Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: can your CAS help proving inequalities?
Replies: 19   Last Post: Mar 11, 2013 12:00 PM

 Messages: [ Previous | Next ]
 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: can your CAS help proving inequalities?
Posted: Mar 11, 2013 11:46 AM

Peter Pein schrieb:
>
> [...]
>
> Show that for all real a,b,c,d three inequalities hold. Direct
> transation into Mathematica code yields (in version 9):
>
> (* def. of r,s,t omitted *)
> In[4]:= InputForm[
> tests=ForAll[{a,b,c,d},Element[{a,b,c,d},Reals],And@@
> (#1[#2[#3@@@{{a,b,c,d},{a,c,b,d},{a,d,c,b}}],0]&@@@{
> {LessEqual,Min,r},{GreaterEqual,Max,s},{LessEqual,Min,t}})]
> ]
> Out[4]//InputForm=
> ForAll[{a, b, c, d}, Element[a | b | c | d, Reals],
> Min[(a - c)*(b - c)*(a - d)*(b - d), (a - b)*(-b + c)*(a - d)*(c - d),
> (a - b)*(a - c)*(-b + d)*(-c + d)] <= 0 &&
> Max[(b + c)*(a + d) - 2*(b*c + a*d) - Abs[(-b + c)*(a - d)],
> (a + c)*(b + d) - 2*(a*c + b*d) - Abs[(a - c)*(b - d)],
> (a + b)*(c + d) - 2*(a*b + c*d) - Abs[(a - b)*(c - d)]] >= 0 &&
> Min[(b + c)*(a + d) - 2*(b*c + a*d) + Abs[(-b + c)*(a - d)],
> (a + c)*(b + d) - 2*(a*c + b*d) + Abs[(a - c)*(b - d)],
> (a + b)*(c + d) - 2*(a*b + c*d) + Abs[(a - b)*(c - d)]] <= 0]
>
> and on my not too fast box it lasts ~22 seconds to get:
>
> In[5]:= AbsoluteTiming[FullSimplify[tests]]
> Out[5]= {21.934255,True}
>

You are posing this as a quantifier elimination problem. Does the
simpler

FullSimplify[Max[
(b + c)*(a + d) - 2*(b*c + a*d) - Abs[(-b + c)*(a - d)],
(a + c)*(b + d) - 2*(a*c + b*d) - Abs[(a - c)*(b - d)],
(a + b)*(c + d) - 2*(a*b + c*d) - Abs[(a - b)*(c - d)]]
>= 0] /; Element[{a,b,c,d},Reals]

fail? In any event, the problem is presumably again addressed by CAD.

Martin.

Date Subject Author
3/8/13 clicliclic@freenet.de
3/8/13 Mate
3/9/13 clicliclic@freenet.de
3/9/13 clicliclic@freenet.de
3/8/13 Nasser Abbasi
3/9/13 Mate
3/9/13 Nasser Abbasi
3/9/13 A N Niel
3/9/13 Mate
3/10/13 A N Niel
3/10/13 Mate
3/9/13 Nasser Abbasi
3/10/13 Mate
3/10/13 Nasser Abbasi
3/10/13 Mate
3/10/13 clicliclic@freenet.de
3/10/13 Nasser Abbasi
3/11/13 Peter Pein
3/11/13 clicliclic@freenet.de
3/11/13 Peter Pein