*To*: Johannes HÃlzl <hoelzl at in.tum.de>*Subject*: Re: [isabelle] a new power operator*From*: noam neer <noamneer at gmail.com>*Date*: Fri, 18 Sep 2015 10:00:58 +0300*Cc*: "cl-isabelle-users at lists.cam.ac.uk" <cl-isabelle-users at lists.cam.ac.uk>*In-reply-to*: <1442558980.21178.27.camel@in.tum.de>*References*: <CAGOKs0955JAEf75HD4oBTn0L+4nwvLVS1_O-qyn_qw7RXCvhog@mail.gmail.com> <1442558980.21178.27.camel@in.tum.de>

hi, as far as I understand now, (THE x. False) is treated by the system as some real number, fixed but unknown. u can prove things about it only if they hold for all reals. this is quite satisfactory for me. also, I think the behavior of 'powr' for negative base is quite similar- it is defined using the logarithm of the base which is "undefined" in a very similar way. this is different in Coq, where the real power behaves like u said and for negative base returns 1. thanx, noam On Fri, Sep 18, 2015 at 9:49 AM, Johannes HÃlzl <hoelzl at in.tum.de> wrote: > Hi Noam, > > for 2. you can not enforce anything, if you chose (THE x. False) it may > be also = 0 or = 1, you just can not prove it. This makes only sense if > you want to have the option to change the value later on. In a total > logic like HOL we usually fix it to a default value, like 0 (or in this > case 1). > > I think a simpler definition of mrpow would be: > x ** y = (if y : Nats then x ^ nat (floor y) > else if - y : Nats then (1 / x) ^ nat (floor (-y)) > else x powr y) > > Then you can prove: > 0 < x ==> x ** y = x powr y > x ** real n = x ^ n > x ** - real n = (1 / x) ^ n > > Of course you get x ** 0 = 1, but just imagine that > (THE x. False) = 1 ;-) > > Another question is: > What are your application for this operator? > > Maybe an extension of ^ to integers would be enough? > > ipow :: 'a::field => int => 'a > ipow x 0 = 1 > ipow x (- n) = (1 / x) ^ n > ipow x (+ n) = x ^ n > > - Johannes > > > > Am Mittwoch, den 16.09.2015, 20:29 +0300 schrieb noam neer: > > hi, > > > > I'm looking for a power operator for real numbers that is closer to the > > mathematical conventions of real analysis than either ^ or powr. if we > > denote it by **, my requirements from it are > > > > 1. its type is "real => real => real". > > 2. 0**y is undefined for y<=0. > > (formally it can be (THE x. False). note that the definition of > '0^0=1' > > is appropriate in algebra, combinatorics and set theory, but not in > > analysis.) > > 3. for negative base and real integer exponent (that is, reals in the set > > Ints) the result should be the expected one. > > (powr, which is defined using the logarithm of the base, doesn't > > satisfy this.) > > > > I'd like to know if anybody have already developed such an operator. If > not > > I guess I'll try it myself. a possible definition is > > > > definition mrpow :: "real â real â real" > > (infixr "**" 80) > > where "x ** y == if x>0 > > then (x powr y) > > else (if x=0 > > then (if y>0 > > then 0 > > else (THE z::real. False)) > > else (if y â Ints > > then (if y â 0 > > then x ^(nat( floor y)) > > else (1/x)^(nat(- floor y))) > > else (THE z::real. False) > > ) > > )" > > > > and if necessary I'll work on proving its properties until it is easy to > > use. again any comments or suggestions are welcome. > > >

**Follow-Ups**:**Re: [isabelle] a new power operator***From:*Johannes Hölzl

**References**:**[isabelle] a new power operator***From:*noam neer

**Re: [isabelle] a new power operator***From:*Johannes Hölzl

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