Compact Normalisation Trace via Lazy Rewriting

Innermost strategy is usually used in compiling term rewriting systems (TRSs) because it allows to build efficiently the result term in a bottom-up fashion. However, the innermost strategy does not always give the shortest normalising derivation. In many cases, using an appropriate laziness annotation on the arguments of the function symbols, we evaluate the lazy arguments only if it is necessary and hence, get a shorter derivation to the normal form while avoiding the non-terminating reductions. We provide in this work a transformation of the annotated TRSs, that allows to compute the normal form using an innermost strategy and to extract a lazy derivation in the original TRS from the normalising derivation in the transformed TRS. We apply our result to improve the efficiency of equational reasoning in the {\Coq} proof assistant using {\ELAN} as an external rewriting engine.