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From: lseward@randvax.UUCP (Larry Seward)
Newsgroups: net.math.symbolic
Subject: Re: Responce to Fatemans article "Eleven Proofs of sin^2+cos^2=1"
Message-ID: <21@randvax.UUCP>
Date: Tue, 11-Feb-86 12:03:35 EST
Article-I.D.: randvax.21
Posted: Tue Feb 11 12:03:35 1986
Date-Received: Fri, 14-Feb-86 03:23:46 EST
References: <258@wacsvax.OZ> <20@randvax.UUCP>
Reply-To: Michael B. Monagan <ihnp4!watmath!watmum!mbmonagan@rand-unix.ARPA>
Organization: Rand Corp., Santa Monica
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From: Michael B. Monagan <ihnp4!watmath!watmum!mbmonagan@rand-unix.ARPA>
Message-Id: <8602062238.AA09557@watmum.UUCP>
Subject: Pattern matching in SMP
Cc: mbmonagan@rand-unix.ARPA

   ...
   It is possible with current pattern matching techniques to

Making use of the identity can be quite interesting.  Your SMP program
simplifies sin(x)^2+cos(x)^2 to 1, but does it also simplify
sin(x)^4-cos(x)^4  to  sin(x)^2-cos(x)^2   and
======================================================================
The message got cut off but I agree 100% with what you are trying
to point out.  There is a limit to what one can do with syntactic
pattern matching as provided by SMP.  The example given of telling
SMP that sin(x)^2 + cos(x)^2 = 1 is full of holes.  SMP will not
be able to deduce that  (sin(x)^4 + 2*sin(x)^2*cos(x)^2 + cos(x)^4)
is also equal to 1.  Neither will it be able to simplify your example
above.  Thus I feel that Kevin's implication that pattern matching
a la SMP is the way to solve problems is wrong.  Syntactic pattern
matching is simply not powerful enough to do anything non-trivial.

In any case, on the topic of implementing sin(x)^2 + cos(x)^2 = 1,
we (Maple) have a good solution in sight - though not automatic.

        simplify( <expr>, {sin(x)^2 + cos(x)^2 = 1} );

Will simplify the expression <expr> with respect to the side relations
sin(x)^2 + cos(x)^2 = 1.  I would also have included  i^2 + 1 = 0  if
I was dealing with Gaussians.  Maple will compute a grobner basis for
the side-relations and reduce the expression with respect to the
Grobner basis.  In this case, sin(x)^2 + cos(x)^2 - 1  is the Grobner
trivially.