This paper applies Dehornoy et al.’s Z theorem and its variant, called the compositional Z theorem, to prove confluence of Parigot’s \(\lambda \mu \)-calculi extended by the simplification rules. First, it is proved that Baba et al.’s modified complete developments for the call-by-name and the call-by-value variants of the \(\lambda \mu \)-calculus with the renaming rule, which is one of the simplification rules, satisfy the Z property. It gives new confluence proofs for them by the Z theorem. Secondly, it is shown that the compositional Z theorem can be applied to prove confluence of the call-by-name and the call-by-value \(\lambda \mu \)-calculi with both simplification rules, the renaming and the \(\mu \eta \)-rules, whereas it is hard to apply the ordinary parallel reduction technique or the original Z theorem by one-pass definition of mappings for these variants.

本文应用Dehornoy等人的Z定理及其变体（称为合成Z定理），证明了由简化规则扩展的Parigot的\（\ lambda \ mu \）-计算的合流。首先，证明了Baba等人使用重命名规则对\（\ lambda \ mu \）-演算的按名称调用和按值调用变体的完整开发进行了修改，该重命名规则是的简化规则，满足Z属性。Z定理为它们提供了新的融合证明。其次，证明了组合Z定理可以用简化规则，重命名和\来证明按名称调用和按值调用\（\ lambda \ mu \）-演算的融合。 （\ mu \ eta \）-规则，而通过这些变量的映射的一遍式定义很难应用普通的并行归约技术或原始Z定理。