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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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clicliclic@freenet.de

Posts: 995
Registered: 4/26/08
Re: can your CAS help proving inequalities?
Posted: Mar 10, 2013 1:57 PM
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"Nasser M. Abbasi" schrieb:
>
> On 3/9/2013 6:07 AM, A N Niel wrote:

> >
> > What CAS did you use, and what does {} mean for it?
> >

>
> Mathematica. It has a special command called
>
> "FindInstance[expr, vars]
>
> finds an instance of vars that makes the statement expr be True.
> gives results in the same form as Solve: if an instance exists,
> and {} if it does not. "
>
> http://reference.wolfram.com/mathematica/ref/FindInstance.html
>
> So that is what I used:
>
> -----------------------------
> Remove["Global`*"]
> r[a_, b_, c_, d_] := (a - c)*(a - d)*(b - c)*(b - d)
> s[a_, b_, c_, d_] := (a + b)*(c + d) - 2*(a*b + c*d) -
> Abs[(a - b)*(c - d)]
> t[a_, b_, c_, d_] := (a + b)*(c + d) - 2*(a*b + c*d) +
> Abs[(a - b)*(c - d)]
> -----------------------------
>
> and now
>
> ----------------------
> FindInstance[
> Min[r[a, b, c, d], r[a, c, b, d], r[a, d, c, b]] > 0, {a, b, c, d}]
>
> {}
>
> FindInstance[
> Min[r[a, b, c, d], r[a, c, b, d], r[a, d, c, b]] > 0, {a, b, c, d}]
>
> {}
>
> FindInstance[
> Min[t[a, b, c, d], t[a, c, b, d], t[a, d, c, b]] > 0, {a, b, c, d}]
>
> {}
> -------------------------
>
> But I was not sure this qualifies as "proof" that is why I asked
> first.
>


Since the Mathematica variables are complex by default, they must here
perhaps be explicitly restricted to real. Judging form the "Notes On
Internal Implementation" it looks like Mathematica's Reduce[] might be
able to do reduce such inequalities: among other techniques, it fields
cylindrical algebraic decomposition (CAD).

Martin.



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