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TypeBased Termination of Recursive Definitions
, 2002
"... This article The purpose of this paper is to introduce b, a simply typed calculus that supports typebased recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and CoupetGrimal 1998), the techn ..."
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Cited by 53 (4 self)
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This article The purpose of this paper is to introduce b, a simply typed calculus that supports typebased recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and CoupetGrimal 1998), the technical machinery behind our system puts a slightly different emphasis on the interpretation of types. More precisely, we formalize the notion of typebased termination using a restricted form of type dependency (a.k.a. indexed types), as popularized by (Xi and Pfenning 1998; Xi and Pfenning 1999). This leads to a simple and intuitive system which is robust under several extensions, such as mutually inductive datatypes and mutually recursive function definitions; however, such extensions are not treated in the paper
MendlerStyle Inductive Types, Categorically
 NORDIC JOURNAL OF COMPUTING 6(1999), 343 361
, 1999
"... We present a basis for a categorytheoretic account of Mendlerstyle inductive types. The account is based on suitably defined concepts of Mendlerstyle algebra and algebra homomorphism; Mendlerstyle inductive types are identified with initial Mendlerstyle algebras. We use the identification to ob ..."
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Cited by 8 (4 self)
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We present a basis for a categorytheoretic account of Mendlerstyle inductive types. The account is based on suitably defined concepts of Mendlerstyle algebra and algebra homomorphism; Mendlerstyle inductive types are identified with initial Mendlerstyle algebras. We use the identification to obtain a reduction of conventional inductive types to Mendlerstyle inductive types and a reduction in the presence of certain restricted existential types of Mendlerstyle inductive types to conventional inductive types.
On the formalization of the modal µcalculus in the Calculus of Inductive Constructions
 Information and Computation
, 2000
"... This paper is part of an ongoing research programme at the Computer Science Department of the University of Udine on proof editors, started in 1992, based on HOAS encodings in dependent typed #calculus for program logics [15, 21, 16]. In this paper, we investigate the applicability of this approach ..."
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Cited by 6 (0 self)
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This paper is part of an ongoing research programme at the Computer Science Department of the University of Udine on proof editors, started in 1992, based on HOAS encodings in dependent typed #calculus for program logics [15, 21, 16]. In this paper, we investigate the applicability of this approach to the modal calculus. Due to its expressive power, we adopt the Calculus of Inductive Constructions (CIC), implemented in the system Coq. Beside its importance in the theory and verification of processes, the modal calculus is interesting also for its syntactic and proof theoretic peculiarities. These idiosyncrasies are mainly due to a) the negative arity of "" (i.e., the bound variable x ranges over the same syntactic class of x#); b) a contextsensitive grammar due the condition on x#; c) rules with complex side conditions (sequentstyle "proof " rules). These anomalies escape the "standard" representation paradigm of CIC; hence, we need to accommodate special techniques for enforcing these peculiarities. Moreover, since generated editors allow the user to reason "under assumptions", the designer of a proof editor for a given logic is urged to look for a Natural Deduction formulation of the system. Hence, we introduce a new proof system N # K in Natural Deduction style for K. This system should be more natural to use than traditional Hilbertstyle systems; moreover, it takes best advantage of the possibility of manipulating assumptions o#ered by CIC in order to implement the problematic substitution of formul for variables. In fact, substitutions are delayed as much as possible, and are kept in the derivation context by means of assumptions. This mechanism fits perfectly the stack discipline of assumptions of Natural Deduction, and it is neatly formalized in CIC. Bes...
Coding Recursion a la Mendler (Extended Abstract)
 Department of Computer Science, Utrecht University
, 2000
"... Abstract We advocate the Mendler style of coding terminating recursion schemes as combinators by showing on the example of two simple and much used schemes (courseofvalue iteration and simultaneous iteration) that choosing the Mendler style can sometimes lead to handier constructions than followin ..."
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Cited by 4 (1 self)
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Abstract We advocate the Mendler style of coding terminating recursion schemes as combinators by showing on the example of two simple and much used schemes (courseofvalue iteration and simultaneous iteration) that choosing the Mendler style can sometimes lead to handier constructions than following the construction style of cata and para like combinators. 1 Introduction This paper is intended as an advert for something we call the Mendler style. This is a not too widely known style of coding terminating recursion schemes by combinators that di ers from the construction style of the famous cata and para combinators (for iteration and primitiverecursion, respectively) [Mal90,Mee92], here called the conventional style. The paper ar...
Under consideration for publication in Math. Struct. in Comp. Science Typebased termination of recursive
, 2000
"... The paper introduces λ ̂ , a simply typed lambda calculus supporting inductive types and recursive function definitions with termination ensured by types. The system is shown to enjoy subject reduction, strong normalization of typable terms and to be stronger than a related system λG in which termin ..."
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The paper introduces λ ̂ , a simply typed lambda calculus supporting inductive types and recursive function definitions with termination ensured by types. The system is shown to enjoy subject reduction, strong normalization of typable terms and to be stronger than a related system λG in which termination is ensured by a syntactic guard condition. The system can, at will, be extended to also support coinductive types and corecursive function definitions. 1.
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.
MendlerStyle Inductive Types, Categorically (Extended Abstract)
"... We present a basis for a categorical account of Mendlerstyle inductive types by introducing a notion of initial Mendlerstyle algebras and use it for giving a reduction of (conventional) inductive types to Mendlerstyle inductive types and two reductions of Mendlerstyle inductive types to (convent ..."
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We present a basis for a categorical account of Mendlerstyle inductive types by introducing a notion of initial Mendlerstyle algebras and use it for giving a reduction of (conventional) inductive types to Mendlerstyle inductive types and two reductions of Mendlerstyle inductive types to (conventional) inductive types.