Information and Computation - Semantics of Data Types
Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
An algorithm for testing conversion in type theory
Logical frameworks
Semantics of type theory: correctness, completeness, and independence results
Semantics of type theory: correctness, completeness, and independence results
Handbook of logic in computer science (vol. 2)
Information and Computation
Computation and reasoning: a type theory for computer science
Computation and reasoning: a type theory for computer science
PAL+: a lambda-free logical framework
Journal of Functional Programming
Expansion postponement for normalising pure type systems
Journal of Functional Programming
Expansion postponement via cut elimination in sequent calculi for pure type systems
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
A Logical Framework with Explicit Conversions
Electronic Notes in Theoretical Computer Science (ENTCS)
Untyped Algorithmic Equality for Martin-Löf's Logical Framework with Surjective Pairs
Fundamenta Informaticae - Typed Lambda Calculi and Applications 2005, Selected Papers
TYPES'06 Proceedings of the 2006 international conference on Types for proofs and programs
Proceedings of the 15th ACM SIGPLAN international conference on Functional programming
Formalized metatheory with terms represented by an indexed family of types
TYPES'04 Proceedings of the 2004 international conference on Types for Proofs and Programs
Towards normalization by evaluation for the βη-calculus of constructions
FLOPS'10 Proceedings of the 10th international conference on Functional and Logic Programming
Untyped Algorithmic Equality for Martin-Löf's Logical Framework with Surjective Pairs
Fundamenta Informaticae - Typed Lambda Calculi and Applications 2005, Selected Papers
Explicit convertibility proofs in pure type systems
Proceedings of the Eighth ACM SIGPLAN international workshop on Logical frameworks & meta-languages: theory & practice
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In a typing system, there are two approaches that may be taken to the notion of equality. One can use some external relation of convertibility defined on the terms of the grammar, such as $\beta$-convertibility or $\beta \eta$-convertibility; or one can introduce a judgement form for equality into the rules of the typing system itself. For quite some time, it has been an open problem whether the two systems produced by these two choices are equivalent. This problem is essentially the problem of proving that the Subject Reduction property holds in the system with judgemental equality. In this paper, we shall prove that the equivalence holds for all functional Pure Type Systems (PTSs). The proof essentially consists of proving the Church-Rosser Theorem for a typed version of parallel one-step reduction. This method should generalise easily to many typing systems which satisfy the Uniqueness of Types property.