An Infinitary and a Cyclic Sequent Calculus for Non-Monotone Inductive Definitions

Abstract: Inductive definitions are an important form of knowledge in mathematics and computer science. Two common techniques to prove theorems about inductive definitions are the principle of mathematical induction and the principle of infinite descent. To formalize these principles, Brotherston and Simpson introduced the sequent calculus proof systems LKID, for mathematical induction, and LKIDω and CLKIDω , for infinite descent. LKIDω is an infinitary system, in which proofs are infinite trees, and CLKIDω a cyclic system, in which proofs are finite graphs. However, these calculi restrict to monotone definitions, while inductive definitions are generally non-monotone. The logic FO(ID) extends classical first-order logic with non-monotone inductive definitions. In earlier work, we provided a formalization of the principle of mathematical induction for non-monotone definitions by extending LKID to a sequent calculus SCFO(ID) for FO(ID). In this paper, we provide a formalization of the principle of infinite descent for non-monotone definitions by extending LKIDω and CLKIDω to sequent calculi SCFO(ID)-inf resp. SCFO(ID)-cyc for FO(ID). Furthermore, we extend several proof-theoretic results for LKIDω and CLKIDω to SCFO(ID)-inf and SCFO(ID)-cyc regarding soundness, completeness, cut-elimination and the relation with SCFO(ID).