       Re: LessEqual vs Inequality, was ..Re: Replacement Rule with Sqrt in denominator

• To: mathgroup at smc.vnet.net
• Subject: [mg114734] Re: LessEqual vs Inequality, was ..Re: Replacement Rule with Sqrt in denominator
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Tue, 14 Dec 2010 06:57:33 -0500 (EST)
• References: <ie2971\$mqh\$1@smc.vnet.net> <4D050013.8050105@cs.berkeley.edu> <928BCB32-AEF9-4D13-87E0-BDDACF1BF878@mimuw.edu.pl> <4D055126.6080209@cs.berkeley.edu> <ie4mt7\$9a7\$1@smc.vnet.net> <4D0645BE.6000205@cs.berkeley.edu> <7F38BE28-6AE1-4BBC-A115-14B9CA3D9BF0@mimuw.edu.pl> <4D06C7C9.8030803@cs.berkeley.edu>

``` Well, you have worn me out, so basically I quit. Not that I agree with
much. Your long post consist of a mixture of the obvious, which you seem
to believe is a revelation and other things that reveal your own
ignorance of Mathematica and even certain areas of algorithmic algebra.
I will point out just one:
>
> You are asserting that "Refine" can be used.  Thus Refine, in the
computer algebra terminology, is
> a Zero Equivalence algorithm.   (or a Logical Tautology algorithm, in
some sense).
>

Not at all. Refine does not deal with logical tautologies or booleans.
In fact Mathematica has EquivalentQ and Tautology for this purpose:

Equivalent[a && (b || c), a && b || a && c] // TautologyQ

True

What Refine does (Like Simplify and FullSimplify) is to perform
simplifications of what are called first order formulas in the language
of real fields with parameters in the real numbers. For example
x^2>y>z+1 is such a formula and so is
z<y-1<x^2-1. But (x^2>y>z+1) - (z<y-1<x^2-1) is not a forumla but a
meaningless construct, a kind of accidental consequence of the way
Mathematica language is constructed. It does not make sense to write

Simplify[x^2>y>z+1 - z<y-1<x^2-1]; even when this sort of thing seems to
work sometimes, it is by accident (it will work when the two expressions
that you are subtracting evaluate to the same thing but not otherwise
since Mathematica will not assign any meaning to the difference. On the
other hand

Refine[x^2 > y > z, z - 1 < y - 1 < x^2 - 1] &&
Refine[z - 1 < y - 1 < x^2 - 1, x^2 > y > z]

True

but it is certainly not a "logical tautology" algorithm. It belongs to
the same "field" as the Tarski-Seidneberg Theorem, Quantifier
Elimination, Cylindirical ALgebraic Decomposition etc. I am not sure if
this is "computer science" but there are advantages to an education that
includes such things which apparently you have not benefitted from.

```

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