In this paper we focus on a set of abstract lemmas that are easy to apply and turn out to be quite valuable in order to establish confluence and/or normalization modularly, especially when adding rewriting rules for extensional equalities to various calculi. We show the usefulness of the lemmas by applying them to various systems, ranging from simply typed lambda calculus to higher order lambda...

We prove that any decision procedure for a modest fragment of L. Henkin's theory of pure propositional types 7, 12, 15, 11] requires time exceeding a tower of 2's of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linearly high towers of 2's and since mid-seventies it was an open problem whether natural decidable th...

In 1970 Friedman proved completeness of beta eta conversion in the simply-typed lambda calculus for the set-theoretical model. Recently Krishnaswami and Benton have captured the essence of Hudak’s reactive programs in an extension of simply typed lambda calculus with causal streams and a temporal modality and provided this typed lambda calculus for reactive programs with a sound ultrametric sem...

We extend the study of the relationship between behavioural equivalence and the indistinguishability relation[4, 7] to the simply typed lambda calculus, where higher-order types are available. The relationship between these two notions is established in terms of factorisability[4]. The main technical tool of this study is pre-logical relations[8], which give a precise characterisation of behavi...

A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the least lambda-theory λβ or the least sensible lambda-theory H (generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, there is a minimal lambda-theory represented by it. In this paper, we give a...

When presenting logical systems and type theories, it is common to write finite lists of things using a comma to indicate the cons-operation and with the head of the list at the right. With this convention there is no common notation for the empty list; we will use the symbol “ ”. Thus ML-style list notation nil a :: nil b :: a :: nil etc becomes , a , a, b etc For non-empty lists, it is very c...

The collection of isomorphism classes of nitely generated rank-ordered sets is shown to be a nitely generated rank-ordered set again. This is used to construct a model of the simply typed lambda calculus extended by the assumption Type: Type. Beside this, the structure of rank-ordered sets is studied. They can be represented as inverse limits of !-cochains of substructures, each being a retract...

We introduce a notion of Grothendieck logical relation and use it to characterise the deenability of morphisms in stable bicartesian closed categories by terms of the simply-typed lambda calculus with nite products and nite sums. Our techniques are based on concepts from topos theory, however our exposition is elementary.