## Stream: maths

### Topic: real powers

#### Chris Hughes (Nov 04 2018 at 15:27):

If we did has_pow real real, what would (-1)^(1/3) be?

nonono

#### Kevin Buzzard (Nov 04 2018 at 15:28):

real^real is surely not a good idea

#### Kevin Buzzard (Nov 04 2018 at 15:28):

what would (-1)^(1/4) be?

#### Kevin Buzzard (Nov 04 2018 at 15:29):

If you want real^real then I would suggest setting $a^b=0$ if $a\leq 0$

#### Kevin Buzzard (Nov 04 2018 at 15:29):

You can make sense of $a^b$ for $a$ a positive real and $b$ any complex number, it should be $\exp(b\log(a))$

#### Johan Commelin (Nov 04 2018 at 15:30):

If you want real^real then I would suggest setting $a^b=0$ if $a\leq 0$

I am quite sure you made a typo. That should have been $a^b = 37$.

#### Kevin Buzzard (Nov 04 2018 at 15:30):

you can make sense of $a^b$ for $a$ any non-zero complex number and $b$ any integer

#### Kevin Buzzard (Nov 04 2018 at 15:30):

I would have put 37 but maybe $0^{37}$ should be 0...

#### Johannes Hölzl (Nov 04 2018 at 16:19):

@Chris Hughes the usual thing: we totalize it. I think for log we can force it to by symmetric. either a long the $y$-axis, or through the origin. For power on $\mathbb{R}$ you might want to have $a^b = |a|^b$ or similar, and of course $0^a = 0$.

#### Chris Hughes (Nov 04 2018 at 16:21):

Don't we want $0^0 = 1$?

#### Kevin Buzzard (Nov 04 2018 at 16:22):

I think you want 0^0 = undefined if both those 0's are real numbers

#### Johannes Hölzl (Nov 04 2018 at 16:22):

I don't think there is a natural choice to totalize it. Theorems about log and pow will always assume that the argument is non-negative.

#### Kevin Buzzard (Nov 04 2018 at 16:22):

and undefined := 0 in Lean

#### Johannes Hölzl (Nov 04 2018 at 16:22):

Undefined means we are free to choose a value

#### Johannes Hölzl (Nov 04 2018 at 16:22):

and maybe for pow, $0^0 = 1$ makes more sense. I don't know.

#### Kevin Buzzard (Nov 04 2018 at 16:22):

There's certainly a case for (0 : real) ^ (0 : nat) = 1

#### Johannes Hölzl (Nov 04 2018 at 16:23):

Yes: $a^{n : \mathbb{R}}= a^n$ should hold

#### Johannes Hölzl (Nov 04 2018 at 16:24):

so 'real'-power should extend 'int'-power should extend 'nat'-power

#### Chris Hughes (Nov 04 2018 at 16:24):

But that doesn't work if $a^n = \mid a\mid^n$

#### Mario Carneiro (Nov 04 2018 at 16:28):

0^a = 0 unless a = 0, otherwise 1

#### Johannes Hölzl (Nov 04 2018 at 16:28):

right it doesn't extend int-power. And I don't see a sensible default where it would. we could check if floor or ceil of the argument is odd/even, but this feels too forced

#### Mario Carneiro (Nov 04 2018 at 16:28):

I have this in the metamath definition, with the if and everything

#### Mario Carneiro (Nov 04 2018 at 16:29):

for negative powers I used complexes, because there was only one definition complex ^ complex, of course that's not an option here

#### Chris Hughes (Nov 04 2018 at 16:29):

In metamath is $(-1)^{1/3} = -1$

#### Mario Carneiro (Nov 04 2018 at 16:29):

no, it's e^2pi i/3

#### Kevin Buzzard (Nov 04 2018 at 16:30):

that ain't no real

#### Mario Carneiro (Nov 04 2018 at 16:30):

like I said, complex

#### Mario Carneiro (Nov 04 2018 at 16:30):

I don't think we should try for the rational extension, it's crazy and not at all complete

#### Johannes Hölzl (Nov 04 2018 at 16:30):

okay, what about using a^b = Re (a ^ b)?

#### Kevin Buzzard (Nov 04 2018 at 16:30):

You want to define the Riemann Zeta function for complex s with Re(s)>1 as the infinite sum of n^{-s}

#### Johannes Hölzl (Nov 04 2018 at 16:31):

where the left one is on reals and right of the equality is the complex power?

#### Mario Carneiro (Nov 04 2018 at 16:31):

that could work...

