Question on “smart” replacing in mathematica

Question

How do I tell mathematica to do this replacement smartly? (or how do I get smarter at telling mathematica to do what i want)

expr = b + c d + ec + 2 a;
expr          


        

Answer 1

For the first case, you might consider:

expr = b + c d + ec + 2 a

PolynomialReduce[expr, {a + b - 1}, {b, a}][[2]]

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For the second case, consider:

expr = b + c (1 - a) + (-1 + b) (a - 1) + (1 - a - b) d + 2 a;

PolynomialReduce[expr, {x + b - 1}][[2]]

(% /. x -> 1 - b) == expr // Simplify

and:

PolynomialReduce[expr, {a + b - 1}][[2]]

Simplify[% == expr /. a -> 1 - b]