Consistency of Recursive Definitions via Shallow Confluence of Non-Orthogonal Non-Terminating Conditional Term Rewriting Systems with any kind of Extra Variables

Recursive definitions can be adequately and conveniently modeled with left-linear conditional term rewriting systems, provided that non-termination, non-trivial critical pairs, and extra variables are admitted. Confluence of such systems guarantees the object-level consistency of the underlying data types. We present a novel sufficient criterion for shallow confluence, a property which is logically stronger than level confluence and confluence, and which is not only needed as a generalization for the hard inductive proof of the sufficiency of the criterion, but has other applications also, e.g. in functional logic programming. By restricting the introduction of extra variables to binding equations that mirror local variable-declarations and constructor-matching variable-introduction constructs in programming languages (such as and in ML), and by requiring the critical pairs to have somehow complementary literals in their conditions, we arrive at the first decidable confluence criterion with which we can write recursive function definitions adequately and conveniently: It admits non-termination; non-trivial critical pairs; extra variables in right-hand-sides introduced by binding equations and arbitrary extra variables in conditions; and non-proper-orientation, non-right-stability, and non-normality of conditional equations.