#### Johannes Hölzl (Nov 04 2018 at 16:31):

AFAIU it is an extension of int-power and it is continuous

#### Mario Carneiro (Nov 04 2018 at 16:32):

except of course where it isn't

#### Mario Carneiro (Nov 04 2018 at 16:32):

I think the power function is discontinuous at 0,0 no matter what you do

argh, yes

#### Kevin Buzzard (Nov 04 2018 at 16:32):

not if you restrict the power to be in nat

#### Mario Carneiro (Nov 04 2018 at 16:33):

and there is probably a branch cut somewhere that will survive in the real version

#### Kevin Buzzard (Nov 04 2018 at 16:33):

for example, when evaluating a polynomial, it's essential that x^0 gets sent to 1

#### Kevin Buzzard (Nov 04 2018 at 16:33):

but that 0 is 0:nat

#### Mario Carneiro (Nov 04 2018 at 16:33):

right, I am 100% of the view that 0^0 = 1

#### Kevin Buzzard (Nov 04 2018 at 16:33):

but my point is that this makes things continuous in this domain of real x nat -> real

oh

#### Mario Carneiro (Nov 04 2018 at 16:34):

a lot of crazy mathematicians use "everything is continuous" ism to justify claiming that it is undefined there

#### Johannes Hölzl (Nov 04 2018 at 16:34):

in this case one wants to use pow : nat -> real -> real anyway. I think the case we are discussing now is how to define pow : real -> real -> real

#### Kevin Buzzard (Nov 04 2018 at 16:34):

but one issue is whether it should extend the nat version

#### Mario Carneiro (Nov 04 2018 at 16:34):

of course, that should be easy

#### Johannes Hölzl (Nov 04 2018 at 16:35):

I agree, it should extend the nat-version. and it would be nice if we could extend the int version in a sensible way

#### Mario Carneiro (Nov 04 2018 at 16:35):

extending int should also be possible

#### Kevin Buzzard (Nov 04 2018 at 16:35):

I know a lot of crazy computer scientists use "everything must extend what we already have in every case even if answers are junk"ism...

#### Mario Carneiro (Nov 04 2018 at 16:35):

extending to rat is possible but a bad idea in my view

#### Kevin Buzzard (Nov 04 2018 at 16:35):

I think (0 : real) ^ (0 : real) is junk so should be 0

lolno

#### Kevin Buzzard (Nov 04 2018 at 16:36):

but (0 : real) ^ (0 : nat) is not junk so should be 1

Noooo

:-)

#### Johannes Hölzl (Nov 04 2018 at 16:36):

we don't extend everything. In Isabelle the logarithm of a negative argument is undefined in the sense of a fixed but unknown value

#### Chris Hughes (Nov 04 2018 at 16:36):

But then cast_pow requires proving things are non zero.

#### Mario Carneiro (Nov 04 2018 at 16:36):

I was not sure about log of negatives either

#### Mario Carneiro (Nov 04 2018 at 16:37):

log of 0 is even worse

junk!

37

#### Mario Carneiro (Nov 04 2018 at 16:37):

log x = log |x| makes some calculus stuff very slightly slicker

#### Mario Carneiro (Nov 04 2018 at 16:38):

but I don't think it will come up much anyway

#### Johannes Hölzl (Nov 04 2018 at 16:38):

What about sgn x * log |x|? I think both have their advantages/disadvantages.

whoa

#### Mario Carneiro (Nov 04 2018 at 16:38):

what is that for?

#### Mario Carneiro (Nov 04 2018 at 16:39):

it looks cool on the graph paper in my head...

#### Johannes Hölzl (Nov 04 2018 at 16:40):

I think there were some cases where it could have been helpful. But I don't remember the exact statements where it would save some non-negativity assumption. But yes the graph looks nice :-)

#### Johannes Hölzl (Nov 04 2018 at 16:43):

But x /= 0 -> y /= 0 -> log (x * y) = log x + log y only works for the log x = log |x| case

#### Kevin Buzzard (Nov 04 2018 at 16:43):

that's a junk theorem!

