Re: [HoTT] Impredicative set + function extensionality + proof irrelevance consistent?

[Michael Shulman <shu...@sandiego.edu>]

On Sat, Dec 16, 2017 at 7:21 AM, Thorsten Altenkirch
<Thorsten....@nottingham.ac.uk> wrote:
> Not really: you can prove ³PropExt -> False² in the current system and you
> shouldn¹t be able to do this.

Ah, I see.  I didn't realize that PropExt was something you could
hypothesize inside of Lean; I thought you were proposing it as a
modification to the underlying type theory.  In that case, yes, I
agree, the implementation is incorrect.  (Are any Lean developers
listening?)

> By definitional proof-irrelevance I mean that we have a ³static² universe
> of propositions and the principle that any tow proofs of propositions are
> definitionally equal. That is what I suggested in my LICS 99 paper.
> However, it seems (following your comments) that we can¹t prove ³PropExt
> -> False² in this system.
>
> One could argue that Lean¹s type theory is defined by its implementation
> but then it may be hard to say anything about it, including wether it is
> consistent.
>
>     > I still wonder what exactly is the difference between a static
> )(efnitionally proof-irrelvant) Prop which seems to correspond to Omega in
> a topos and set-level HoTT (i.e. using HProp). Hprop is also a subobject
> classifier (with some predicativity proviso) but the HoTT view gives you
> some extra power.
>
>     A prime example of that "extra power" is that with HProp you can prove
>     function comprehension (unique choice).  This goes along with a
>     reduction in the class of models: I believe that a static Prop can
>     also be modeled by the strong-subobject classifier in a quasitopos, in
>     which case unique choice is false.
>
> Ok, so you are saying that a static Prop only gives rise to a quasitopos
> which fits with the observation that we don't get unique choice in this
> case. On the other hand set level HoTT gives rise to a topos?
>
> Thorsten
>
>     > Ok, once we also allow QITs we know that this goes beyond the usual
> topos logic (cf. the example in your paper with Peter).
>     >
>     > Thorsten
>     >
>     >
>     > On 12/12/2017, 23:14, "homotopyt...@googlegroups.com on behalf
> of Michael Shulman" <homotopyt...@googlegroups.com on behalf of
> shu...@sandiego.edu> wrote:
>     >
>     >     This is really interesting.  It's true that all toposes satisfy
> both
>     >     unique choice and proof irrelevance.  I agree that one
> interpretation
>     >     is that definitional proof-irrelevance is incompatible with the
>     >     HoTT-style *definition* of propositions as (-1)-truncated types,
> so
>     >     that you can *prove* something is a proposition, rather than
> having
>     >     "being a proposition" being only a judgment.  But could we
> instead
>     >     blame it on the unjustified omission of type annotations?
> Morally, a
>     >     pairing constructor
>     >
>     >     (-,-) : (a:A) -> B(a) -> Sum(x:A) B(x)
>     >
>     >     ought really to be annotated with the types it acts on:
>     >
>     >     (-,-)^{(a:A). B(a)} : (a:A) -> B(a) -> Sum(x:A) B(x)
>     >
>     >     and likewise the projection
>     >
>     >     first : (Sum(x:A) B(x)) -> A
>     >
>     >     should really be
>     >
>     >     first^{(a:A). B(a)} : (Sum(x:A) B(x)) -> A.
>     >
>     >     If we put these annotations in, then your "x" is
>     >
>     >     (true,refl)^{(b:Bool). true=b}
>     >
>     >     and your two apparently contradictory terms are
>     >
>     >     first^{(b:Bool). true=b} x == true
>     >
>     >     and
>     >
>     >     second^{(b:Bool). false=b} x : first^{(b:Bool). false=b} x =
> false
>     >
>     >     And we don't have "first^{(b:Bool). false=b} x == true", because
>     >     beta-reduction requires the type annotations on the projection
> and the
>     >     pairing to match.  So it's not really the same "first x" that's
> equal
>     >     to true as the one that's equal to false.
>     >
>     >     In many type theories, we can omit these annotations on pairing
> and
>     >     projection constructors because they are uniquely inferrable.
> But if
>     >     we end up in a type theory where they are not uniquely
> inferrable, we
>     >     are no longer justified in omitting them.
>     >
>     >
>     >     On Tue, Dec 12, 2017 at 4:21 AM, Thorsten Altenkirch
>     >     <Thorsten....@nottingham.ac.uk> wrote:
>     >     > Good point.
>     >     >
>     >     > OK, in a topos you have a static universe of propositions.
> That is wether something is a proposition doesn¹t depend on other
> assumptions you make.
>     >     >
>     >     > In set-level HoTT we define Prop as the types which have at
> most one inhabitant. Now wether a type is a proposition may depend on
> other assumptions. (-1)-univalence i.e. propositional extensionality turns
> Prop into a subobject classifier (if you have resizing otherwise you get
> some sort of predicative topos).
>     >     >
>     >     > However, the dynamic interpretation of propositions gives you
> some additional power, in particular you can proof unique choice, because
> if you can prove Ex! x:A.P x , where Ex! x:A.P x is defined as Sigma x:A.P
> x /\ Pi y:A.P y -> x=y then this is a proposition even though A may not
