A constructive characterization is given of the isomorphisms which must hold in all models of the typed lambda calculus with surjective pairing. By the close relation between closed Cartesian categories and models of these calculi, we also produce a characterization of those isomorphisms which hold in all CCC's. By the correspondence between these calculi and proofs in intuitionistic positive propositional logic, we thus provide a characterization of equivalent formulae of this logic, where the definition of equivalence of terms depends on having "invertible" proofs between the two terms. Rittri (1989), on types as search keys in program libraries, provides an interesting example of use of these characterizations.

This thesis concerns rewriting in the typed -calculus. Traditional categorical models of typed -calculus use concepts such as functor, adjunction and algebra to model type constructors and their associated introduction and elimination rules, with the natural categorical equations inherent in these structures providing an equational theory for -terms. One then seeks a rewrite relation which, by transforming terms into canonical forms, provides a decision procedure for this equational theory. Unfortunately the rewrite relations which have been proposed, apart from for the most simple of calculi, either generate the full equational theory but contain no decision procedure, or contain a decision procedure but only for a subtheory of that required. Our proposal is to unify the semantics and reduction theory of the typed -calculus by generalising the notion of model from categorical structures based on term equality to categorical structures based on term reduction. This is accomplished via...

) Delia Kesner CNRS and LRI, B at 490, Universit e Paris-Sud - 91405 Orsay Cedex, France. e-mail:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and non-extensional #-calculi with explicit substitutions, where extensionality is interpreted by #-expansion. For that, we propose a general scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our general scheme makes it possible to treat at the same time many well-known calculi such as ## , ## # and ## , or some other new calculi that we propose in this paper. We also show for those calculi not fitting in the general scheme that can be translated to another one fitting the scheme, such as #s , how to reason about confluence properties of their extensional and non-extensional versions. 1 Introduction The #-calculus is a convenient framework to study functional programming, where the evaluation process is modeled by #-reduction. The...

. We consider the simply typed -calculus with primitive recursion operators and types corresponding to categorical products and coproducts.. The standard equations corresponding to extensionality and to surjectivity of pairing and its dual are oriented as expansion rules. Strong normalization and ground (base-type) confluence is proved for the full calculus; full confluence is proved for the calculus omitting the rule for strong sums. In the latter case, fixed-point constructors may be added while retaining confluence. 1 Introduction The systems investigated here are simply typed -caluli whose types include pairs, unit, sums, an empty type, and a type of natural numbers supporting constructions by primitive recursion. In the core system the types behave as categorical product and coproducts, so the subject at hand is equivalently ([LS86]) the equational theory of the free bicartesian closed category (generated by objects for the base types) with weak natural numbers object. Su...

We add extensional equalities for the functional and product types to the typed -calculus with not only products and terminal object, but also sums and bounded recursion (a version of recursion that does not allow recursive calls of infinite length). We provide a confluent and strongly normalizing (thus decidable) rewriting system for the calculus, that stays confluent when allowing unbounded recursion. For that, we turn the extensional equalities into expansion rules, and not into contractions as is done traditionally. We first prove the calculus to be weakly confluent, which is a more complex and interesting task than for the usual -calculus. Then we provide an effective mechanism to simulate expansions without expansion rules, so that the strong normalization of the calculus can be derived from that of the underlying, traditional, non extensional system. These results give us the confluence of the full calculus, but we also show how to deduce confluence directly form our simulation...

This paper studies confluence of extensional and non-extensional -calculi with explicit substitutions, where extensionality is interpreted by j-expansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our method makes it possible to treat at the same time many well-known calculi such as oe , oe * , OE , s , AE , f , d and dn . Keywords: functional programming, -calculi, explicit substitutions, confluence, extensionality. 1 Introduction The -calculus is a convenient framework to study functional programming, where the evaluation process is modeled by fi-reduction. The main mechanism used to perform fi-reduction is substitution, which consists of the replacement of formal parameters by actual arguments. The correctness of substitution is guaranteed by a systematic renaming of bound variables, inconvenient which can be simply avoided in the -calculus `a la de Bruijn by using natur...

. We prove the confluence and strong normalization properties for second order lambda calculus equipped with an expansive version of j-reduction. Our proof technique, based on a simple abstract lemma and a labelled -calculus, can also be successfully used to simplify the proofs of confluence and normalization for first order calculi, and can be applied to various extensions of the calculus presented here. 1 Introduction The typed lambda calculus provides a convenient framework for studying functional programming and offers a natural formalism to deal with proofs in intuitionistic logic. It comes traditionally equipped with the fi equality (x:M)N = M [N=x] as fundamental computational mechanism, and with the j (extensional) equality x:Mx = M as a tool for reasoning about programs. This basic calculus can then be extended by adding further types, like products, unit and second order types, each coming with its own computational mechanism and/or its extensional equalities. To reason abou...

It is well known that confluence and strong normalization are preserved when combining algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual contraction rule for #, or recursion together with the usual contraction rule for surjective pairing. We show that confluence and strong normalization are modular properties for the combination of algebraic rewriting systems with typed lambda calculi enriched with expansive extensional rules for # and surjective pairing. We also show how to preserve confluence in a modular way when adding fixpoints to di#erent rewriting systems. This result is also obtained by a simple translation technique allowing to simulate bounded recursion. 1 Introduction Confluence and strong normalization for the combination of lambda calculus and algebraic rewriting systems have been the object of many studies [BT88, JO91, BTG94, HM90], where the modularity of these properties is s...

<F NaN><F NaN> A A ×B<F NaN><F NaN> oo # 1<F NaN><F NaN> // # 2 B That is: h = #f, g# = ## 1 # h, # 2 # h# Case C A<F NaN><F NaN> // in 1<F NaN><F NaN> << f y y y y y y y y y y y y y y y y y y y A +B<F NaN><F NaN> OO [f,g] h<F NaN><F NaN> OO B<F NaN><F NaN> oo in 2<F NaN><F NaN> bb g E E E E E E E E E E E E E E E E E E E That is: h = [f, g] = [h # in 1 , h # in 2 ] Roberto Di Cosmo Glasgow, September 96 4 From equations to rewriting Two choices to orient

. 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 calculi, for which we can establish systematically confluence and/or normalization (or decidability of equality) in a simple way. Many result are new, but we also discuss systems for which our technique allows to provide a much simpler proof than what can be found in the literature. 1 Introduction During a recent investigation of confluence and normalization properties of polymorphic lambda calculus with an expansive version of the # rule, we came across a nice lemma that gives a simple but quite powerful sufficient condition to check the Church Rosser property for a compound rewriting system...