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On Formalised Proofs of Termination of Recursive Functions
 In Proceedings of the Int. Conf. on Principles and Practice of Declarative Programming, volume 1702 of LNCS
, 1999
"... In proof checkers and theorem provers (e.g. Coq [4] and ProPre [13]) recursive de nitions of functions are shown to terminate automatically. ..."
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In proof checkers and theorem provers (e.g. Coq [4] and ProPre [13]) recursive de nitions of functions are shown to terminate automatically.
Types complets dans une extension du système AF2
"... In this paper, we extend the system AF2 in order to have the subject reduction for the βηreduction. We prove that the types with positive quantifiers are complete to models stable by weakheadexpansion. 1 ..."
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In this paper, we extend the system AF2 in order to have the subject reduction for the βηreduction. We prove that the types with positive quantifiers are complete to models stable by weakheadexpansion. 1
Church numerals, twice!
, 2002
"... This paper explains Church numerals, twice. The first explanation links Church numerals to Peano numerals via the wellknown encoding of data types in the polymorphic λcalculus. This view suggests that Church numerals are folds in disguise. The second explanation, which is more elaborate, but also ..."
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This paper explains Church numerals, twice. The first explanation links Church numerals to Peano numerals via the wellknown encoding of data types in the polymorphic λcalculus. This view suggests that Church numerals are folds in disguise. The second explanation, which is more elaborate, but also more insightful, derives Church numerals from first principles, that is, from an algebraic specification of addition and multiplication. Additionally, we illustrate the use of the parametricity theorem by proving exponentiation as reverse application correct. 1
CC+: An extension of the Calculus of Constructions with fixpoints
, 1993
"... We follow an original idea suggested by Constable and Smith [6, 7] providing a way for reasoning about non terminating computations in a typed framework. A former study has been worked out within NuPrl by Smith [21]. We investigate how these ideas can be developed within the Calculus of Construct ..."
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We follow an original idea suggested by Constable and Smith [6, 7] providing a way for reasoning about non terminating computations in a typed framework. A former study has been worked out within NuPrl by Smith [21]. We investigate how these ideas can be developed within the Calculus of Constructions (CC). The adaptation provides an conservative extension, denoted CC+. Strong normalisation for fireductions is preserved. We recover the alternate "recursive" coding for integers introduced in AF2 by Parigot [12, 13]. Thus, the computational behaviour for terms coding integers is improved. Moreover, as expected, all partial recursive functions are now definable. Relationships with primitive coding through "Church" integers within the pure Calculus is studied, giving some insights into logical expressiveness issue. All these results easily generalize to all the usual data structures.
LogiCal Project
, 2004
"... This document is the Reference Manual of version 8.0 of the COQ proof assistant. A companion volume, the COQ Tutorial, is provided for the beginners. It is advised to read the Tutorial first. A new book [13] on practical uses of the COQ system will be published in 2004 and is a good support for both ..."
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This document is the Reference Manual of version 8.0 of the COQ proof assistant. A companion volume, the COQ Tutorial, is provided for the beginners. It is advised to read the Tutorial first. A new book [13] on practical uses of the COQ system will be published in 2004 and is a good support for both the beginner and the advanced user.
A presentation of the CurryHoward Correspondance.
, 1997
"... These notes are extracted from the rst version of the paper iFrom Computation to Foundations: the calculus and its webbed modelsj. They nearly disappeared in the revised version of that paper, and we make them available separately. 0.1 calculus as a foundation for Programming Theory. calculu ..."
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These notes are extracted from the rst version of the paper iFrom Computation to Foundations: the calculus and its webbed modelsj. They nearly disappeared in the revised version of that paper, and we make them available separately. 0.1 calculus as a foundation for Programming Theory. calculus came back to the front of the scene in the sixties with the development of Computer Science, under the impulse of Landin [21] and Backus (cf. [1]) and generated the family of functional languages (Lisp [McCarthy 1960], Haskell, Miranda, ML, Caml, ...). In functional languages functions and functionals may be passed as arguments to a program as easily as concrete datas, which is not the case with imperative languages (Fortran, Pascal, C ) (the other conceptual dioeerences between imperative programming (founded by Von Neumann) and functional programming are clearly explained e.g. in the rst pages of [1]). See barendregt's survey [2]: The other main conceptual contribution of calculus to P...
Least and Greatest Fixed Points in Intuitionistic Natural Deduction
, 2002
"... This paper is a comparative study of a number of (intensionalsemantically distinct) least and greatest fixed point operators that naturaldeduction proof systems for intuitionistic logics can be extended with in a prooftheoretically defendable way. Eight pairs of such operators are analysed. The e ..."
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This paper is a comparative study of a number of (intensionalsemantically distinct) least and greatest fixed point operators that naturaldeduction proof systems for intuitionistic logics can be extended with in a prooftheoretically defendable way. Eight pairs of such operators are analysed. The exposition is centered around a cubeshaped classification where each node stands for an axiomatization of one pair of operators as logical constants by intended proof and reduction rules and each arc for a proof and reductionpreserving encoding of one pair in terms of another. The three dimensions of the cube reflect three orthogonal binary options: conventionalstyle vs. Mendlerstyle, basic (``[co]iterative'') vs. enhanced (``primitive[co]recursive''), simple vs. courseofvalue [co]induction. Some of the axiomatizations and encodings are wellknown; others, however, are novel; the classification into a cube is also new. The differences between the least fixed point operators considered are illustrated on the example of the corresponding natural number types.
The Coq Proof Assistant  Reference Manual V 5.10
, 1995
"... ion All Axiom Begin Cd Chapter Check CheckGuard CoFixpoint Compute Defined Definition Drop Elimination End Eval Explain Extraction Fact Fixpoint Focus for Go Goal Hint Hypothesis Immediate Induction Inductive Infix Inspect Lemma Let Local Minimality ML Module Modules Mutual Node Opaque Parameter Par ..."
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ion All Axiom Begin Cd Chapter Check CheckGuard CoFixpoint Compute Defined Definition Drop Elimination End Eval Explain Extraction Fact Fixpoint Focus for Go Goal Hint Hypothesis Immediate Induction Inductive Infix Inspect Lemma Let Local Minimality ML Module Modules Mutual Node Opaque Parameter Parameters Print Proofs Prop Pwd Qed Remark Require Restart Resume Save Scheme Script Search Section Set Show Silent States Suspend Syntactic Theorem Token Transparent Tree Type TypeSet Undo Unfocus Variable Variables Write Other keywords and user's tokens The following sequences of characters are also keywords:  : := = ? ?? !? !! ! ? ; # * , ? @ :: / ! You can add new tokens with the command Token (see section 5.7.4). New tokens must be sequences, without blanks, of characters taken from the following list: ! ? / "  + = ; ,  ! @ # % & ? * : ~ $ a..z A..Z ' 0..9 that do not start with a character from $ a..z A..Z ' 0..9 Lexical ambiguities are resolved according to the "longest m...