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Syntax and Semantics of Dependent Types
 Semantics and Logics of Computation
, 1997
"... ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe ..."
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Cited by 40 (4 self)
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ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe is written Set instead of U . The Eloperator is omitted. For example the \Pitype is described by the following constant and equality declarations (understood in every valid context): ` \Pi : (oe: Set; : (oe)Set)Set ` App : (oe: Set; : (oe)Set; m: \Pi(oe; ); n: oe) (m) ` : (oe: Set; : (oe)Set; m: (x: oe) (x))\Pi(oe; ) oe: Set; : (oe)Set; m: (x: oe) (x); n: oe ` App(oe; ; (oe; ; m); n) = m(n) Notice, how terms with free variables are represented as framework abstractions (in the type of ) and how substitution is represented as framework application (in the type of App and in the equation). In this way the burden of dealing correctly with variables, substitution, and binding is s...
Internal Type Theory
 Lecture Notes in Computer Science
, 1996
"... . We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with f ..."
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Cited by 37 (7 self)
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. We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with families formally in MartinLof's intensional intuitionistic type theory. Finally, we discuss the coherence problem for these internal categories with families. 1 Introduction In a previous paper [8] I introduced a general notion of simultaneous inductiverecursive definition in intuitionistic type theory. This notion subsumes various reflection principles and seems to pave the way for a natural development of what could be called "internal type theory", that is, the construction of models of (fragments of) type theory in type theory, and more generally, the formalization of the metatheory of type theory in type theory. The present paper is a first investigation of such an internal type theor...
Constructions, Inductive Types and Strong Normalization
, 1993
"... This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and typechecking, based on the equalityasjudgement presentation. We present a settheoretic notio ..."
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Cited by 31 (2 self)
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This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and typechecking, based on the equalityasjudgement presentation. We present a settheoretic notion of model, CCstructures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to nonalgebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a nontrivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the...
From semantics to rules: A machine assisted analysis
 Proceedings of CSL '93, LNCS 832
, 1999
"... this paper is similar to the one in [2]. In this paper they define a normalization function for simply typed ..."
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Cited by 29 (0 self)
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this paper is similar to the one in [2]. In this paper they define a normalization function for simply typed
A Calculus of Substitutions for IncompleteProof Representation in Type Theory
, 1997
"... : In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the CurryHoward isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambdaterms. In order to see the pr ..."
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Cited by 16 (1 self)
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: In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the CurryHoward isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambdaterms. In order to see the process of proof construction as an incremental process of term construction, it is necessary to extend the lambdacalculus with new operators. First, we consider typed metavariables to represent the parts of a proof that are under construction, and second, we make explicit the substitution mechanism in order to deal with capture of variables that are bound in terms containing metavariables. Unfortunately, the theory of explicit substitution calculi with typed metavariables is more complex than that of lambdacalculus. And worse, in general they do not share the same properties, notably with respect to confluence and strong normalization. A contribution of this thesis is to show that the pr...
Normalization for Typed Lambda Calculi with Explicit Substitution
 University of Cambridge, Computer Laboratory, Technical Report
, 1994
"... This paper shows that the strong normalization property holds for a restricted class of reductions: those which push a substitution under a abstraction only if this is the outermost constructor. All standard implementations of the typed calculus, like those using a lazy or eager strategy, have th ..."
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This paper shows that the strong normalization property holds for a restricted class of reductions: those which push a substitution under a abstraction only if this is the outermost constructor. All standard implementations of the typed calculus, like those using a lazy or eager strategy, have this property, hence we can conclude that they terminate. Furthermore, this result means that an implementation of a typed 
Dependent Types and Explicit Substitutions
, 1999
"... We present a dependenttype system for a #calculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization. ..."
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We present a dependenttype system for a #calculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.
On Explicit Substitutions and Names
 In Proc. ICALP
, 1997
"... Calculi with explicit substitutions have found widespread acceptance as a basis for abstract machines for functional languages. In this paper we investigate the relations between variants with de Bruijnnumbers, with variable names, with reduction based on raw expressions and calculi with equational ..."
