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There are many posts here and elsewhere asking for algorithms to simplify simple arithmetic expressions. For example, this question asks how to simplify the expression $axc + byc + ayc + bxc$. Symbolics and xcas are two examples among many of Computer Algebra Systems (CAS) that can do this. But Symbolics and xcas perform differently: Symbolics can simplify the expression above and turn it into $cx(a + b) + cy(a + b)$, whereas xcas completely factorizes the expression into $(a + b)c(x + y)$.

I was wondering if there is any CAS that simplifies expressions based on some set of equations that you can input into the system? As an example, assume that we knew $a + b = 2$ and $x + y = -1$: is there any known (implemented) algorithm that given said equations, would factorize the expression above into simply $-2c$ ? Perhaps it's more pertinent to ask: could such an algorithm exist?

In the example above, we can set $y = -1 - x$ and $b = 2 - a$ (as pointed out in a comment), but in more complicated examples it might not be possible.

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    $\begingroup$ Not sure the this is what you want, but setting $b:=2-a$ and $y:=-1-x$ will give you the desired simplification. $\endgroup$
    – user16034
    Commented Sep 23, 2022 at 14:56

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If your knowledge is in the form of linear equations, as in your example, then you can make a simple substitution. In your example, set $b=2-a$ and $y=-1-x$, as Yves Daoust suggests. This will work for any system of linear equations.

If the expression is a polynomial and the "knowledge" is expressed as equations of polynomials, then it is possible that Groebner bases might be useful - I am not sure.

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  • $\begingroup$ Thank you. I'm sorry I didn't explain that I also might use non-linear equations. $\endgroup$
    – Lluís Alemany-Puig
    Commented Sep 24, 2022 at 15:22
  • $\begingroup$ I've accepted your answer because I can see the question is too broad and perhaps I'm not using the right terminology. Thanks! $\endgroup$
    – Lluís Alemany-Puig
    Commented Sep 24, 2022 at 15:26

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