Re: [mizar] Re: proof objects for mizar: already available?

[Josef Urban <josef.urban@gmail.com>]

On 10/9/11, Jesse Alama <jesse.alama@gmail.com> wrote:
> On 2011-10-01 09:26:45 +0000, Josef Urban said:
>
>> On 10/1/11, Jesse Alama <jesse.alama@gmail.com> wrote:
>>
>>> (b) Proofs objects are already available, in some sense, for mizar
>>> proofs.  They do not give precisely what is wanted from the
>>> concept of proof object, though:
>>>
>>> * They are not always available.  With the current transformation
>>> into a vanilla first-order format, and with current ATPs, for
>>> less than half of mizar theorems from the library do we have
>>> deductions.  This is clearly an important result for the
>>> community, but not having deductions is a serious shortcoming.
>>> We want proof objects for all mizar proofs.
>>
>> The "ATP proof object completion" used for ATP cross-verification of
>> Mizar in http://www.springerlink.com/content/b7383x43l55p1183/ does
>> not rely on re-proving whole Mizar theorems.
>>
>> It is based on re-proving the Mizar "by" and "from" steps, where the
>> ATP success rate is (so far) over 99% , and on ATP verification of the
>> Mizar structural natural deduction steps, where the success rate is
>> (so far) 100%.
>
> Nonetheless, the situation is not entirely satisfactory.  What we have
> is a kind of hybrid proof: part is in the mizar proof formalism, and
> the other part is in the proof formalism of whatever ATP is used.  From
> a proof-theoretic perspective from a mizar proof we find a natural
> deduction proof with lots of axioms (corresponding to its "by" steps).

The whole (completed) proofs are in one formalism: TPTP format with
assumptions (see the paper).

>> I totally agree that having detailed proof objects for all of MML
>> would be a great resource for study and data-mining of mathematical
>> proofs. The ATP path to this turned out to be very cheap once I had
>> practically complete ATP export and strong ATP methods.
>>
>> Doing it the hard way inside Mizar is certainly possible (I have
>> partially done it in 2000), but it will be a serious amount of work on
>> Mizar, and even if you do it, it will be in a constant danger of
>> becoming obsolete by later re-implementations of parts of Mizar. Bill
>> McCune told me in 2004 that the detailed Otter proof objects were so
>> much added code that it lead him to reimplement the whole thing as
>> Prover9. My experience from 2000 with the Mizar kernel was similar: it
>> was a large blow-up of the code, and a cleaner complete rewrite would
>> be needed.
>
> I agree that it probably would be a rather complex project to have bona
> fide proof objects for mizar.  In addition to the problem of exposing
> "by"/"from" steps, there remains the problem of exporting a mizar proof
> -- even taking for granted that its "by" and "from" steps can be filled
> in some acceptable way, treating them as black boxes (or as axioms, in
> a natural deduction/sequent calculus context) -- into a vanilla natural
> deduction format.  I think there are lots of design problems here.  For
> example, consider the text fragment:
>
> set X = the set;
> theorem X = X;
>
> Do we translate this as the universal formula:
>
> for X : set (X = the set --> X = X)
>
> Or as the equation:
>
> the set = the set

There are indeed various choice points when translating from Mizar to
pure first-order logic, but their default solutions have been provided
in MPTP since 2005. See
http://www.springerlink.com/content/kw9075426q26pu32/ . "set foo =
bar" is just a syntactic macro in Mizar, so already Mizar semantic
layer is "the set = the set", there is not really much of a choice
here. The choice comes when translating "the" (Hilbert's epsilon) to
first-order logic. This is not described in the paper above, because
"the" came later in Mizar (2009?). But its default translation is very
similar to translation of Faenkel terms (de-anonymization).

To sum up: there is a default TPTP translation both of the Mizar
natural deduction proof format and of the Mizar extensions over
first-order logic, and both have been around and productively used for
a couple of years now.

But I'd be happy to see a continuation of the 2008 cross-verification
paper that would attempt ATP-based cross-verification of the full MML
(not just large random parts of it). Based on that (on the full proof
objects), it would be probably fairly easy to import MML into HOL
Light, Isabelle, Coq, ..., and that could in turn trigger further work
on communication between the libraries of these systems.

Josef



>
> (Put aside for the moment the problem of translating "the set".)  It
> won't do to say that these are "the same thing".  They're equivalent,
> but that requires proof.  The latter is generally taken as an axiom
> when reasoning with equality in natural deduction or sequent calculus;
> the latter is not (though perhaps it could be -- we might need to
> explore extensions of various calculi when deciding how proof objects
> should be represented).  The universal requires an application of
> universal introduction (if we are working in a natural deduction
> setting), implication introduction, and an axiom of equality.
>
> --
> Jesse Alama
> http://centria.di.fct.unl.pt/~alama/
>
>
>