#### Kevin Buzzard (Nov 04 2018 at 16:44):

This islike you trying to figure out what the square root of anegative number should be so that sqrt(ab)=sqrt(a)sqrt(b) always works. Nobody who wants to apply that theorem will have a,b<0

#### Mario Carneiro (Nov 04 2018 at 16:45):

sometimes a,b>0 but it's a hassle to prove

#### Kevin Buzzard (Nov 04 2018 at 16:45):

you can't say "x != 0" is any better than "x > 0". It's still a precondition, so tehe user has to supply something

#### Johannes Hölzl (Nov 04 2018 at 16:45):

but sometimes x !=0 is easier to proof than x > 0.

boggle

#### Mario Carneiro (Nov 04 2018 at 16:46):

I mean it is literally a weaker hypothesis

#### Kevin Buzzard (Nov 04 2018 at 16:46):

if you're in a situation where you're taking logs and you don't have x>0 as a hypothesis then something is seriously wrong with your local context anyway

#### Chris Hughes (Nov 04 2018 at 16:46):

Proving things are $>0$ is cheap on paper but expensive in lean. Keeping track of which theorems are randomly true for some non-intuitive definition is cheap in Lean but expensive on paper.

#### Mario Carneiro (Nov 04 2018 at 16:46):

you had it in your context once, then you did 5 rewrites and it's not obviously true any more

#### Kevin Buzzard (Nov 04 2018 at 16:47):

I did see that in a 1st year's code recently -- "let H2 := H,..." and I thought "wooah what is this fool doing?" and then I understood that they wanted to rewrite H but keep track of it anyway

#### Mario Carneiro (Nov 04 2018 at 16:49):

those could be some combination of rewrites to the log(s) and rewrites to the hypothesis too

#### Mario Carneiro (Nov 04 2018 at 16:49):

What are the preconditions to log (log (log x + 1)) < 3?

x>>0

#### Mario Carneiro (Nov 04 2018 at 16:50):

but not too >> or else it won't be < 3 :)

#### Kevin Buzzard (Nov 04 2018 at 16:50):

If it's too >> then it's false. If it's not >> enough then it makes no sense

it's NaN

#### Mario Carneiro (Nov 04 2018 at 16:51):

unfortunately (or fortunately), real has no NaN

#### Mario Carneiro (Nov 04 2018 at 16:51):

just ask anyone who has dealt with IEEE floats, NaN is a mess

#### Kevin Buzzard (Nov 04 2018 at 16:51):

If you say to a mathematician "this is true for x=20" they will assume you mean that the LHS evaluates to something meaningful in maths

#### Mario Carneiro (Nov 04 2018 at 16:52):

right, and that's one of the more expensive in lean things to do

#### Mario Carneiro (Nov 04 2018 at 16:53):

you might think "we could just have preconditions on log and then it would all make sense" but that would just make the problem worse

#### Kevin Buzzard (Nov 04 2018 at 16:53):

I think we need a with_NaN structure

#### Mario Carneiro (Nov 04 2018 at 16:54):

with_bot?

#### Kevin Buzzard (Nov 04 2018 at 16:54):

but it might not be a bottom

#### Mario Carneiro (Nov 04 2018 at 16:54):

or equal to itself..

#### Kevin Buzzard (Nov 04 2018 at 16:54):

well obviously we'd have to modify eq

obviously

#### Kevin Buzzard (Nov 04 2018 at 16:55):

we need not_a_Type

#### Kevin Buzzard (Nov 04 2018 at 16:55):

or not_a_term or something

#### Mario Carneiro (Nov 04 2018 at 16:56):

it was silly of us to assume that all types have a reflexive relation... we should generalize to semitypes

#### Kevin Buzzard (Nov 04 2018 at 16:56):

rofl that's clearly what these guys have been missing for the last 100 years

#### Kevin Buzzard (Nov 04 2018 at 16:56):

we realised we needed semimodules

#### Kevin Buzzard (Nov 04 2018 at 16:56):

it was only a matter of time before we had the real breakthrough

#### Mario Carneiro (Nov 04 2018 at 16:57):

I joke, but PERs (partial equivalence relations) as types are a thing

nuprl uses it

#### Kevin Buzzard (Nov 04 2018 at 16:57):

is that where you drop reflexivity?

yeah

#### Kevin Buzzard (Nov 04 2018 at 16:58):

so you get equivalence classes and then some wasteland

#### Kevin Buzzard (Nov 04 2018 at 16:58):

of terms which are related to nothing

#### Mario Carneiro (Nov 04 2018 at 16:58):

it's just an equivalence relation on a subtype

right

#### Mario Carneiro (Nov 04 2018 at 16:58):

In type theory it has the purpose of rolling subtyping and quotients into one construct

#### Mario Carneiro (Nov 04 2018 at 16:59):

so you get some nice categorical structure

Last updated: May 10 2021 at 08:14 UTC