Proofs and types
Handbook of logic in computer science (vol. 2)
Basic simple type theory
Basic proof theory (2nd ed.)
Lambda-Calculus and Combinators: An Introduction
Lambda-Calculus and Combinators: An Introduction
A finitary subsystem of the polymorphic λ-calculus
TLCA'01 Proceedings of the 5th international conference on Typed lambda calculi and applications
How to assign ordinal numbers to combinatory terms with polymorphic types
Archive for Mathematical Logic
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This article investigates a theory of type assignment (assigning types to lambda terms) called ETA which is intermediate in strength between the simple theory of type assignment and strong polymorphic theories like Girard's F (Proofs and types. Cambridge University Press, Cambridge, 1989). It is like the simple theory and unlike F in that the typability and type-checking problems are solvable with respect to ETA. This is proved in the article along with three other main results: (1) all primitive recursive functionals of finite type are representable in ETA; (2) every term typable in ETA has a unique normal form; (3) there is a function defined by $${{\varepsilon}_0}$$ -recursion which takes every typable term to a natural number which is an upper bound to the lengths of all β驴-reduction sequences starting with that term.