Dear Michael For x/0, the essential problem is on its definition. I think the division by zero is trivial and clear all. However, we will need a new axiom. So, I would like to ask for your kind help for the axiom problem; Foundation of Mathematics.
Please look the draft: *viXra:1908.0100 <http://vixra.org/abs/1908.0100>* *submitted on 2019-08-06 20:03:01*, Fundamental of Mathematics; Division by Zero Calculus and a New Axiom With best regards, Sincerely yours, Saburou Saitoh 2019.8.10.9:25 2019年8月10日(土) 9:07 Norrish, Michael (Data61, Acton) < michael.norr...@data61.csiro.au>: > It’s still defined inasmuch as it is perfectly legitimate to write x/0 and > use that term to define other things in turn. For example, I can define > foo = x/0 + 1 and then quite successfully prove that x/0 < foo. > > I would avoid the use of the word undefined in this context; rather x/0 > has an unspecified value. All functions are total so all applications of > functions to all possible arguments have values. > > Michael > > > On 9 Aug 2019, at 21:46, Chun Tian (binghe) <binghe.l...@gmail.com> > wrote: > > > > A follow-up of this old topic: > > > > Finally I found the following definitions of `extreal_inv` and > `extreal_div` based on new_specification(): > > > > local > > val lemma = Q.prove ( > > `?i. (i NegInf = Normal 0) /\ > > (i PosInf = Normal 0) /\ > > (!r. r <> 0 ==> (i (Normal r) = Normal (inv r)))`, > > (* proof *) > > Q.EXISTS_TAC `\x. if (x = PosInf) \/ (x = NegInf) then Normal 0 > > else if x = Normal 0 then ARB > > else Normal (inv (real x))` \\ > > RW_TAC std_ss [extreal_not_infty, real_normal]); > > in > > (* |- extreal_inv NegInf = Normal 0 /\ > > extreal_inv PosInf = Normal 0 /\ > > !r. r <> 0 ==> extreal_inv (Normal r) = Normal (inv r) > > *) > > val extreal_inv_def = new_specification > > ("extreal_inv_def", ["extreal_inv"], lemma); > > end; > > > > local > > val lemma = Q.prove ( > > `?d. (!r. d (Normal r) PosInf = Normal 0) /\ > > (!r. d (Normal r) NegInf = Normal 0) /\ > > (!x r. r <> 0 ==> (d x (Normal r) = extreal_mul x (extreal_inv > (Normal r))))`, > > (* proof *) > > Q.EXISTS_TAC `\x y. > > if ((y = PosInf) \/ (y = NegInf)) /\ (?r. x = Normal r) then > Normal 0 > > else if y = Normal 0 then ARB > > else extreal_mul x (extreal_inv y)` \\ > > RW_TAC std_ss [extreal_not_infty, real_normal]); > > in > > (* |- (!r. extreal_div (Normal r) PosInf = Normal 0) /\ > > (!r. extreal_div (Normal r) NegInf = Normal 0) /\ > > !x r. r <> 0 ==> extreal_div x (Normal r) = x * extreal_inv > (Normal r) > > *) > > val extreal_div_def = new_specification > > ("extreal_div_def", ["extreal_div"], lemma); > > end; > > > > In this way, things like `extreal_inv 0` and `extreal_div x 0` are > *really* undefined. > > > > --Chun > > > > Il 20/02/19 06:48, michael.norr...@data61.csiro.au ha scritto: > >> Your right hand side is no better than ARB really. You say that your > aim is to avoid x/0 = y, with y a literal extreal. But if you believe ARB > is a literal extreal, then I will define > >> > >> val pni_def = Define`pni = @x. (x = PosInf) \/ (x = NegInf)`; > >> > >> and then I can certainly prove that x/0 = pni. If ARB is a literal > extreal, surely pni is too. > >> > >> (Recall that ARB's definition is `ARB = @x. T`.) > >> > >> Michael > >> > >> > >> On 20/2/19, 09:31, "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 > >> > > > > _______________________________________________ > hol-info mailing list > hol-info@lists.sourceforge.net > https://lists.sourceforge.net/lists/listinfo/hol-info >
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