Sequent calculus proof systems for inductive definitions
Inductive definitions are the most natural means by which to represent many families of structures occurring in mathematics and computer science, and their corresponding induction / recursion principles provide the fundamental proof techniques by which to reason about such families. This thesis studies formal proof systems for inductive definitions, as needed, e.g., for inductive proof support in automated theorem proving tools. The systems are formulated as sequent calculi for classical first-order logic extended with a framework for (mutual) inductive definitions. The default approach to reasoning with inductive definitions is to formulate the induction principles of the inductively defined relations as suitable inference rules or axioms, which are incorporated into the reasoning framework of choice. Our first system LKID adopts this direct approach to inductive proof, with the induction rules formulated as rules for introducing atomic formulas involving inductively defined predicates on the left of sequents. We show this system to be sound and cut-free complete with respect to a natural class of Henkin models. As a corollary, we obtain cut-admissibility for LKID. The well-known method of infinite descent `a la Fermat, which exploits the fact that there are no infinite descending chains of elements of well-ordered sets, provides an alternative approach to reasoning with inductively defined relations. Our second proof system LKIDw formalises this approach. In this system, the left-introduction rules for formulas involving inductively defined predicates are not induction rules but simple case distinction rules, and an infinitary, global soundness condition on proof trees — formulated in terms of “traces” on infinite paths in the tree — is required to ensure soundness. This condition essentially ensures that, for every infinite branch in the proof, there is an inductive definition that is unfolded infinitely often along the branch. By an infinite descent argument based upon the well-foundedness of inductive definitions, the infinite branches of the proof can thus be disregarded, whence the remaining portion of proof is well-founded and hence sound. We show this system to be cutfree complete with respect to standard models, and again infer the admissibility of cut. The infinitary system LKIDw 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” proof system, CLKIDw, is suitable for formal reasoning since proofs have finite representations and the soundness condition on proofs is thus decidable. We show how the formulation of our systems LKIDw and CLKIDw can be generalised to obtain soundness conditions for a general class of infinite proof systems and their corresponding cyclic restrictions. We provide machinery for manipulating and analysing the structure of proofs in these essentially arbitrary cyclic systems, based primarily on viewing them as generating regular infinite trees, and we show that any proof can be converted into an equivalent proof with a restricted cycle structure. For proofs in this “cycle normal form”, a finitary, localised soundness condition exists that is strictly stronger than the general, infinitary soundness condition, but provides more explicit information about the proof. Finally, returning to the specific setting of our systems for inductive definitions, we show that any LKID proof can be transformed into a CLKIDw proof (that, in fact, satisfies the finitary soundness condition). We conjecture that the two systems are in fact equivalent, i.e. that proof by induction is equivalent to regular proof by infinite descent.