Hendrik P. Barendregt. The Lambda Calculus, Its Syntax and Semantics. North-Holland, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(fn x = Var(x 1) t This definition is slightly unusual and also rather inefficient, but it is more general than the usual definition (see [3] for example) which uses two recursive functions instead of one. The added generality enables us to easily state the substitution lemma of calculus [1]: Proposition 1. Let t be a term and f and g functions of type int term. Then: subst f (subst g t) subst ( subst f) o g) t This proposition does not even require totality of f and g, in the sense that a = b holds iff either the expressions a and b are both undefined or both are defined and ....

Hendrik P. Barendregt. The Lambda-Calculus, its Syntax and Semantics. NorthHolland, 1984.

....form is an implicit operation, e.g. when applying a substitution to a term. A substitution is in long fij normal form if all terms in the image of are in long fij normal form. The convention that ff equivalent terms are identified and that free and bound variables are kept disjoint (see also [5]) is used in the following. Furthermore, we assume that bound variables with different binders have different names. Define Dom( fX j X 6= Xg and Rng( S X2Dom( FV( X) Two substitutions are equal on a set of variables W , written as =W 0 , if ff = 0 ff for all ff 2 W . The ....

Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

....form is an implicit operation, e.g. when applying a substitution to a term. A substitution is in long fij normal form if all terms in the image of are in long fij normal form. The convention that ff equivalent terms are identified and that free and bound variables are kept disjoint (see also [5]) is used in the following. Furthermore, we assume that bound variables with different binders have different names. Define Dom( fX j X 6= Xg and Rng( S X2Dom( FV( X) Two substitutions are equal on a set of variables W , written as =W 0 , if ff = 0 ff for all ff 2 W . The ....

Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

....The Classical Proof In this section we generalize Aczel s [1] confluence proof from his consistent sets of contraction schemes to arbitrary OHRS. Note that the former are a proper subset of the latter. The proof proceeds roughly like the one for the untyped calculus due to Tait and Martin Lof [3]. Although we want to prove the confluence of OHRS, the first steps towards the standard proof work just as well for GHRS. Given a fixed GHRS R, parallel reduction w.r.t. R is the smallest relation R on terms which is closed under the following inference rules: s i R t i (i = 1; n) ....

Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984. 26

....in j normal form, e.g. X instead of xn:X(xn ) We assume that the transformation into long fij normal form is an implicit operation, e.g. when applying a substitution to a term. The convention that ff equivalent terms are identified and that free and bound variables are kept disjoint (see also [2]) is used in the following. Furthermore, we assume that bound variables with different binders have different names. Define Dom( fX j X 6= Xg and Rng( S X2Dom( FV( X) Two substitutions are equal on a set of variables W , written as =W 0 , if ff = 0 ff for all ff 2 W . A ....

Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

....by lazy narrowing. In that way we have a modular structure, and higherorder lazy narrowing or an equally powerful method is used only where it is really needed. 2 Preliminaries We follow the standard notation of term rewriting, see e.g. 5 ] For the standard theory of calculus we refer to [ 8, 1 ] and for higher order unification we refer to [ 29 ] We assume the following variable conventions: ffl F; G; H;P;X;Y free variables, ffl a; b; c; f; g (function) constants, ffl x; y; z bound variables and ffl ff; fi; type variables. Type judgements are written as t : Further, s and t ....

....to ffl function constants of order n 1 and ffl variables of order n. A substitution is in long fij normal form if all terms in the image of are in long fij normal form. The convention that ff equivalent terms are identified and that free and bound variables are kept disjoint (see also [ 1 ] ) is used in the following. Furthermore, we assume that bound variables with different binders have different names. Define Dom( fX j X 6= Xg and Rng( S X2Dom( FV( X) Two substitutions are equal on a set of variables W , written as =W 0 , if ff = 0 ff for all ff 2 W . The ....

Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

....p in A by B. A structural equality relation of HOL terms is induced by reduction ( X A)B [B=X ]A ( X CX) C where X is not free in C. It is well known, that the reduction relations , and are terminating and con uent on w ( so that there are unique normal forms (cf. Bar84] for an introduction) In HOL, the set of base types is fo; g for truth values and individuals. We will call a formula of type o a proposition, and a sentence if it is closed. We will assume that the signature contains logical constants for negation : o o , conjunction o o o , quanti cation ....

Hendrik P. Barendregt. The Lambda Calculus { Its Syntax and Semantics. North Holland, 1984.

....by a list of instructions. Applying these instructions to a term (at its root) yields either the empty list (rule not applicable) or a singleton list of terms containing the result of the application. This view is independent from the choice of a one step strategy (concerning strategies see [1] chapter 13, or [7] because the search for a redex has to be done elsewhere. For many step strategies such as lazy evaluation this is not quite true. But strategies do not affect the type anyway. Not all elements of the type [instruction] are proper instruction sequences, e.g. a Pop or a Right ....

Hendrik P. Barendregt. The Lambda-Calculus, its Syntax and Semantics. North-Holland, 1984.

....by a list of instructions. Applying these instructions to a term (at its root) yields either the empty list (rule not applicable) or a singleton list of terms containing the result of the application. This view is independent from the choice of a one step strategy (concerning strategies see [1] chapter 13, or [6] because the search for a redex has to be done elsewhere. For many step strategies such as lazy evaluation this is not quite true. But strategies do not affect the type anyway. Not all elements of the type [instruction] are proper instruction sequences, e.g. a Pop or a Right ....

