spirit.  But I find it even more difficult to
imagine how to define this phenomenon precisely in a way that would include
them...

On Fri, Nov 18, 2022 at 8:59 AM Jon Sterling <jon@jonmsterling.com> wrote:

> On 18 Nov 2022, at 11:56, Michael Shulman wrote:
>
> > Thanks.  It does sound like we mostly agree.  I would probably argue that
> > even for type theories in development, where we don't yet know that full
> > definitional equality is decidable -- or intrinsically ill-behaved type
> > theories like Lean, or Agda with non-confluent rewrite rules, where
> > definitional equality *isn't* decidable -- there is still value in being
> > able to reduce just substitutions.  But I think that's a relatively minor
> > point.
> >
> > Maybe we can work out some way to use words so that this underlying
> > agreement is evident.  For instance, can we find a third word to use
> > alongside "admissible" and "derivable" to refer to operations/equalities
> > like substitution and its theory, which hold in all reasonable models,
> and
> > can be made admissible in some presentations, but more importantly can be
> > eliminated in an equality-checking algorithm?
> >
>
> Cool, it's a relief to start getting this cleared up! I really agree with
> you that there is a need for terminology to describe the situation you
> mention, and maybe even more, there is a need to define this phenomenon...
>
> I wonder if we can think of more concrete examples of this. For instance,
> in many versions of syntax (like bidirectional ones) we can choose to drop
> certain annotations because they will be available as part of the flow of
> information in the elaboration algorithm. Would these be an example of the
> phenomenon you are describing, or is it something different?
>
> Best,
> Jon
>

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