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"... : In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between propositions and types: the Curry-Howard Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to rep ..."

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: In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between propositions and types: the Curry-Howard Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to represent incomplete proofs as terms describing a state of knowledge where some part of the proof is built, but another part remains undeveloped. The pieces of proof terms that are unknown are called places-holders. We present a theoretical approach to place-holders in the typed -calculus. In this approach place-holders are represented by metavariables and terms are built incrementally by instantiation of metavariables. We show how an appropriate extension to typed -calculus with explicit substitutions and explicit typing of metavariables allows to identify terms construction and proofs development activities. Preliminaries We recall a few concepts about intuitionistic logic, typed -calculus...

### Closed Reductions in the λ-calculus (Extended Abstract)

, 1999

"... Closed reductions in the lamda-calculus is a strategy for a calculus of explicit substitutions which overcomes many of the usual syntactical problems of substitution. This is achieved by only moving closed substitutions through certain constructs, which gives a weak form of reduction, but is rich en ..."

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Closed reductions in the lamda-calculus is a strategy for a calculus of explicit substitutions which overcomes many of the usual syntactical problems of substitution. This is achieved by only moving closed substitutions through certain constructs, which gives a weak form of reduction, but is rich enough to capture the usual strategies in the lambda-calculus (call-by-value, call-by-need, etc.) and is adequate for the evaluation of programs. An interesting point is that the calculus permits substitutions to move through abstractions, and reductions are allowed under abstractions, if certain conditions hold. The calculus naturally provides an efficient notion of reduction (with a high degree of sharing), which can easily be implemented....

### Postponement, Conservation and Preservation of Strong Normalisation for Generalised Reduction

"... Postponement of K -contractions and the conservation theorem do not hold for ordinary but have been established by de Groote for a mixture of with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation e which generalises . We show morever, t ..."

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Postponement of K -contractions and the conservation theorem do not hold for ordinary but have been established by de Groote for a mixture of with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation e which generalises . We show morever, that e has the Preservation of Strong Normalisation property. Keywords: Generalised -reduction, Postponement of K-contractions, Generalised Conservation, Preservation of Strong Normalisation. 1 The -calculus with generalized reduction In the term (( x : y :N)P )Q, the abstraction starting with x and the argument P form the redex ( x : y :N)P . When this redex is contracted, the abstraction starting with y and Q will in turn form a redex. It is important to note that Q (or some residual of Q) is the only argument that the abstraction (or some residual of the abstraction) starting with y can ever have. This fact has been exploited by many researchers. Reduction has been ex...

### Meta-theoretical properties of ...: A left-linear variant of ...

, 1997

"... : In this paper we consider explicit substitutions calculi that allow open terms. In particular, we propose a variant of the oe -calculus, that we call OE . For this calculus and its simply-typed version, we study its meta-theoretical properties. The OE -calculus enjoys the same general character ..."

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: In this paper we consider explicit substitutions calculi that allow open terms. In particular, we propose a variant of the oe -calculus, that we call OE . For this calculus and its simply-typed version, we study its meta-theoretical properties. The OE -calculus enjoys the same general characteristics as oe , i.e. a simple and finitary first-order presentation, confluent on terms with meta-variables, with a composition operator and with simultaneous substitutions. However, OE does not have the non-left-linear surjective pairing rule of oe which raises technical problems in some frameworks. (R'esum'e : tsvp) Cesar.Munoz@inria.fr Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33 1) 39 63 55 11 -- T'el'ecopie : (33 1) 39 63 53 30 Propri'et'es m'eta-th'eoriques de OE : Une variante lin'eaire `a gauche de oe R'esum'e : Dans cet article, on s'int'eresse aux calculs avec substitutions explicites q...

### On Explicit Binding and Substitution Preserving Strong Normalisation (Extended Abstract)

, 1996

"... In recent years a large number of `explicit substitution calculi' have been proposed with various combinations of properties. One property that has attracted special attention is `PSN:' whether the set of fi-strongly normalising terms is still strongly normalising with explicit substitution. Sever ..."

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In recent years a large number of `explicit substitution calculi' have been proposed with various combinations of properties. One property that has attracted special attention is `PSN:' whether the set of fi-strongly normalising terms is still strongly normalising with explicit substitution. Several calculi with this property have been found: we discuss AE, Ø, s, t, and x; in this note we add two new variants: AE1 and AE0. We show that these calculi all have essentially the same reductions, or put differently: the renaming overhead is negligible with respect to normalisation. Furthermore x -- the only one of the lot with implicit binding using usual -calculus variables -- is a least common denominator in the sense that all the others are (strict) conservative extensions of it. A consequence of this is that all the PSN results proven for these calculi are equivalent (and follow from PSN for x).

### A Left-linear Variant of λσ

, 1997

"... In this paper we consider -calculi of explicit substitutions that admit open expressions, i.e. expressions with meta-variables. In particular, we propose a variant of the oe-calculus that we call L . For this calculus and its simply-typed version, we study its meta-theoretical properties. The L-cal ..."

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In this paper we consider -calculi of explicit substitutions that admit open expressions, i.e. expressions with meta-variables. In particular, we propose a variant of the oe-calculus that we call L . For this calculus and its simply-typed version, we study its meta-theoretical properties. The L-calculus enjoys the same general characteristics as oe, i.e. a simple and finitary first-order presentation, confluent on expressions with meta-variables of terms and weakly normalizing on typed expressions. Moreover, L does not have the non-left-linear surjective pairing rule of oe which raises technical problems in some frameworks.

### SUBSTITUTIONS

"... Abstract. Typed λ-terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the Curry-Howard isomorphism relates proof trees with typed λ-terms. The proofs-as-terms principle can be used to check a proof by type checking the λ-term extracted from ..."

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Abstract. Typed λ-terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the Curry-Howard isomorphism relates proof trees with typed λ-terms. The proofs-as-terms principle can be used to check a proof by type checking the λ-term extracted from the complete proof tree. However, proof trees and typed λ-terms are built differently. Usually, an auxiliary representation of unfinished proofs is needed, where type checking is possible only on complete proofs. In this paper we present a proof synthesis method for dependent-type systems where typed open terms are built incrementally at the same time as proofs are done. This way, every construction step, not just the last one, may be type checked. The method is based on a suitable calculus where substitutions as well as meta-variables are first-class objects.

### Operated by Universities Space Research Association

"... CÉSAR MUÑOZ∗ Abstract. We present a dependent-type system for a λ-calculus with explicit substitutions. In this system, meta-variables, as well as substitutions, are first-class objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak ..."

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CÉSAR MUÑOZ∗ Abstract. We present a dependent-type system for a λ-calculus with explicit substitutions. In this system, meta-variables, as well as substitutions, are first-class objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.

### Redexes in Item Notation Classical Notation Item Notation

, 2009

"... • I(λx.B) = [x]I(B) and I(AB) = (I(B))I(A) • I((λx.(λy.xy))z) ≡ (z)[x][y](y)x. The items are (z), [x], [y] and (y). • applicator wagon (z) and abstractor wagon [x] occur NEXT to each other. • A term is a wagon followed by a term. • (β) ..."

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• I(λx.B) = [x]I(B) and I(AB) = (I(B))I(A) • I((λx.(λy.xy))z) ≡ (z)[x][y](y)x. The items are (z), [x], [y] and (y). • applicator wagon (z) and abstractor wagon [x] occur NEXT to each other. • A term is a wagon followed by a term. • (β)