Overlapping and Order-Independent Patterns - Definitional Equality for All

Dependent pattern matching is a safe and efficient way to write programs and proofs in dependently typed languages. Current languages with dependent pattern matching treat overlapping patterns on a first-match basis, hence the order of the patterns can matter. Perhaps surprisingly, this order-dependence can even occur when the patterns do not overlap. To fix this confusing behavior, we developed a new semantics of pattern matching which treats all clauses as definitional equalities, even when the patterns overlap. A confluence check guarantees correctness in the presence of overlapping patterns. Our new semantics has two advantages. Firstly, it removes the order-dependence and thus makes the meaning of definitions clearer. Secondly, it allows the extension of existing definitions with new (consistent) evaluation rules. Unfortunately it also makes pattern matching harder to understand theoretically, but we give a theorem that helps to bridge this gap. An experimental implementation in Agda shows that our approach is feasible in practice too.