[Agda] [lean-user] Re: [Coq-Club] Why dependent type theory?

[Bas Spitters]

Dear Kevin,

The excitement about HoTT is that it has brought together several
communities. Some are interested in homotopy theory and higher
category theory, some (like Vladimir) want a new foundation for modern
mathematics.
Some combine those two by higher toposes.

Some are trying to improve the previous generation of proof
assistants. E.g. this influenced the design of quotients types in
lean.
By Curry-Howard this also influences the design of programming
languages, like the cubical agda programming language
(https://pure.itu.dk/portal/files/84649948/icfp19main_p164_p.pdf)

If we consider HoTT as an extension of type theory with the univalence
axiom, then *of course* everything that was done before can still be
done.
E.g. the proof of Feit-Thompson is constructive and thus also works in
HoTT. (I can elaborate on this if needed.)

In fact, classical logic is valid in the simplicial set model
(https://www.math.uwo.ca/faculty/kapulkin/notes/LEM_in_sSet.pdf).
Moreover, that model also interprets strict propositions, so one could
even extend lean with univalence (I believe).
It would be interesting to know whether this simplifies the definition
of perfectoid spaces.

Best regards,

Bas

On Thu, Mar 5, 2020 at 12:25 PM Kevin Buzzard <kevin.m.buzzard at gmail.com> wrote:
>
>
>
> On Wed, 4 Mar 2020 at 07:18, Martin Escardo <m.escardo at cs.bham.ac.uk> wrote:
>>
>> Dependent types are good for pure mathematics (classical or
>> constructive). They are the natural home to define group, ring, metric
>> space, topological space, poset, lattice, category, etc, and study them.
>> Mathematicians that use(d) dependent types include Voevodsky (in Coq)
>> and Kevin Buzzard (in Lean), among others. Kevin and his team defined,
>> in particular, perfectoid spaces in dependent type theory. Martin
>
>
> The BCM (Buzzard, Commelin, Massot) paper defined perfectoid spaces in Lean
> and looking forwards (in the sense of trying to attract "working mathematicians"
> into the area of formalisation) I think it's an interesting question as to whether this definition
> could be made in other systems in a way which is actually usable. My guess: I don't see why it couldn't
> be done in Coq (but of course the type theories of Lean and Coq are similar), although
> there is a whole bunch of noncomputable stuff embedded in the mathematics.
> I *suspect* that it would be a real struggle to do it in any of the HOL systems
> because a sheaf is a dependent type, but these HOL people are good at tricks
> for working around these things -- personally I would start with seeing whether
> one can set up a theory of sheaves of modules on a locally ringed space in a HOL
> system, because that will be the first stumbling block. And as for the HoTT systems,
> I have no feeling as to whether it is possible to do any serious mathematics other than
> category theory and synthetic homotopy theory -- my perception is that
> the user base are more interested in other kinds of questions.
>
> In particular, connecting back to the original question, a sheaf of modules on a
> locally-ringed space is a fundamental concept which shows up in a typical MSc
> or early PhD level algebraic geometry course (they were in the MSc algebraic
> geometry course I took), and if one wants to do this kind of mathematics in a
> theorem prover (and I do, as do several other people in the Lean community)
> then I *suspect* that it would be hard without dependent types. On the other hand
> I would love to be proved wrong.
>
> Kevin
>>
>>
>> On 03/03/2020 19:43, jasongross9 at gmail.com wrote:
>> > I'm in the process of writing my thesis on proof assistant performance
>> > bottlenecks (with a focus on Coq), and there's a large class of
>> > performance bottlenecks that come from (mis)using the power of dependent
>> > types.  So in writing the introduction, I want to provide some
>> > justification for the design decision of using dependent types, rather
>> > than, say, set theory or classical logic (as in, e.g., Isabelle/HOL).
>> > And the only reasons I can come up with are "it's fun" and "lots of
>> > people do it"
>> >
>> > So I'm asking these mailing lists: why do we base proof assistants on
>> > dependent type theory?  What are the trade-offs involved?
>> > I'm interested both in explanations and arguments given on list, as well
>> > as in references to papers that discuss these sorts of choices.
>> >
>> > Thanks,
>> > Jason
>> >
>> > _______________________________________________
>> > Agda mailing list
>> > Agda at lists.chalmers.se
>> > https://lists.chalmers.se/mailman/listinfo/agda
>> >
>>
>> --
>> Martin Escardo
>> http://www.cs.bham.ac.uk/~mhe
>
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