The transformation of a symbolic representation using an inverted match that represents an inverted form of a specific kind of non-linear match. The match is inverted around a particular reaction variable and merged on that variable after determining that the match represents a specific kind of non-linear identity that has at least one associative commutative operator, and in which each variable has one instance, except for the reaction variable, which has multiple instances in the match, at least one of the instances not being a direct argument of an associative commutative operator. The inverted match may then be applied to the symbolic expression to thereby transform the symbolic expression.