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Topic: Formal Proof Language Example - Human-Readable?
Replies: 39   Last Post: Jun 27, 2009 11:09 PM

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 Jan Burse Posts: 1,472 Registered: 4/12/05
Re: Formal Proof Language Example - Human-Readable?
Posted: Jun 25, 2009 6:30 PM
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Andrew Tomazos schrieb:
> On Jun 22, 4:01 am, Jan Burse <janbu...@fastmail.fm> wrote:
>> b*b = (c+a) * (c-a)
>> n*q*m*n*q*m = n*q*n * q*m*m (factorization of b, c+a, c-a sic!)

>
> In this step you have used:
>
> b = n*q*m
> c+a = q*n*n
> c-a = q*m*m
>
> How do you know that there exists n, q and m such that these three
> equalities hold?
> -Andrew.

By some pingeonhole principle.

n, q and m are classes of factors of b. If a class c is empty
then c=1. If a class c contains factors j, k, etc.. then c=j*k*...
So the n, q and m are not itself necessary prime, they group
certain prime number factors.

Each factor of b occurs twice in b*b, so b*b=n*q*m*n*q*m is trivial.

And thus (c+a)*(c-a)=n*q*m*n*q*m follows also, since factors are unique,
the same classes also occure in the product (c+a)*(c-a).

1) Now if a factor does not occur in (c+a) then it occurs twice in
(c-a), this is the class m.

2) Further if a factor does not occur in (c-a) then it occurs twice in
(c+a), this is the class n.

3) If a factor does not (not occur in (c+a) and not occur in (c-a)),
then it occurs in (c+a) and it occurs in (c-a), this is the class q.

Since the cases 1, 2 and 3 are exhaustive (*), therefore (c+a)=q*n*n and
(c-a)=q*m*m.

Bye

(*) This is usually a proof figure called "proof by cases",

A -> B ~A -> B
-------------------
B

For 1), 2) and 3) it is applied multiple times, plus some more stuff.

Date Subject Author
6/21/09 Andrew Tomazos
6/21/09 Jan Burse
6/21/09 Andrew Tomazos
6/21/09 Jan Burse
6/21/09 Andrew Tomazos
6/21/09 Jan Burse
6/21/09 Jan Burse
6/21/09 Jan Burse
6/21/09 Andrew Tomazos
6/21/09 Jan Burse
6/21/09 Andrew Tomazos
6/21/09 Jan Burse
6/21/09 Jan Burse
6/24/09 Andrew Tomazos
6/25/09 MeAmI.org
6/25/09 Jan Burse
6/26/09 Andrew Tomazos
6/27/09 Jan Burse
6/27/09 Andrew Tomazos
6/27/09 Jan Burse
6/27/09 Andrew Tomazos
6/27/09 Joshua Cranmer
6/27/09 Andrew Tomazos
6/21/09 Marshall
6/21/09 Spiros Bousbouras
6/24/09 Tim Smith
6/21/09 Charlie-Boo
6/21/09 William Elliot
6/22/09 MeAmI.org
6/22/09 MeAmI.org
6/23/09 Slawomir
6/24/09 David Bernier
6/24/09 MeAmI.org
6/24/09 MeAmI.org
6/24/09 Andrew Tomazos
6/24/09 Andrew Tomazos
6/25/09 Slawomir

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