> be. However using projections you also get Sigma x:A.P x.
>     >     >
>     >     > Hence I guess I should have said a topos with unique choice (I
> am not sure wether this is enough). Btw, set-level HoTT also gives you
> QITs which eliminate many uses of choice (e.g. the definition of the
> Cauchy Reals and the partiality monad).
>     >     >
>     >     > Thorsten
>     >     >
>     >     >
>     >     >
>     >     >
>     >     >
>     >     >
>     >     > On 12/12/2017, 12:02, "Thomas Streicher"
> <stre...@mathematik.tu-darmstadt.de> wrote:
>     >     >
>     >     >>But very topos is a model of extensional type theory when
> taking Prop
>     >     >>= Omega. All elements of Prop are proof irrelevant and
> equivalent
>     >     >>propositions are equal.
>     >     >>
>     >     >>Since it is a model of extensional TT there is no difference
> between
>     >     >>propsoitional and judgemental equality.
>     >     >>
>     >     >>Thomas
>     >     >>
>     >     >>
>     >     >>> If you have proof-irrelevance in the strong definitional
> sense then you cannot be in a topos. This came up recently in the context
> of Lean which is a type-theory based interactive proof system developed at
> microsoft and which does implement proof-irrelvance. Note that any topos
> has extProp:
>     >     >>>
>     >     >>> Given a:A define Single(a) = Sigma x:A.a=x. We have
> Single(a) : Prop and
>     >     >>>
>     >     >>> p : Single(true) <-> Single(false)
>     >     >>>
>     >     >>> since both are inhabited. Hence by extProp
>     >     >>>
>     >     >>> extProp p : Single(true) = Single(false)
>     >     >>>
>     >     >>> now we can use transport on (true,refl) : Single(true) to
> obtain
>     >     >>>
>     >     >>> x = (extProp p)*(true,refl) : Single(false)
>     >     >>>
>     >     >>> and we can show that
>     >     >>>
>     >     >>> second x : first x = false
>     >     >>>
>     >     >>> but since Lean computationally ignores (extProp p)* we also
> get (definitionally):
>     >     >>>
>     >     >>> first x == true
>     >     >>>
>     >     >>> My conclusion is that strong proof-irrelvance is a bad idea
> (note that my ???99 paper on Extensionality in Intensional Type Theory
> used exactly this). It is more important that our core theory is
> extensional and something pragmatically close to definitional
> proof-irrelevance can be realised as some tactic based sugar. It has no
> role in a foundational calculus.
>     >     >>>
>     >     >>>
>     >     >>> Thorsten
>     >     >>>
>     >     >>>
>     >     >>>
>     >     >>>
>     >     >>> On 12/12/2017, 10:15, "Andrea Vezzosi" <sanz...@gmail.com>
> wrote:
>     >     >>>
>     >     >>> >On Mon, Dec 11, 2017 at 3:23 PM, Thorsten Altenkirch
>     >     >>> ><Thorsten....@nottingham.ac.uk> wrote:
>     >     >>> >> Hi Kristina,
>     >     >>> >>
>     >     >>> >> I guess you are not assuming Prop:Set because that would
> be System U and hence inconsistent.
>     >     >>> >>
>     >     >>> >> By proof-irrelevance I assume that you mean that any two
> inhabitants of a proposition are definitionally equal. This assumption is
> inconsistent with it being a tops since in any Topos you get propositional
> extensionality, that is P,Q : Prop, (P <-> Q) <-> (P = Q), which is indeed
> an instance of univalence.
>     >     >>> >>
>     >     >>> >
>     >     >>> >I don't know if it's relevant to the current discussion,
> but I thought
>     >     >>> >the topos of sets with Prop taken to be the booleans would
> support
>     >     >>> >both proof irrelevance and propositional extensionality,
> classically
>     >     >>> >at least. Is there some extra assumption I am missing here?
>     >     >>> >
>     >     >>> >
>     >     >>> >> It should be possible to use a realizability semantics
> like omega-sets or Lambda-sets to model the impredicative theory and
> identify the propositions with PERs that are just subsets.
>     >     >>> >>
>     >     >>> >> Cheers,
>     >     >>> >> Thorsten
>     >     >>> >>
>     >     >>> >>
>     >     >>> >> On 11/12/2017, 04:22,
> "homotopyt...@googlegroups.com on behalf of Kristina Sojakova"
> <homotopyt...@googlegroups.com on behalf of
> sojakova...@gmail.com> wrote:
>     >     >>> >>
>     >     >>> >>     Dear all,
>     >     >>> >>
>     >     >>> >>     I asked this question last year on the coq-club
> mailing list but did not
>     >     >>> >>     receive a conclusive answer so I am trying here now.
> Is the theory with
>     >     >>> >>     a proof-relevant impredicative universe Set,
> proof-irrelevant
>     >     >>> >>     impredicative universe Prop, and function
> extensionality (known to be)
>     >     >>> >>     consistent? It is known that the proof-irrelevance of
> Prop makes the Id
>     >     >>> >>     type behave differently usual and in particular,
> makes the theory
>     >     >>> >>     incompatible with univalence, so it is not just a
> matter of tacking on
>     >     >>> >>     an interpretation for Prop.
>     >     >>> >>
>     >     >>> >>     Thanks in advance for any insight,
>     >     >>> >>
>     >     >>> >>     Kristina
>     >     >>> >>
>     >     >>> >>
>     >     >>> >>
>     >     >>> >>
>     >     >>> >>
>     >     >>> >>
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