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Cited by 2 (1 self)
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Calculi with explicit substitutions have found widespread acceptance as a basis for abstract machines for functional languages. In this paper we investigate the relations between variants with de Bruijnnumbers, with variable names, with reduction based on raw expressions and calculi with equational judgements. We show the equivalence between these variants, which is crucial in establishing the correspondence between the semantics of the calculus and its implementations. 1 Introduction Explicit substitution calculi (or oecalculi for short) first appeared in a seminal paper by Abadi et al. [1]. The basic idea is that instead of having substitutions as a metalevel operation, as in traditional calculus, we should make them part of the objectlevel calculus. The advantages of this approach are twofold. Firstly, it makes it possible to design much more efficient abstract machines as we are allowed to delay substitutions, and secondly it makes it much easier to prove them correct since...
Syntactic Multicategories and Categorical Combinators for Linear Logic
, 1993
"... ) Eike Ritter Valeria de Paiva Computer Laboratory University of Cambridge 1 Introduction This paper contributes to the area of "categorical combinatory logic" or "categorical combinators", following the steps of Curien [Cur93] and Ritter [Rit92]. We provide a precise syntactic formulation of the n ..."
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) Eike Ritter Valeria de Paiva Computer Laboratory University of Cambridge 1 Introduction This paper contributes to the area of "categorical combinatory logic" or "categorical combinators", following the steps of Curien [Cur93] and Ritter [Rit92]. We provide a precise syntactic formulation of the notion of a multicategory (a recent reference is Lambek [Lam89]) and based on that we give categorical combinators for (multiplicative) Intuitionistic Linear Logic, following the general approach of Ritter for the Calculus of Constructions [Rit92]. Multicategories are usually thought of as "just like categories, except that instead of arrows A ! B one has multiarrows An ; : : : ; A 1 ! B". There is a wellknown correspondence between multicategories and (rudimentary) linear logic in such a way that a multimap f : An ; : : : ; A 1 ! B above corresponds to a term denoting a proof of B from the assumptions x i : A i . But the picture of a multicategory referred to above is lacking in two accou...
Rewriting Properties of Combinators for Intuitionistic Linear Logic
 In Proceedings of the Workshop on Higher Order Algebra, Logic and Term Rewriting
, 1994
"... this paper we investigate the possibility of developing a (semi)automatic rewriting tool for manipulating and reasoning about combinators for Intuitionistic Linear Logic. In particular, we develop a canonical (i.e. confluent and terminating) term rewriting system associated to a theory of categoric ..."
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this paper we investigate the possibility of developing a (semi)automatic rewriting tool for manipulating and reasoning about combinators for Intuitionistic Linear Logic. In particular, we develop a canonical (i.e. confluent and terminating) term rewriting system associated to a theory of categorical combinators for (rudimentary) Linear Logic. In order to do that, we make use of the KnuthBendix completion algorithm [3] to transform the equational theory for the combinators into an equivalent canonical rewrite system. This means that a set of categorical combinators for Linear Logic has first to be derived, and then the resulting system of combinators can be checked for rewriting properties using rewriting techniques. The process of deriving categorical combinators has been a relatively long one, with an interesting interaction between the more abstract side of the work (reported in [7]) and its mechanized version. We started with a first (theoretically correct) formulation of the combinators for each of the several fragments of the logic under consideration. As usual, the use of an automatic rewriting tool to derive and check properties such as local confluence, termination and canonicity of a system, has led us to "improve" the first axiomatic characterization of the categorical combinators, until an equivalent presentation was derived with the nice feature that rewriting techniques may be applied to it successfully. This process of transformation of the data provided by the theoretical considerations follows a definite pattern that is described in Section 2.3. Intuitionistic Linear Logic was introduced by Girard and Lafont in [4]. The basic assumption of Linear Logic is that one should be able to have a logical control of the resources available for a derivation. Th...