Re: [isabelle] a new power operator
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
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
Am Mittwoch, den 16.09.2015, 20:29 +0300 schrieb noam neer:
> 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
> 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.
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