[TYPES] Types-list Digest, Vol 155, Issue 3

[Andrew Polonsky]

Yes, the initial model of System F satisfies the identity extension lemma,
assuming that by "initial model" you mean the extensional quotient of
closed terms as treated e.g. by Hasegawa [1].

The identity relation in this model is the extensional equality defined by
induction on type structure with the usual logical relation conditions.
This identifies terms that are not beta(-eta) convertible, like the two
successors on the polymorphic church numerals.

Nat = \forall A. (A -> A) -> (A -> A)
S, S' : Nat -> Nat
S = \nfs.f(nfs)
S' = \nfs.nf(fs)

If by "initial model" you mean something intensional like a lambda algebra
with beta conversion, then IEL fails.  We can use the same type as the
counterexample.

T(X) = X -> X
A1 = A2 = Nat
R \subset A1xA2 is =beta
T(R) identifies S and S'

Best,
Andrew

P.S. Regarding the other question raised in the Zulip chat you linked,
whether every type A is isomorphic to "forall X. (A -> X) -> X" in this
extensional model, I believe the answer to be "yes" based on comments by
Thierry Coquand in [2], slide 52.

[1] https://urldefense.com/v3/__https://doi.org/10.1007/3-540-54415-1_61__;!!IBzWLUs!S8aJh356wvmDdcHzuNn_x-n_pAINEuB9OorQS4vpVVPxzpk6f1Frz6bXU2iKOmWU2eJnfiof4UvCQ7Pz7EY2e8iQYns02p8_gTB7LQ$ 
[2] https://urldefense.com/v3/__https://youtu.be/6Ao9zXwyteY?si=5FdC5t5Mnyajmchz&t=4660__;!!IBzWLUs!S8aJh356wvmDdcHzuNn_x-n_pAINEuB9OorQS4vpVVPxzpk6f1Frz6bXU2iKOmWU2eJnfiof4UvCQ7Pz7EY2e8iQYns02p8az4OKSg$ 


On Tue, Aug 12, 2025 at 12:02 PM <types-list-request at lists.seas.upenn.edu>
wrote:

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>    1.  Identity extension for the system F term model (Ryan Wisnesky)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Mon, 11 Aug 2025 21:25:19 -0700
> From: Ryan Wisnesky <wisnesky at gmail.com>
> To: "types-list at lists.seas.upenn.edu"
>         <types-list at LISTS.SEAS.UPENN.EDU>
> Subject: [TYPES] Identity extension for the system F term model
> Message-ID: <CC3F8356-0B05-4A68-9E87-DD181A587DB4 at gmail.com>
> Content-Type: text/plain;       charset=utf-8
>
> Hi All,
>
> I'm hoping the folks on this list can settle a question of folklore;
> myself and Mike Shulman and others have been discussing it on the applied
> category theory zulip channel but have yet to reach a conclusion:
> https://urldefense.com/v3/__https://categorytheory.zulipchat.com/*narrow/channel/229199-learning.3A-questions/topic/Continuations.2C.20parametricity.2C.20and.20polymorphism/with/528479188__;Iw!!IBzWLUs!Uf89Hb7uWjtbz4qUeFbMCVwfxkuye2lpB27Oi4vChhMV6yMPpA57jl2oZnBt3TaOOYa7UhjQTz16dPMUZoGtGubQesXT$
>
> The question is whether the initial (term) model of system F (2nd order
> impredicative polymorphic lambda calculus) satisfies the "identity
> extension lemma", which is one of the primary lemmas characterizing
> (Reynolds) parametric models.  To be clear, this question is about the
> initial (term) model of system of F, its initial model of contexts and
> substitutions.
>
> There are many places that state the system F term model should obey
> identity extension, for example, this remark by Andy Kovacs:
> https://urldefense.com/v3/__https://cs.stackexchange.com/questions/136359/rigorous-proof-that-parametric-polymorphism-implies-naturality-using-parametrici/136373*136373__;Iw!!IBzWLUs!Uf89Hb7uWjtbz4qUeFbMCVwfxkuye2lpB27Oi4vChhMV6yMPpA57jl2oZnBt3TaOOYa7UhjQTz16dPMUZoGtGpQSw3Ii$
> .
>
> However, neither myself nor Mike nor anyone on the zulip chat or anyone
> we?ve asked has been able to find a proof (or disproof).
>
> Anyway, do please let me know if you know of a clear proof or disproof of
> the identity extension lemma for the initial (term) model of system F!
>
> Thanks,
> Ryan Wisnesky
>
> PS Here's a statement from Mike about the exact definition of this
> question:
>
> Let C be the initial model of system F.  Let R(C) be the relational model
> built from C, so its objects are objects of C equipped with a binary
> relation.  There is a strict projection functor R(C) -> C, and since C is
> initial this projection has a section C -> R(C), which is "external
> parametricity".  In a theory like System F that has type variables, a
> "model" includes information about types in a context of type variables, so
> external parametricity sends every type in context to a "relation in
> context".  Do the relations-in-context resulting in this way from types of
> System F always map identity relations to identity relations?
>
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> End of Types-list Digest, Vol 155, Issue 3
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