Superdevelopments for Weak Reduction

We study superdevelopments in the weak lambda calculus of Cagman and Hindley, a confluent variant of the standard weak lambda calculus in which reduction below lambdas is forbidden. In contrast to developments, a superdevelopment from a term M allows not only residuals of redexes in M to be reduced but also some newly created ones. In the lambda calculus there are three ways new redexes may be created; in the weak lambda calculus a new form of redex creation is possible. We present labeled and simultaneous reduction formulations of superdevelopments for the weak lambda calculus and prove them equivalent.