Equivalences. Re: Checking equivilence of equalities

On 2/8/2012 7:50 AM, Richard Fateman wrote:
> On 2/8/2012 3:10 AM, Edd Barrett wrote:
>> On Wed, Feb 08, 2012 at 06:01:31AM -0500, Stavros Macrakis wrote:
....

I think that what you want is this:   for x being a single variable or a 
vector of variables...

e1: A(x)=B(x)     and e2: C(x)=D(x)   are "equivalent"  if

for all values of x ,    A(x)=B(x)  implies C(x)=D(x)   AND
C(x)=B(x)  implies A(x)=B(x).

As Stavros pointed out, if you solve(e1,x)   and solve(e2,x)  and the 
solutions are the same,
then you are done, and Maxima has done the job.

If you want another way of saying this,  you could ask

zeroequivalence of (C-D) subject to the relation   A-B=0   and

zeroequivalence of (A-B) subject to the relation C-D=0.

A potentially useful goal that may have more general traction is  
finding an algorithm
where you could reduce both  e1 and e2 to
a simpler form and then, even though you could not solve that, you could 
show they
are the same.

for example,
e1:  f(x)+4= g(x)+6
e2:  f(x)-2 =g(x)

could both be reduced to (say)

  f(x)=g(x)+2

For some classes of functions f and g this makes sense, e.g. finding 
minimal polynomials.

RJF