Re: [isabelle] 1/0 = 0?





There is a spectrum of possibilities here.  

At one end you have the attitude that takes OpenTheory to leave n - m undefined on the natural numbers when m > n (and pre 0 also undefined). At the other, you can try to give "reasonable" values to everything.

Note that if you want to have things be ‘properly’ unspecified, you would definitely take René’s second approach: that consequence (5/0 = 3/0) is horrid. 

As with all definitions, the question is what consequences you want to be true.

Michael


On 27/2/18, 22:32, "cl-isabelle-users-bounces at lists.cam.ac.uk on behalf of Thiemann, Rene" <cl-isabelle-users-bounces at lists.cam.ac.uk on behalf of Rene.Thiemann at uibk.ac.at> wrote:

    Dear Manfred,
    
    I did not look in the history on why it was defined like this but just make
    some comments.
    
    As you already said, Isabelle is a logic of total function, so you have to totally specify
    division. Therefore, there also is a special constant “undefined” of each type
    which stands for some arbitrary fixed value of that type. So, “undefined :: real” 
    is a real number, but we don’t know whether it is 0, 1, pi, e, or ….
    
    So in principle, you could define division as
      “x / y = (if y = 0 then undefined else THE-unique z such that y * z = x)”
    
    As a consequence, several nice equalities now get preconditions.
    For instance, the current lemma times_divide_eq_left
    
    "b / c * a = b * a / c” 
    
    with the above definition will only be provable if c is not 0.
    
    This will definitely make formal proofs more verbose, because one now
    has to keep track that all divisions are not by 0. 
    
    And on the contrary, even if you define “x / y” as above, then you still will be able to prove
    that “5 / 0 = 2 / 0” since both sides simplify to the same constant “undefined”.
    
    So even “undefined” behaviour is fully specified and can be exploited for proofs.
    (though you can also define “x / y = (if y = 0 then undefined x else …)” where
     then “5 / 0 = 2 / 0” simplifies to “undefined 5 = undefined 2” which as far as
     I know impossible to proof or to disprove; here, “undefined” is a fixed function
     of type real to real.).
    
    I hope this clarifies the situation a bit,
    René
    
    
    
    > Am 27.02.2018 um 12:02 schrieb Manfred Kerber <mnfrd.krbr at gmail.com>:
    > 
    > Hi,
    > 
    > I was recently confused about expressions such as value "(1::nat)/0"
    > for which I get the result 0 :: real.
    > 
    > I understand that the logic used in Isabelle/HOL is total. However, I
    > thought that usually partiality is approximated by leaving the value
    > of expressions such as 1/0 as unspecified. What is the reason that
    > this seems not to be the case for 1/0?
    > 
    > Kind regards
    > Manfred
    > 
    
    





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