symbolic manipulation -- insufficient simplification

[Robert Samal]

I observed the following weird behavior of the symbolic engine.

sage: x/x
1
sage: x^2/x
x
sage: (x^2+x)/x
(x^2 + x)/x
sage: assume(x>0)
sage: assume(x,'real')
sage: assumptions()
[x > 0, x is real]
sage: (x^2+x)/x
(x^2 + x)/x

To clarify: first, I consider the first two simplifications slightly incorrect (x/x is undefined if x=0, or possibly if x is in some weird algebraic structure). However, if x/x==1 and x^2/x==x then I wonder why simplification of (x^2+x)/x is not done?





In this case, .simplify() does not help, .full_simplify() does. In my original example though, .full_simplify() did something crazy, so I was led to this.

In general, is there some explanation regarding which simplifications one can expect to be done automatically and which by the two simplify-functions?




Thanks!





[Emmanuel Charpentier]

From the docstrings:

sage: x.simplify?

   Return a simplified version of this symbolic expression.

   Note: Currently, this just sends the expression to Maxima and

     converts it back to Sage.

   See also: "simplify_full()", "simplify_trig()",

     "simplify_rational()", "simplify_rectform()"

     "simplify_factorial()", "simplify_log()", "simplify_real()",

     "simplify_hypergeometric()", "canonicalize_radical()"

Not very informative, ... and Maxima's simplifications are numerous, complex and somewhat difficult to follow.

sage: x.simplify_full?

   Apply "simplify_factorial()", "simplify_rectform()",

   "simplify_trig()", "simplify_rational()", and then "expand_sum()"

   to self (in that order).

Note that you have also:

sage: assumptions()

[]

sage: ((x^2+x)/x).collect_common_factors()

x + 1

sage: x.collect_common_factors?

   This function does not perform a full factorization but only looks

   for factors which are already explicitly present.

   Polynomials can often be brought into a more compact form by

   collecting common factors from the terms of sums. This is

   accomplished by this function.

and

sage: ((x^2+x)/x).canonicalize_radical()

x + 1

sage: x.collect_common_factors?

   This function does not perform a full factorization but only looks

   for factors which are already explicitly present.

   Polynomials can often be brought into a more compact form by

   collecting common factors from the terms of sums. This is

   accomplished by this function.

And, indeed, these functions do more or less undocumented assumptions...

HTH,