[Agda] Encoding higher lambda calculus in agda

[András Kovács]

You can represent the syntax of pretty much every dependent and
impredicative theory in Agda. However, that does not settle whether a
particular proof or construction using a syntax is a) possible in base Agda
b) feasible or convenient in base Agda.

System F and F-omega syntax is easy to represent in Agda, in either
intrinsic or extrinsic manner. Showing consistency or normalization would
require an impredicative universe in Agda, so that's not possible without
postulating stuff, but even without that there are many interesting things
which you could do with the syntax.

Predicative dependent type theories can be represented and proven
consistent; this is <http://www.cse.chalmers.se/~abela/popl18.pdf> the most
advanced development I know of which only uses native Agda features (in
this case, induction-recursion). Note though that formal handling of
dependent types tends to be highly technically challenging.

For dependent type itself, I do not think it's possible, due to Goedel's
> incompleteness theorem.


Gödel doesn't enter the picture in any of the developments I know of,
because the metatheories are stronger than object theories, due to having
inductive families and/or induction/recursion. Also, Gödel would only
prevent consistency proofs, but doesn't preclude representing the syntax or
doing a number of interesting things with it which don't imply consistency.

is Prop type in Coq is impredicative, and therefore the calculus underneath
> (caculus of inductive constructions, CIC) is stronger than calculus of
> constructions(CoC)


CoC has impredicative base Set.

2018-04-05 5:01 GMT+02:00 Jason -Zhong Sheng- Hu <fdhzs2010 at hotmail.com>:

> my intuition was there is nothing special about system F, and all terms
> and types should be just able to fit into Set along. and indeed, someone
> has done the work: https://github.com/sstucki/system-f-agda
>
>
> regarding stronger calculus, i do not see why they are not possible,
> except for dependent type itself(i.e. the strongest calculus of the cube).
> For example, it's straightforward to formalize Fsub in Coq:
> https://github.com/plclub/metalib/tree/master/Fsub
>
>
> The modelling calculus of Scala, (dependent object types, DOT), for
> example, has type dependent types, type dependent values, and a weak form
> of value dependent types, which can also be proved with terms and types in
> Set: https://github.com/samuelgruetter/dot-calculus ht
> tps://github.com/amaurremi/dot-calculus
>
>
> Though in Coq, the terms and types are defined in Set, which roughly
> corresponds to Set in Agda, and that hopefully means you won't need to go
> above Set 2 to prove it's sound.
>
>
> For dependent type itself, I do not think it's possible, due to Goedel's
> incompleteness theorem. However, someone did it in Coq (which i've not
> checked, but, well, it's there) https://github.com/coq-contribs/coq-in-coq .
> The reason why it might be doable in Coq, is Prop type in Coq is
> impredicative, and therefore the calculus underneath (caculus of inductive
> constructions, CIC) is stronger than calculus of constructions(CoC).
> Indeed, the author represents typ to be Prop because he had no choice.
> https://github.com/coq-contribs/coq-in-coq/blob/master/Types.v
>
>
> Hope that helps.
>
>
>
> *Sincerely Yours, *
>
> * Jason Hu*
> ------------------------------
> *From:* Agda <agda-bounces at lists.chalmers.se> on behalf of How Si Yu <
> howseeu at gmail.com>
> *Sent:* April 4, 2018 10:36:21 PM
> *To:* agda at lists.chalmers.se
> *Subject:* [Agda] Encoding higher lambda calculus in agda
>
> Hi,
>
> There are numerous implementations of untyped lambda calculus and
> simply typed lambda calculus in Agda online. However, I can't find any
> mention of encoding of other "higher" lambda calculus in the lambda
> cube like λ2 or λP in Agda. Is it because they are inherently
> impredicative and can't be represented in Agda?
>
> Thanks,
> Si Yu
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