Shallow confluence of conditional term rewriting systems

Recursion can be conveniently modeled with left-linear positive/negative-conditional term rewriting systems, provided that non-termination, non-trivial critical overlaps, non-right-stability, non-normality, and extra variables are admitted. For such systems we present novel sufficient criteria for shallow confluence and arrive at the first decidable confluence criterion admitting non-trivial critical overlaps. To this end, we restrict the introduction of extra variables of right-hand sides to binding equations and require the critical pairs to have somehow complementary literals in their conditions. Shallow confluence implies [level] confluence, has applications in functional logic programming, and guarantees the object-level consistency of the underlying data types in the inductive theorem prover QuodLibet.