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

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Peter Pein

Posts: 1,147
Registered: 5/4/05
Re: can your CAS help proving inequalities?
Posted: Mar 11, 2013 12:00 PM
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Am 11.03.2013 16:46, schrieb clicliclic@freenet.de:
>
> 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.
>

No, it succeeds - but it is by two orders of magnitude faster :)

In[1]:= 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]
] // AbsoluteTiming

Out[1]= {0.187011, True}

Peter



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