Soundness of Unravelings for Deterministic Conditional Term Rewriting Systems via Ultra-Properties Related to Linearity

Unravelings are transformations from a conditional term rewriting system (CTRS, for short) over an original signature into an unconditional term rewriting systems (TRS, for short) over the extended signature. They are not always sound for every CTRS w.r.t. the reduction, while they are complete. Here, soundness w.r.t. the reduction means that every reduction sequence of the corresponding unraveled TRS, of which the initial and end terms are over the original signature, can be simulated by the reduction of the original CTRS. In this paper, we show that an optimized variant of Ohlebusch’s unraveling for deterministic CTRSs is sound for a CTRS w.r.t. the reduction if the corresponding unraveled TRS is either left-linear or both right-linear and non-erasing. We also show that soundness of the variant implies soundness of Ohlebusch’s unraveling.