Hendrik P. Barendregt. The Lambda-Calculus, its Syntax and Semantics. North-Holland, 1984.

....[23] G. Dowek 92 [7] Lucchesi 72 [20] 1 . decidable . D. Miller 91[21] Wolfram 92 [31] Fig. 1. Decidability of Higher Order Unification 2 Notation and Basic Definitions The following notation of calculus are used in the sequel. For the standard theory of calculus we refer to [ 15, 2 ] . We assume the following variable conventions: F; G; H;P;X;Y free variables, a; b; c; f; g (function) constants, x; y; z bound variables, ff; fi types. The following grammar defines the syntax for terms, t = F j x j c j x:t j (t 1 t 2 ) A list of syntactic objects s 1 ; ....

....variable. This restriction is necessary for unitary unification, but for our purpose this is not relevant and we will henceforth work with relaxed higher order patterns and call these patterns for brevity. We identify ff equivalent terms and assume that free and bound variables are kept disjoint [ 2 ] . Furthermore, we assume that bound variables with different binders have different names. 3 Pre unification by Transformations We present in the following a version of the transformation system PT for higherorder unification of Snyder and Gallier [ 30 ] More precisely, we use the primed ....

Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

....form is an implicit operation, e.g. when applying a substitution to a term. A substitution is in long fij normal form if all terms in the image of are in long fij normal form. The convention that ff equivalent terms are identified and that free and bound variables are kept disjoint (see also [4]) is used in the following. Furthermore, we assume that bound variables with different binders have different names. Define Dom( fX j X 6= Xg and Rng( S X2Dom( FV( X) Two substitutions are equal on a set of variables W , written as =W 0 , if ff = 0 ff for all ff 2 W . A ....

Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

....The Classical Proof In this section we generalize Aczel s [1] confluence proof from his consistent sets of contraction schemes to arbitrary OPRSs. Note that the former are a proper subset of the latter. The proof proceeds roughly like the one for the untyped calculus due to Tait and Martin Lof [3]. Although we want to prove the confluence of OPRSs, the first steps towards the standard proof work just as well for HRSs. Given a fixed HRS R, parallel reduction w.r.t. R is the smallest relation on terms which is closed under the following inference rules: s i t i (i = 1; n) a(s n ....

Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

....form is an implicit operation, e.g. when applying a substitution to a term. A substitution is in long fij normal form if all terms in the image of are in long fij normal form. The convention that ff equivalent terms are identified and that free and bound variables are kept disjoint (see also [3]) is used in the following. Furthermore, we assume that bound variables with different binders have different names. Define Dom( fX j X 6= Xg and Rng( S X2Dom( FV( X) Two substitutions are equal on a set of variables W , written as =W 0 , if ff = 0 ff for all ff 2 W . ....

Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

....of the Danish National Research Foundation) Bloomington, IN 47405, USA. 1 Introduction 1.1 Numeral Systems in the Calculus Numbers are traditionally represented on computers with a size proportional to their logarithm. Traditional numeral systems in the calculus, such as Church numerals [1, 3] and Barendregt numerals [1] however, typically involve linear representations of numbers. In such systems, the size of the representation of a number n is proportional to n. In this paper, we present an adequate binary numeral system for the calculus, where the successor function, the ....

....Foundation) Bloomington, IN 47405, USA. 1 Introduction 1.1 Numeral Systems in the Calculus Numbers are traditionally represented on computers with a size proportional to their logarithm. Traditional numeral systems in the calculus, such as Church numerals [1, 3] and Barendregt numerals [1], however, typically involve linear representations of numbers. In such systems, the size of the representation of a number n is proportional to n. In this paper, we present an adequate binary numeral system for the calculus, where the successor function, the predecessor function, and the test ....

[Article contains additional citation context not shown here]

Hendrik P. Barendregt. The Lambda Calculus, Its Syntax and Semantics. North-Holland, 1984.

....of the BRICS Technical Report RS 95 42 [6] Bloomington, IN 47405, USA. 1 Introduction 1.1 Numeral Systems in the Calculus Numbers are traditionally represented on computers with a size proportional to their logarithm. Traditional numeral systems in the calculus, such as Church numerals [1, 4] and Barendregt numerals [1] however, typically involve linear representations of numbers. In such systems, the size of the representation of a number n is proportional to this number. In this paper, we present an adequate binary numeral system for the calculus, where the successor function, ....

....RS 95 42 [6] Bloomington, IN 47405, USA. 1 Introduction 1.1 Numeral Systems in the Calculus Numbers are traditionally represented on computers with a size proportional to their logarithm. Traditional numeral systems in the calculus, such as Church numerals [1, 4] and Barendregt numerals [1], however, typically involve linear representations of numbers. In such systems, the size of the representation of a number n is proportional to this number. In this paper, we present an adequate binary numeral system for the calculus, where the successor function, the predecessor function, and ....

[Article contains additional citation context not shown here]

Hendrik P. Barendregt. The Lambda Calculus, Its Syntax and Semantics. North-Holland, 1984.

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Hendrik Pieter Barendregt. The Lambda Calculus, Its syntax and Semantics. North Holland, 1981.

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Hendrik P. Barendregt. The Lambda Calculus, Its Syntax and Semantics. North-Holland, 1984.

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Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

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Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

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Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

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Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics.North Holland, 2nd edition, 1984.

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Hendrik Pieter Barendregt. The Lambda Calculus, Its syntax and Semantics. North Holland, 1981.

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Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

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