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31
Indexed InductionRecursion
, 2001
"... We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in ..."
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Cited by 50 (17 self)
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We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in intuitionistic type theory. The more restricted of the two axiomatization arises naturally by considering indexed inductiverecursive de nitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.
Modelling General Recursion in Type Theory
 Mathematical Structures in Computer Science
, 2002
"... Constructive type theory is an expressive programming language where both algorithms and proofs can be represented. However, general recursive algorithms have no direct formalisation in type theory since they contain recursive calls that satisfy no syntactic condition guaranteeing termination. ..."
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Cited by 44 (6 self)
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Constructive type theory is an expressive programming language where both algorithms and proofs can be represented. However, general recursive algorithms have no direct formalisation in type theory since they contain recursive calls that satisfy no syntactic condition guaranteeing termination.
General recursion via coinductive types
 Logical Methods in Computer Science
"... Vol. 1 (2:1) 2005, pp. 1–28 ..."
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Recursive coalgebras from comonads
 Inform. and Comput
, 2006
"... The concept of recursive coalgebra of a functor was introduced in the 1970s by Osius in his work on categorical set theory to discuss the relationship between wellfounded induction and recursively specified functions. In this paper, we motivate the use of recursive coalgebras as a paradigm of struct ..."
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Cited by 15 (3 self)
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The concept of recursive coalgebra of a functor was introduced in the 1970s by Osius in his work on categorical set theory to discuss the relationship between wellfounded induction and recursively specified functions. In this paper, we motivate the use of recursive coalgebras as a paradigm of structured recursion in programming semantics, list some basic facts about recursive coalgebras and, centrally, give new conditions for the recursiveness of a coalgebra based on comonads, comonadcoalgebras and distributive laws of functors over comonads. We also present an alternative construction using countable products instead of cofree comonads.
Inductive and coinductive components of corecursive functions
 in Coq. ENTSC 203
, 2008
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QArith: Coq formalisation of lazy rational arithmetic
 Types for Proofs and Programs, volume 3085 of LNCS
, 2003
"... Abstract. In this paper we present the Coq formalisation of the QArith library which is an implementation of rational numbers as binary sequences for both lazy and strict computation. We use the representation also known as the SternBrocot representation for rational numbers. This formalisation use ..."
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Cited by 10 (2 self)
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Abstract. In this paper we present the Coq formalisation of the QArith library which is an implementation of rational numbers as binary sequences for both lazy and strict computation. We use the representation also known as the SternBrocot representation for rational numbers. This formalisation uses advanced machinery of the Coq theorem prover and applies recent developments in formalising general recursive functions. This formalisation highlights the rôle of type theory both as a tool to verify handwritten programs and as a tool to generate verified programs. 1
Tait in one big step
 In MSFP 2006
, 2006
"... We present a Taitstyle proof to show that a simple functional normaliser for a combinatory version of System T terminates. The main interest in our construction is methodological, it is an alternative to the usual smallstep operational semantics on the one side and normalisation by evaluation on t ..."
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Cited by 10 (6 self)
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We present a Taitstyle proof to show that a simple functional normaliser for a combinatory version of System T terminates. The main interest in our construction is methodological, it is an alternative to the usual smallstep operational semantics on the one side and normalisation by evaluation on the other. Our work is motivated by our goal to verify implementations of Type Theory such as Epigram. Keywords: Normalisation,Strong Computability 1.
Bigstep Normalisation
 UNDER CONSIDERATION FOR PUBLICATION IN J. FUNCTIONAL PROGRAMMING
, 2007
"... Traditionally, decidability of conversion for typed λcalculi is established by showing that smallstep reduction is confluent and strongly normalising. Here we investigate an alternative approach employing a recursively defined normalisation function which we show to be terminating and which reflec ..."
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Cited by 7 (3 self)
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Traditionally, decidability of conversion for typed λcalculi is established by showing that smallstep reduction is confluent and strongly normalising. Here we investigate an alternative approach employing a recursively defined normalisation function which we show to be terminating and which reflects and preserves conversion. We apply our approach to the simplytyped λcalculus with explicit substitutions and βηequality, a system which is not strongly normalising. We also show how the construction can be extended to System T with the usual βrules for the recursion combinator. Our approach is practical, since it does verify an actual implementation of normalisation which, unlike normalisation by evaluation, is first order. An important feature of our approach is that we are using logical relations to establish equational soundness (identity of normal forms reflects the equational theory), instead of the usual syntactic reasoning using the ChurchRosser property of a term rewriting system.
Inductive invariants for nested recursion
 Theorem Proving in Higher Order Logics (TPHOLS'03), volume 2758 of LNCS
, 2003
"... Abstract. We show that certain inputoutput relations, termed inductive invariants are of central importance for termination proofs of algorithms defined by nested recursion. Inductive invariants can be used to enhance recursive function definition packages in higherorder logic mechanizations. We d ..."
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Cited by 5 (2 self)
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Abstract. We show that certain inputoutput relations, termed inductive invariants are of central importance for termination proofs of algorithms defined by nested recursion. Inductive invariants can be used to enhance recursive function definition packages in higherorder logic mechanizations. We demonstrate the usefulness of inductive invariants on a large example of the BDD algorithm Apply. Finally, we introduce a related concept of inductive fixpoints with the property that for every functional in higherorder logic there exists a largest partial function that is such a fixpoint. 1
Coinductive Field of Exact Real Numbers and General Corecursion
, 2006
"... In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be se ..."
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Cited by 3 (0 self)
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In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be seen as coalgebra maps on the coalgebra of streams and hence they will be formalised as general corecursive functions. We use the machinery of Coq proof assistant for coinductive types to present the formalisation.