Handle substitutions of partial sums and products

[videlec]

Sage is not able to identify partial sum in a substitution

sage: var('x,y')
sage: f = x + x^2 + x^4
sage: f.subs(x^2 == y)             # one term is fine
x^4 + x + y
sage: f.subs(x + x^2 == y)         # partial sum does not work
x^4 + x^2 + x
sage: f.subs(x + x^2 + x^4 == y)   # whole sum is fine
y

Similarly with products

sage: f = x * cos(x) * sin(x)
sage: f.subs( cos(x) * sin(x) == y)
x*cos(x)*sin(x)

As mentioned in the doc, this is the same behavior as in Maple but differ from Mathematica. We should be clearer on the semantic of substitute and potentially implement partial sum and product substitutions.

See also #10049 and ​http://ask.sagemath.org/question/25972/substitute-xy-by-u/ (with related ticket #17879)

CC: @orlitzky @mezzarobba

Component: symbolics

Issue created by migration from https://trac.sagemath.org/ticket/18396

[nbruin]

comment:2

I'm not so sure we have to do more than document it. Obviously you cannot expect substitutions to happen on any "equal" subexpression, since that concept isn't well-defined.

The thing is: x+x^2 isn't a syntactical subunit of x + x^2 + x^4 for the internal representation, which is roughly ('+',x,('^',x,2)) versus ('+',x,('<sup>',x,2),('</sup>',x,4))

You'll have to decide how much tricks are worthwhile to implement before you just add the relation y-(x^2+x) and ask for elimination of x via groebner bases.

[videlec]

comment:3

Replying to @nbruin:

I'm not so sure we have to do more than document it. Obviously you cannot expect substitutions to happen on any "equal" subexpression, since that concept isn't well-defined.

I do not want to substitute "equal" subexpression but only identical ones. And doing so, I want to consider 'a+c' as a unit of 'a+b+c+d' and 'ac' as a unit in 'abcd'. This is perhaps not desirable though.

The thing is: x+x^2 isn't a syntactical subunit of x + x^2 + x^4 for the internal representation, which is roughly ('+',x,('^',x,2)) versus ('+',x,('<sup>',x,2),('</sup>',x,4))

I know, and this is precisely the purpose of the ticket.

You'll have to decide how much tricks are worthwhile to implement before you just add the relation y-(x^2+x) and ask for elimination of x via groebner bases.

Note that x + y - (u + v) does not exist. But I agree that there is an ambiguous +/- issue (as far as the ticket description is concerned).

[rwst]

comment:5

However, even the whole matching does not work consistently. See pynac/pynac#269

[EmmanuelCharpentier]

Of note :

sage: f=x+x^2+x^4
sage: f.subs(x+x^2==y)
x^4 + x^2 + x
sage: f._sympy_().subs(sympy.sympify({x+x^2:y}))._sage_()
x^4 + y

This suggests that an algorithm="sympy" keyword (or alternatively deep=True) might switch to the behavious expected by Maple/Sympy/Mathematica users...