On the Church-Rosser property for expressive type systems and its consequences for their metatheoretic study

We consider two alternative definitions for the conversion rule in pure type systems. We study the consequences of this choice for the metatheory and point out the related implementation issues. We relate two open problems by showing that if a PTS allows the construction of a fixed point combinator, then Church-Rosser for βη-reduction fails. We present a new formalization of Russell´s paradox in a slight extension of Martin-Lof´s inconsistent theory with Type:Type and show that the resulting term leads to a fix-point construction. The main consequence is that the corresponding system is non-confluent. This example shows that in some typed λ-calculi, the Church-Rosser proof for the βη-reduction is not purely combinatorial anymore, as in pure λ-calculus, but relies on the normalization and thus the logical consistency of the system