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Termination Checking with Types
, 1999
"... The paradigm of typebased termination is explored for functional programming with recursive data types. The article introduces , a lambdacalculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types ..."
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Cited by 36 (6 self)
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The paradigm of typebased termination is explored for functional programming with recursive data types. The article introduces , a lambdacalculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types are used to track the size of function arguments and return values. The system is shown to be type safe and strongly normalizing. The main novelty is a bidirectional type checking algorithm whose soundness is established formally.
Semicontinuous sized types and termination
 In Zoltán Ésik, editor, Computer Science Logic, 20th International Workshop, CSL 2006, 15th Annual Conference of the EACSL
"... Abstract. Some typebased approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is onl ..."
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Abstract. Some typebased approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is only sound if the type of the recursive function is admissible, i.e., depends on the size index in a certain way. To explore the space of admissible functions in the presence of higherkinded data types and impredicative polymorphism, a semantics is developed where sized types are interpreted as functions from ordinals into sets of strongly normalizing terms. It is shown that upper semicontinuity of such functions is a sufficient semantic criterion for admissibility. To provide a syntactical criterion, a calculus for semicontinuous functions is developed. 1.
Implementing a Normalizer Using Sized Heterogeneous Types
 Journal of Functional Programming, MSFP’06 special issue
"... In the simplytyped lambdacalculus, a hereditary substitution replaces a free variable in a normal form r by another normal form s of type a, removing freshly created redexes on the fly. It can be defined by lexicographic induction on a and r, thus, giving rise to a structurally recursive normalize ..."
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Cited by 13 (2 self)
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In the simplytyped lambdacalculus, a hereditary substitution replaces a free variable in a normal form r by another normal form s of type a, removing freshly created redexes on the fly. It can be defined by lexicographic induction on a and r, thus, giving rise to a structurally recursive normalizer for the simplytyped lambdacalculus. We generalize this scheme to simultaneous substitutions, preserving its simple termination argument. We further implement hereditary simultaneous substitutions in a functional programming language with sized heterogeneous inductive types, Fωb, arriving at an interpreter whose termination can be tracked by the type system of its host programming language.
Stop when you are AlmostFull Adventures in constructive termination
"... Disjunctive wellfoundedness (used in Terminator), sizechange termination, and wellquasiorders (used in supercompilation and termrewrite systems) are examples of techniques that have been successfully applied to automatic proofs of program termination and online termination testing, respectively ..."
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Disjunctive wellfoundedness (used in Terminator), sizechange termination, and wellquasiorders (used in supercompilation and termrewrite systems) are examples of techniques that have been successfully applied to automatic proofs of program termination and online termination testing, respectively. Although these works originate in different communities, there is an intimate connection between them – they rely on closely related principles and both employ similar arguments from Ramsey theory. At the same time there is a notable absence of these techniques in programming systems based on constructive type theory. In this paper we’d like to highlight the aforementioned connection and make the core ideas widely accessible to theoreticians and Coq programmers, by offering a Coq development which culminates in some novel tools for performing induction. The benefit is nice composability properties of termination arguments at the cost of intuitive and lightweight user obligations. Inevitably, we have to present some Ramseylike arguments: Though similar proofs are typically classical, we offer an entirely constructive development standing on the shoulders of Veldman and Bezem, and Richman and Stolzenberg. 1.
Standardization for the Coinductive
, 2002
"... In the calculus Λ co of possibly nonwellfounded λterms, standardization is proved for a parallel notion of reduction. For this system confluence has recently been established by means of a bounding argument for the number of reductions provoked by the joining function which witnesses the confluenc ..."
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In the calculus Λ co of possibly nonwellfounded λterms, standardization is proved for a parallel notion of reduction. For this system confluence has recently been established by means of a bounding argument for the number of reductions provoked by the joining function which witnesses the confluence statement. Similarly, bounds have to be introduced in order to turn the proof of standardization for the wellfounded λcalculus into a sound coinductive argument, thus limiting the number of reduction steps arising in the process of standardization. This leads to elementary complexity bounds for the length of the resulting standard reduction sequence in terms of the length of the input sequence. A fortiori, these bounds also apply to the usual wellfounded λcalculus, strengthening previous results by Xi.
Standardization for the Coinductive LambdaCalculus
, 2002
"... In the calculus of possibly nonwellfounded terms, standardization is proved for a parallel notion of reduction. For this system confluence has recently been established by means of a bounding argument for the number of reductions provoked by the joining function which witnesses the conflue ..."
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In the calculus of possibly nonwellfounded terms, standardization is proved for a parallel notion of reduction. For this system confluence has recently been established by means of a bounding argument for the number of reductions provoked by the joining function which witnesses the confluence statement. Similarly,