How to Give Local Strategies to Function Symbols for Equality of Two Implementations of the E-strategy with and without Evaluated Flags

The evaluation strategy (abbr. the E-strategy) initiated by OBJ2 is one of compromises between outermost and innermost reduction strategies. Although rewriting with the E-strategy is intuitively easy-to-understand, the straightforward implementation cannot lead to e cient rewriting. Then, the implementations of OBJ3 and CafeOBJ use evaluated ags to implement reasonably rewriting with the E-strategy. However, rewriting with evaluated ags is not always equivalent to that without evaluated ags, which depends on local strategies given to function symbols. So far we did not have any useful guidelines on how to give local strategies to function symbols. In this paper, we present a su cient but reasonable condition (i.e. how to give local strategies to function symbols) that two implementations of the E-strategy with and without evaluated ags are equivalent, and give its proof.