Proofs and types
Theoretical Computer Science
A finitary version of the calculus of partial inductive definitions
ELP'91 Conference Proceedings on Extensions of logic programming
Basic proof theory
Cut-elimination for a logic with definitions and induction
Theoretical Computer Science - Special issue on proof-search in type-theoretic languages
Reasoning with higher-order abstract syntax in a logical framework
ACM Transactions on Computational Logic (TOCL)
A logical framework for reasoning about logical specifications
A logical framework for reasoning about logical specifications
A proof theory for generic judgments
ACM Transactions on Computational Logic (TOCL)
Combining Generic Judgments with Recursive Definitions
LICS '08 Proceedings of the 2008 23rd Annual IEEE Symposium on Logic in Computer Science
The Abella Interactive Theorem Prover (System Description)
IJCAR '08 Proceedings of the 4th international joint conference on Automated Reasoning
Reasoning in Abella about Structural Operational Semantics Specifications
Electronic Notes in Theoretical Computer Science (ENTCS)
Least and Greatest Fixed Points in Linear Logic
ACM Transactions on Computational Logic (TOCL)
On the specification of sequent systems
LPAR'05 Proceedings of the 12th international conference on Logic for Programming, Artificial Intelligence, and Reasoning
Combining Deduction Modulo and Logics of Fixed-Point Definitions
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
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Proof systems for logics with recursive definitions typically impose a strict syntactic stratification on the body of a definition to ensure cut elimination and consistency of the logics, i.e., by forbidding any negative occurrences of the predicate being defined. Often such a restriction is too strong, as there are cases where such negative occurrences do not lead to inconsistency. Several logical frameworks based on logics of definitions have been used to mechanise reasoning about properties of operational semantics and type systems. However, some of the uses of these frameworks actually go beyond what is justified by their logical foundations, as they admit definitions which are not strictly stratified, e.g., in the formalisation of logical-relation type of arguments in typed λ-calculi. We consider here a more general notion of stratification, which allows one to admit some definitions that are not strictly stratified. We outline a novel technique to prove consistency and a partial cut elimination result, showing that every derivation can be transformed into a certain head normal form, by simulating its cut reductions in an infinitary proof system. We demonstrate this technique for a specific logic, but it can be extended to other richer logics.