Abstraction and invariance for algebraically indexed types
POPL '13 Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Dependent Type Theory for Verification of Information Flow and Access Control Policies
ACM Transactions on Programming Languages and Systems (TOPLAS)
Proceedings of the 18th ACM SIGPLAN international conference on Functional programming
Names for free: polymorphic views of names and binders
Proceedings of the 2013 ACM SIGPLAN symposium on Haskell
Proceedings of the 15th Symposium on Principles and Practice of Declarative Programming
Proceedings of the 15th Symposium on Principles and Practice of Declarative Programming
A relationally parametric model of dependent type theory
Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages
From parametricity to conservation laws, via Noether's theorem
Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages
Logical relations for a logical framework
ACM Transactions on Computational Logic (TOCL)
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Reynolds' abstraction theorem (Reynolds, J. C. (1983) Types, abstraction and parametric polymorphism, Inf. Process.83(1), 513-523) shows how a typing judgement in System F can be translated into a relational statement (in second-order predicate logic) about inhabitants of the type. We obtain a similar result for pure type systems (PTSs): for any PTS used as a programming language, there is a PTS that can be used as a logic for parametricity. Types in the source PTS are translated to relations (expressed as types) in the target. Similarly, values of a given type are translated to proofs that the values satisfy the relational interpretation. We extend the result to inductive families. We also show that the assumption that every term satisfies the parametricity condition generated by its type is consistent with the generated logic.