Proofs and types
Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
POPL '90 Proceedings of the 17th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
An algorithm for testing conversion in type theory
Logical frameworks
Higher-Order and Symbolic Computation
Typed lambda-calculi with explicit substitutions may not terminate
TLCA '95 Proceedings of the Second International Conference on Typed Lambda Calculi and Applications
TLCA '95 Proceedings of the Second International Conference on Typed Lambda Calculi and Applications
Categorical Reconstruction of a Reduction Free Normalization Proof
CTCS '95 Proceedings of the 6th International Conference on Category Theory and Computer Science
Normalization by Evaluation for Typed Lambda Calculus with Coproducts
LICS '01 Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science
Intuitionistic model constructions and normalization proofs
Mathematical Structures in Computer Science
Journal of Functional Programming
Extensional rewriting with sums
TLCA'07 Proceedings of the 8th international conference on Typed lambda calculi and applications
Epigram: practical programming with dependent types
AFP'04 Proceedings of the 5th international conference on Advanced Functional Programming
MSFP'06 Proceedings of the 2006 international conference on Mathematically Structured Functional Programming
Electronic Notes in Theoretical Computer Science (ENTCS)
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Traditionally, decidability of conversion for typed λ-calculi is established by showing that small-step reduction is confluent and strongly normalising. Here we investigate an alternative approach employing a recursively defined normalisation function which we show to be terminating and which reflects and preserves conversion. We apply our approach to the simply typed λ-calculus with explicit substitutions and βη-equality, a system which is not strongly normalising. We also show how the construction can be extended to system T with the usual β-rules for the recursion combinator. Our approach is practical, since it does verify an actual implementation of normalisation which, unlike normalisation by evaluation, is first order. An important feature of our approach is that we are using logical relations to establish equational soundness (identity of normal forms reflects the equational theory), instead of the usual syntactic reasoning using the Church–Rosser property of a term rewriting system.