Nonterminating rewritings with head boundedness

We define here the concept of head boundedness, head normal form and head confluence of term rewriting systems that allow infinite derivations. Head confluence is weaker than confluence, but sufficient to guarantee the correctness of lazy implementations of equational logic programming languages. Then we prove several results. First, if a left-linear system is locally confluent and head-bounded, then it is head-confluent. Second, head-confluent and head-bounded systems have the head Church-Rosser property. Last, if an orthogonal system is head-terminating, then it is head-bounded. These results can be applied to generalize equational logic programming languages.