Hi Chun, I run some experiments so to check if it is violating any fundamental rule. So far it went good.
On Tue, Feb 19, 2019 at 5:31 PM Chun Tian (binghe) <binghe.l...@gmail.com> wrote: > Some further updates: > > With my last definition of `extreal_div`, I still have: > > |- !x. x / 0 = ARB > > and > > |- 0 / 0 = ARB > > trivially holds (by definition). This is still not satisfied to me. > > Now I tried the following new definition which looks more reasonable: > > val extreal_div_def = Define > `extreal_div x y = if y = Normal 0 then > (@x. (x = PosInf) \/ (x = NegInf)) > else extreal_mul x (extreal_inv y)`; > > literally, it says anything (well, let's ignore zero) divides zero is > equal to either +Inf or -Inf. But actually the choice of +Inf/-Inf is > irrelevant, as the sole purpose is to prevent any theorem like ``|- x / > 0 = y`` being proved, in which y is a literal extreal. For example, if I > try to prove ``!x. x / 0 = ARB``: > > (* with the new definition, ``x / 0 = ARB`` (or any other extreal) can't > be proved, e.g. > val test_div = prove ( > `!x. extreal_div x (Normal 0) = ARB`, > RW_TAC std_ss [extreal_div_def] > >> Suff `(\f. f = ARB) (@x. (x = PosInf) ∨ (x = NegInf))` > >- RW_TAC std_ss [] > >> MATCH_MP_TAC SELECT_ELIM_THM > >> RW_TAC std_ss [] (* 3 gubgoals *) > NegInf = ARB > > PosInf = ARB > > ∃x. (x = PosInf) ∨ (x = NegInf)); > *) > > at the end I got 3 subgoals like above. I *believe*, no matter what > value I put instead of ARB, at least one of the subgoals must be false. > Thus the theorem should be unprovable. (am I right?) > > Can I adopt this new definition of `extreal_div` in the future? Any > objection or suggestion? > > --Chun > > Il 15/02/19 23:37, Chun Tian (binghe) ha scritto: > > I'm going to use the following definition for `extreal_div`: > > > > (* old definition of `extreal_div`, which allows `0 / 0 = 0` > > val extreal_div_def = Define > > `extreal_div x y = extreal_mul x (extreal_inv y)`; > > > > new definition of `extreal_div`, excluding the case `0 / 0`: *) > > val extreal_div_def = Define > > `extreal_div x y = if (y = Normal 0) then ARB > > else extreal_mul x (extreal_inv y)`; > > > > previously ``|- !x. inv x = 1 / x`` holds, now I have to add `x <> 0` as > > antecedent, which makes perfectly senses. > > > > P.S. the division of extended reals in HOL4 are not used until the > > statement and proof of Radon-Nikodým theorem, then the conditional > > probability. > > > > --Chun > > > > Il 15/02/19 17:39, Mark Adams ha scritto: > >> I think there is definitely an advantage in keeping ``x/y`` undefined > >> (even granted that it will always be possible to prove ``?y. x/0 = y``), > >> namely that it means that your proofs are much more likely to directly > >> translate to other formalisms of real numbers where '/' is not total. > >> > >> Of course there is also a disadvantage, in that it makes proof harder. > >> But then, arguably, being forced to justify that we are staying within > >> the "normal" domain of the function is probably a good thing (in a > >> similar way as being forced to conform to a type system is a good > >> thing). I understand that, historically, it is this disadvantage of > >> harder proofs that was the reason for making ``0/0=0`` in HOL. It's > >> much easier for automated routines if they don't have to consider side > >> conditions. > >> > >> Mark. > >> > >> On 15/02/2019 08:56, Chun Tian (binghe) wrote: > >>> Thanks to all kindly answers. > >>> > >>> Even I wanted ``0 / 0 = 0`` to be excluded from at least > >>> "extreal_div_def" (in extrealTheory), I found no way to do that. All I > >>> can do for now is to prevent ``0 / 0`` in all my proofs - whenever it's > >>> going to happen, I do case analysis instead. > >>> > >>> --Chun > >>> > >>> Il 14/02/19 18:12, Konrad Slind ha scritto: > >>>> It's a deliberate choice. See the discussion in Section 1.2 of > >>>> > >>>> > http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=775DBF504F7EE4EE28CC5169488F4190?doi=10.1.1.56.4692&rep=rep1&type=pdf > >>>> > >>>> > >>>> > >>>> > >>>> On Thu, Feb 14, 2019 at 10:40 AM Chun Tian (binghe) > >>>> <binghe.l...@gmail.com <mailto:binghe.l...@gmail.com>> wrote: > >>>> > >>>> Hi, > >>>> > >>>> in HOL's realTheory, division is defined by multiplication: > >>>> > >>>> [real_div] Definition > >>>> > >>>> ⊢ ∀x y. x / y = x * y⁻¹ > >>>> > >>>> and zero multiplies any other real number equals to zero, which is > >>>> definitely true: > >>>> > >>>> [REAL_MUL_LZERO] Theorem > >>>> > >>>> ⊢ ∀x. 0 * x = 0 > >>>> > >>>> However, above two theorems together gives ``0 / 0 = 0``, as shown > >>>> below: > >>>> > >>>> > REWRITE_RULE [REAL_MUL_LZERO] (Q.SPECL [`0`, `0`] real_div); > >>>> val it = ⊢ 0 / 0 = 0: thm > >>>> > >>>> How do I understand this result? Is there anything "wrong"? > >>>> > >>>> (The same problems happens also in extrealTheory, since the > >>>> definition > >>>> of `*` finally reduces to `*` of real numbers) > >>>> > >>>> Regards, > >>>> > >>>> Chun Tian > >>>> > >>>> _______________________________________________ > >>>> hol-info mailing list > >>>> hol-info@lists.sourceforge.net > >>>> <mailto:hol-info@lists.sourceforge.net> > >>>> https://lists.sourceforge.net/lists/listinfo/hol-info > >>>> > >>> > >>> > >>> _______________________________________________ > >>> hol-info mailing list > >>> hol-info@lists.sourceforge.net > >>> https://lists.sourceforge.net/lists/listinfo/hol-info > > > > _______________________________________________ > hol-info mailing list > hol-info@lists.sourceforge.net > https://lists.sourceforge.net/lists/listinfo/hol-info > -- Waqar Ahmad, Ph.D. Post Doc at Hardware Verification Group (HVG) Department of Electrical and Computer Engineering Concordia University, QC, Canada Web: http://save.seecs.nust.edu.pk/waqar-ahmad/
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