[Agda] strict unit type

Vladimir Voevodsky vladimir at ias.edu
Sun Apr 29 18:08:24 CEST 2012

On Apr 29, 2012, at 11:38 AM, Guillaume Brunerie wrote:

> Hello,
> 2012/4/29 Vladimir Voevodsky <vladimir at ias.edu>:
>> Hello,
>> let Pt be the unit type ("one point"). In Coq-like systems it is introduced through an eliminator and the associated iota-reduction rule. This is in practice sufficient for proving all its expected properties modulo propositional equality (or identity types) but does not create a terminal object in the category of contexts.
> I’m not sure if it is your question, but you do not need a unit type
> to have a terminal object in the category of contexts, the empty
> context is already a terminal object (there is exactly one context
> morphism from any context to the empty context : the empty morphism).
>> I wonder if any one has considered type theories where Pt is introduced as having a distinguished object tt together with reduction rules of the following form (in the absence of dependent sums, otherwise one needs more):
>> 1. any object of Pt other than tt reduces to tt,
> This is just the eta-expansion rule for Pt. For example Agda has it if
> you implement Pt as a record.
> You only need this (and beta/iota reduction) for (Pt) to be terminal
> in the category of contexts (it will be isomorphic to the empty
> context).

I wonder how it works in Agda. This reduction for the unit type creates non-confluet situation when combined with iota/eta reduction for dependent sums. Additional reductions are needed to recover confluence. Are they implemented in Agda?

>> 2. Prod x : T , Pt reduces to Pt,
>> 3. Prod x : Pt , T reduces to T.
> Sorry, I don’t know about these two, I have never seen those reduction
> rules but you should wait for the answer of an actual type theorist.
> Guillaume
>> Thanks,
>> Vladimir.

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