Cut Elimination for a Logic with Induction and Co-induction

Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles are based on a proof theoretic (rather than set-theoretic) notion of definition. Definitions are akin to logic programs, where the left and right rules for defined atoms allow one to view theories as “closed” or defining fixed points. The use of definitions and free equality makes it possible to reason intentionally about syntax. We add in a consistent way rules for pre and post fixed points, thus allowing the user to reason inductively and co-inductively about properties of computational system making full use of higher-order abstract syntax. Consistency is guaranteed via cut-elimination, where we give the first, to our knowledge, cut-elimination procedure in the presence of general inductive and co-inductive definitions.

💡 Research Summary

The paper addresses a long‑standing gap in proof‑theoretic frameworks for specifying and reasoning about computational systems: the simultaneous treatment of inductive and co‑inductive definitions within a single sequent calculus. The authors begin by observing that proof search, especially when encoded as a logic programming language, provides a natural way to describe operational semantics, type systems, and other formal specifications. However, traditional sequent calculi either lack fixed‑point operators altogether or support only one of the two proof principles (induction or co‑induction), which limits their expressive power for systems that involve both finite and infinite behaviours.

To overcome this limitation, the authors introduce a calculus in which definitions are first‑class citizens. A definition is a collection of clauses that resemble a logic program; each defined atom has a left rule (L‑def) that expands the atom on the left‑hand side of a sequent and a right rule (R‑def) that contracts it on the right‑hand side. This dual treatment allows a theory to be viewed as a closed fixed point: the left rule corresponds to unfolding the definition, while the right rule corresponds to using the definition as a derived rule. Because the calculus also includes a free equality predicate, syntactic objects can be manipulated directly, enabling intentional reasoning about syntax (e.g., α‑equivalence, capture‑avoiding substitution) without resorting to external encodings.

The core technical contribution is the addition of pre‑fixed point (inductive) and post‑fixed point (co‑inductive) rules. The pre‑fixed point rule captures the least fixed point of a definition, supporting standard mathematical induction. The post‑fixed point rule captures the greatest fixed point, supporting co‑induction and reasoning about potentially infinite structures such as streams, processes, or infinite derivations. Both rules are formulated in a way that respects higher‑order abstract syntax (HOAS), so that binders in the object language are represented by binders in the meta‑language without loss of fidelity.

Having defined the syntax and inference rules, the authors turn to the metatheory. The central theorem is cut‑elimination: any derivation that uses the cut rule can be transformed into a cut‑free derivation. Proving cut‑elimination in the presence of general inductive and co‑inductive definitions is non‑trivial because the usual reduction steps can interact badly with the unfolding of definitions and with the fixed‑point operators. The authors resolve this by introducing a sophisticated measure on derivations that combines three components: (1) the depth of definition unfolding, (2) the number of pre‑fixed point expansions, and (3) the number of post‑fixed point expansions. Each reduction step is shown to strictly decrease this measure, guaranteeing termination of the reduction process.

The reduction procedure proceeds by case analysis on the principal formulas of the cut. When the cut formula is a defined atom, the L‑def and R‑def rules are permuted past the cut, effectively pushing the cut deeper into the premises where the definition is unfolded. When the cut formula is a pre‑fixed point, the authors use the induction principle to replace the cut with a smaller induction hypothesis; similarly, for a post‑fixed point, a co‑induction hypothesis is introduced. Equality cuts are handled by a dedicated equality propagation rule that distributes the equality over the surrounding context before the cut is eliminated. Throughout, the measure ensures that no infinite regress occurs, even when definitions are mutually recursive.

The cut‑elimination theorem yields two immediate corollaries: consistency (the empty sequent is not provable) and the subformula property (every formula occurring in a cut‑free proof is a subformula of the end‑sequent). These properties are essential for the soundness of any logic used as a foundation for automated reasoning tools.

To demonstrate the practical relevance of the system, the paper presents several case studies. One example encodes the simply‑typed λ‑calculus with inductive typing rules and proves type preservation using the inductive fixed‑point rule. Another example models a process calculus with infinite transition sequences; co‑inductive definitions capture bisimulation, and the post‑fixed point rule is used to prove that two processes are bisimilar. In both cases, the higher‑order abstract syntax representation allows the authors to avoid explicit handling of α‑conversion, and the definition‑based rules provide a concise, declarative specification of the semantics.

In the related work discussion, the authors compare their approach to earlier systems such as the µ‑calculus, the logic of fixed points (LFP), and proof assistants that support inductive types (Coq, Agda) or co‑inductive types (Coq’s co‑fix). They argue that, unlike those systems, their calculus treats definitions proof‑theoretically rather than set‑theoretically, which enables a uniform treatment of both inductive and co‑inductive reasoning within the same sequent framework. Moreover, the cut‑elimination result is claimed to be the first for a logic that simultaneously admits unrestricted inductive and co‑inductive definitions.

The paper concludes by outlining future directions: extending the calculus with modular definitions, integrating richer equality theories (e.g., nominal equality), and embedding the system into automated proof search engines. The authors anticipate that the combination of definition‑based reasoning, higher‑order abstract syntax, and a robust cut‑elimination theorem will provide a solid foundation for formal verification tools that need to reason about both finite and infinite computational phenomena.