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## Church–Rosser theorem

Church–Rosser theorem A theorem, proved jointly by A. Church and J. B. Rosser, concerning Church's lambda calculus. It states that if a lambda-expression x can be reduced in two ways leading respectively to expressions y1 and y2 then there must be an expression z to which both y1 and y2 can be reduced. The choice of ways to reduce an expression arises from the possibility of separately reducing different “parts” of the expression. The Church–Rosser theorem shows that either part can be worked on first, without the loss of any possibilities obtainable from starting with the other part. A corresponding theorem exists for combinatory logic. More generally, any language for which there is a notion of reduction for expressions within the language is said to have the Church–Rosser property, or to be confluent, if it admits an appropriate version of the Church–Rosser theorem. The property plays an important role in term rewriting with equations.

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