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

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

Excellent observation! So basically the implementation of Lean is incorrect because we shouldn’t be able to derive true = false from the assumption of propositional extensionality if we take account of the type annotations.

The example arose from the question whether we can add propositional extensionality to Lean. That s we define HProp = Sigma P:Type.Pi x,y.P.x=y. Note that the equality we use here is the static Prop valued equality. Now I suggested to add propositional extensionality for HProp as an axiom to Lean but it seemed to lead to the problem.

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. 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 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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