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The Third Homomorphism Theorem

The Third Homomorphism Theorem is a folk theorem of the constructive algorithmics community. It states that a function on lists that can be computed both from left to right and from right to left is necessarily a list homomorphism – it can be computed according to any parenthesization of the list. We formalize and prove the theorem, and use it to improve an O ( n 2 ) sorting algorithm to O ( n log n ).