[TYPES] Identity extension for the system F term model

Thu Aug 28 13:20:58 EDT 2025

What is the definition of a "parametric model"?  And can you give a
reference for the statement that the "extensional" quotient of the
overall-initial model is the initial parametric model?

On Wed, Aug 27, 2025 at 9:20 PM Andrew Polonsky <andrew.polonsky at gmail.com>
wrote:

> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list 
> ]
>
> Put
> Top = \forall X. X -> X
> tt = /\X.\x:X.x : Top
> T(X) = (X -> Top) -> (Top -> Top)
> A1 = A2 = Top
> R \subset A1xA2 is =beta(eta)
> T(R) identifies \x.x, \xy.y, and \xy.tt.
>
> While I agree that the term "initial model" is most often applied to a term
> model with a syntactic notion of equality, in discussions of IEL there's a
> general implicature that you are interested in parametric models. And the
> initial such model is precisely the extensional quotient of the term model,
> for which IEL holds tautologically.  (More precisely, it holds by the
> fundamental theorem of logical relations, the version for open terms that
> is verified in the proof of the theorem.)
>
> Best,
> Andrew
>
>
> On Wed, Aug 27, 2025 at 7:59 PM Ryan Wisnesky <wisnesky at gmail.com> wrote:
>
> > Thanks everyone for the informative discussion!   Andrew’s
> counter-example
> > of T(X) = X->X to the IEL in the initial model is just what was asked
> for.
> >
> > If I may ask a follow-up: does anyone have a counter-example for the IEL
> > in the initial model for a type expression that defines a functor?  Or a
> > proof that IEL always holds for definable functors in the initial model?
> > (T(X) = X->X is well-known not to be a functor.).
> >
> > Thanks again!
> > Ryan
> >
> > > On Aug 14, 2025, at 10:59 PM, Andrew Polonsky <
> andrew.polonsky at gmail.com>
> > wrote:
> > >
> > > [ The Types Forum,
> > http://lists.seas.upenn.edu/mailman/listinfo/types-list   ]
> > >
> > > 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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> > >> Today's Topics:
> > >>
> > >>   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
> > >> ******************************************
> > >>
> >
> >
>