Hi Ted,
As William stated, I think any functionality improving SageMath's appeal
for, say, educating high school students would be very welcome. My main
concern is how valuable what you propose with PRESS-like printing is in
this respect.

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You gave a printout of your current PRESS implementation earlier, for
solving a simple linear equation in one variable. That printout is
extremely verbose and very micro-syntactically oriented. My guess would
be that few high school teachers would wish to teach the simple isolation of a
variable technique that way. And students struggling with the technique
would mostly just become more confused ("undefine the binary '-' operator",
"change the association of +", etc.), and would fail building up
mathematical intuition but rather focus on the syntax.
You gave the following examples of more advanced problems PRESS can
solve:
> 4^(2*x+1) * 5^(x-2) = 6^(1-x) (A.E.B. November 1971)
> cos(x) + cos(3*x) + cos(5*x) = 0 (A.E.B. June 1976)
> 3*tan(3*x) - tan(x) + 2 = 0 (Oxford Autumn 1978)
> log_2 x + 4*log_x 2 = 5 (London January 1978)
> 3*sech^2(x) + 4*tanh(x) + 1 = 0 (A.E.B. June 1971)
> log_e(x+1) + log_e(x-1) = 3
> e^(3*x) - 4*e^x + 3*e^(-x) = 0 (London June 1977)
> cosh(x) - 3*sinh(y) = O & 2*sinh(x) + 6*cosh(y) = 5 (A.E.B. June 1973)"
In Denmark - and my impression is that the same goes for other countries
as well - high school curriculum has largely moved away from solving of
such equations using identities and "tricks". The cos(x+y) identities
etc. and similar are not even taught anymore. In algebra, students
mostly solve linear equations with 1 or 2 unknowns and second-degree
equations. Simple isolation involving logs and exponentials, and there's
also some simplification and/or factoring of e.g. polynomial fractions
involved in function analysis.
Best,
Johan
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