This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cut-free complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (non-well-founded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple case-split rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cut-free complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.

We propose a novel approach to proving the termination of heapmanipulating programs, which combines separation logic with cyclic proof within a Hoare-style proof system. Judgements in this system express (guaranteed) termination of the program when started from a given line in the program and in a state satisfying a given precondition, which is expressed as a formula of separation logic. The proof rules of our system are of two types: logical rules that operate on preconditions; and symbolic execution rules that capture the effect of executing program commands. Our logical preconditions employ inductively defined predicates to describe heap properties, and proofs in our system are cyclic proofs: cyclic derivations in which some inductive predicate is unfolded infinitely often along every infinite path, thus allowing us to discard all infinite paths in the proof by an infinite descent argument. Moreover, the use of this soundness condition enables us to avoid the explicit construction and use of ranking functions for termination. We also give a completeness result for our system, which is relative in that it relies upon completeness of a proof system for logical implications in separation logic. We give examples illustrating our approach, including one example for which the corresponding ranking function is non-obvious: termination of the classical algorithm for in-place reversal of a (possibly cyclic) linked list.