Just to clear some things up in this conversation:

- The Lean kernel is consistent with any form of propositional
extensionality, as far as I know.
- The kernel for the current version of Lean, Lean 3, is not designed for
HoTT and inconsistent with univalence. However, it does *not* ignore any
transport of an equality. The bottom universe in Lean is the universe of
propositions, Prop (not to be confused with hProp). What does hold is that
there is definitional proof irrelevance for proofs of a proposition, i.e.
if P : Prop and p, q : P then p is definitionally equal to q. This means
that if we have a proof p : A = A for some type A, then transporting along
p is the identity function (because p = refl, definitionally). However, if
p : A = B, then transporting along p is not the identity function (which
would be type incorrect to state).
- Lean has an evaluator in a virtual machine, which is used to evaluate
programs and tactics efficiently outside the kernel. This evaluator is not
trusted, and cannot be used when writing any proof in Lean. This evaluator
removes all type information and propositions for efficiency, and can
evaluate expressions to the wrong answer when axioms are added to the type
theory (even if the axioms are consistent). For example, if we add the
following lines to Ben's example
```
constant H : HProp_ext
#eval Sig.fst (x H)
```
we see that the evaluator incorrectly evaluates this expression to tt
(true). Needless to say, the evaluator was not designed for this, but
instead to evaluate tactics like the following quickly.
```
if n < 1000000 then apply tactic1 else apply tactic2
```

I hope this clears up any confusion.

Best,
Floris


On 17 December 2017 at 18:16, Thorsten Altenkirch <
Thorsten....@nottingham.ac.uk> wrote:

> Thank you, Ben! That is good news actually.
>
> Thorsten
>
> From: Ben Sherman <she...@csail.mit.edu>
> Date: Sunday, 17 December 2017 at 18:08
> To: Michael Shulman <shu...@sandiego.edu>
> Cc: Thorsten Altenkirch <psz...@exmail.nottingham.ac.uk>, Homotopy Type
> Theory <homotopyt...@googlegroups.com>
> Subject: Re: [HoTT] Impredicative set + function extensionality + proof
> irrelevance consistent?
>
> I don’t think that HProp extensionality is false in Lean (note that
> regular Prop extensionality is an axiom that is taken to hold in Lean), or
> at the least, I don’t think Thorsten’s argument goes through. Here is Lean
> code that formalizes the argument:
>
> structure HProp : Type (u + 1) :=
>   (car : Type u)
>   (subsingleton : ∀ x y : car, x = y)
>
> structure Sig {A : Type u} (P : A → Prop) : Type u :=
>   (fst : A)
>   (snd : P fst)
>
> def Single {A : Type u} (a : A) : HProp :=
>   ⟨ Sig (λ x : A, x = a)
>   , begin
>     intros, cases x, cases y,
>     subst snd, subst snd_1,
>     end
>   ⟩
>
> structure iffT (A B : Type u) :=
>   (left : A → B)
>   (right : B → A)
>
> def HProp_ext : Prop :=
>   ∀ (P Q : HProp.{u}), (iffT (HProp.car P) (HProp.car Q)) → P = Q
>
> def true_HProp : HProp.{u} := ⟨ punit ,
>   begin intros, cases x, cases y, reflexivity end  ⟩
>
> lemma Single_inh {A : Type u} (a : A) : HProp_ext.{u} → Single
> a = true_HProp :=
> begin
> intros H,
> apply H, constructor; intros,
> constructor, constructor, reflexivity,
> end
>
> lemma Single_bool (H : HProp_ext.{0}) : Single tt = Single ff :=
> begin
> rw Single_inh, rw Single_inh, assumption, assumption,
> end
>
> def x (H : HProp_ext.{0}) :
>   HProp.car (Single ff) :=
>   eq.rec_on (Single_bool H) ⟨ tt, rfl ⟩
>
> lemma snd_x (H : HProp_ext.{0}) : Sig.fst (x H) = ff := Sig.snd (x H)
>
> lemma fst_x (H : HProp_ext.{0}) : Sig.fst (x H) = tt := begin
> dsimp [x], admit,
> end
>
> The proof state at the end of the proof for `fst_x` looks like this:
>
> ⊢ (eq.rec {fst := tt, snd := _} _).fst = tt
>
> and `reflexivity` fails to solve the goal, so I think the `eq.rec` on the
> left-hand side fails to reduce. Note that the equality proof that we
> transport over is a proof that `Single tt = Single ff`; the two sides of
> this equation are not definitionally equal, which I think explains why
> `eq.rec` cannot reduce.
>
> On Dec 17, 2017, at 7:55 AM, Michael Shulman <shu...@sandiego.edu> wrote:
>
> 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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