Zulip Chat Archive

Stream: general

Topic: int * real


view this post on Zulip James Arthur (Jul 01 2020 at 08:14):

I am trying to prove that for a sine function every multiple of π\pi is 0. This is my code thus far:

import tactic
import basic zero_lemmas
import data.real.basic
import data.complex.exponential

lemma sin_npi (n : ) : sin (n * pi) = 0 :=
begin
  {have H : 1 + (n-1) = n,
    {ring},
  rw [H],
  have H2 : (1 + (n-1)) * pi = pi + (n - 1) * pi,

  sorry},

end

and get this error:

type mismatch at application
  (1 + (n - 1)) * pi
term
  pi
has type
  
but is expected to have type
  

I think it may be because I haven't imported the integers? As that was an issue before but I have tried to import:

import data.int.basic init.data.int.basic

but it still throws an error.

view this post on Zulip Johan Commelin (Jul 01 2020 at 08:15):

Lean sees 1 + (n - 1) and thinks you want to work with integers. You need to tell it explicitly that you want to view this as a real.

view this post on Zulip Johan Commelin (Jul 01 2020 at 08:16):

Telling lean to view x as a term of type Y can be done by writing (x : Y).

view this post on Zulip Johan Commelin (Jul 01 2020 at 08:17):

(This only works if someone explained to lean how to move x to this other type, this is called a coercion, or coe. Lean shows you that it coerced x by putting a tiny up-arrow before it.)

view this post on Zulip James Arthur (Jul 01 2020 at 08:20):

So I need something like this?

have H2 : ((1 + (n-1) : )) * pi = pi + (n - 1) * pi,

it works now but I fear that n is no longer an integer.

view this post on Zulip Kevin Buzzard (Jul 01 2020 at 08:20):

What do you mean "n is no longer an integer"? Things don't magically change their type. There are invisible functions in there. Look at the up arrows.

view this post on Zulip James Arthur (Jul 01 2020 at 08:23):

So even if I coerce 1+(n1)1 + (n-1) by using ((1 + (n-1) : ℝ)) , n is still an integer just that whole statement is a real? I see the arrows, they mean that the variable is in a different field to the rest doesn't it?

view this post on Zulip Kevin Buzzard (Jul 01 2020 at 08:26):

n is an integer and this won't ever change. \u n is a real. Coerce as soon as you can. Note that \u (n+1) isn't the same as (\u n) + 1 because those are two different addition functions. Compatible, but different.

view this post on Zulip Kevin Buzzard (Jul 01 2020 at 08:27):

norm_cast will try and write everything as \u (all the maths) and push_cast will try and write everything as (all the maths applied to \u stuff)

view this post on Zulip James Arthur (Jul 01 2020 at 08:27):

OK, that's very useful to know. I have now used the \u in the have and produced have H2 : (↑(1 + (n-1))) * pi = pi + (n - 1) * pi, . It now works, thanks!

view this post on Zulip Bryan Gin-ge Chen (Jul 01 2020 at 08:28):

By the way, I can't tell exactly since I don't know what's in basic and zero_lemmas, but there's something like sin_npi already proved in mathlib: src#real.sin_int_mul_pi

view this post on Zulip Patrick Massot (Jul 01 2020 at 08:29):

Elaborating on Bryan's comment: James, it's good that you tried to include the imports in the code you pasted, but they seem to refer to local files you have. The goal of this #mwe story is that people here can copy paste into their VScode and see your issue. Here we can't.

view this post on Zulip James Arthur (Jul 01 2020 at 08:33):

Sorry, I think the imports didn't explicitly import pi. I have worked everything out and moved my local imports away from the mathlib imports so that shouldn't happen again.


Last updated: May 11 2021 at 13:22 UTC