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

[Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>]

Good point.

OK, in a topos you have a static universe of propositions. That is wether something is a proposition doesn’t 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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