Mutual Coinduction

Abstract

In this paper we present mutual coinduction as a dual of mutual induction and also as a generalization of standard coinduction. In particular, we present a precise formal definition of mutual induction and mutual coinduction. In the process we present the associated mutual induction and mutual coinduction proof principles, and we present the conditions under which these principles hold. In spite of some mention of mutual (co)induction in research literature, but the formal definition of mutual (co)induction and the proof of the mutual (co)induction proof principles we present here seem to be the first such definition and proof. As such, it seems our work is the first to point out that monotonicity of mutual generators seems not sufficient for guaranteeing the existence of least and greatest simultaneous fixed points in complete lattices, and that continuity on the other hand is sufficient for guaranteeing their existence. [The paper has been responsively updated so as to not require the continuity of generators but only require their monotonicity, and it now includes a proof of the sufficiency of monotonicity. A full revision of the paper to reflect the relaxed requirement is currently underway.] In the course of our presentation of mutual coinduction we also discuss some concepts related to standard (also called direct) induction and standard coinduction, as well as ones related to mutual (also called simultaneous or indirect) induction. During the presentation we purposely discuss particular standard concepts so as to help motivate the definitions of their more general counterparts for mutual/ simultaneous/ indirect (co)induction. Greatest simultaneous post-fixed points, in particular, will be abstractions and models of mathematical objects (e.g., points, sets, types, predicates, etc.) that are defined mutually-coinductively.