Two beta-equal lambda-I-terms with no types in common

Two beta-Equal lambda-I-Terms with no Types in Common

It is known that in the Curry-style type-assignment system for simple type-theory there exist two closed lambda-K-terms that are beta-equal but have no types in common. This note extends that result to lambda-I-terms. Simple type theory for the A-calculus is usually formalised in one of two versions, a typed-term version due to Church or a type-assignment version due to Curry. (The former is ou...

Monadic Presentations of Lambda Terms Using Generalized Inductive Types

Reducing Lambda Terms with Traversals

We introduce a method to evaluate untyped lambda terms by combining the theory of traversals, a term-tree traversing technique inspired from Game Semantics, with judicious use of the eta-conversion rule of the lambda calculus. The traversal theory of the simply-typed lambda calculus relies on the eta-long transform to ensure that when traversing an application, there is a subterm representing e...

Lambda Calculus with Types

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author’s classic The Lambda Calculus (1984). The formalism has since been extended with types...

Counting isomorphism classes of $\beta$-normal linear lambda terms

Unanticipated connections between different fragments of lambda calculus and different families of embedded graphs (a.k.a. “maps”) motivate the problem of enumerating β-normal linear lambda terms. In this brief note, it is shown (by appeal to a theorem of Arquès and Beraud) that the sequence counting isomorphism classes of β-normal linear lambda terms up to free exchange of adjacent lambda abst...