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27
Combining Testing and Proving in Dependent Type Theory
 16th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2003
, 2003
"... We extend the proof assistant Agda/Alfa for dependent type theory with a modi ed version of Claessen and Hughes' tool QuickCheck for random testing of functional programs. In this way we combine testing and proving in one system. Testing is used for debugging programs and speci cations before ..."
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Cited by 13 (1 self)
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We extend the proof assistant Agda/Alfa for dependent type theory with a modi ed version of Claessen and Hughes' tool QuickCheck for random testing of functional programs. In this way we combine testing and proving in one system. Testing is used for debugging programs and speci cations before a proof is attempted. Furthermore, we demonstrate by example how testing can be used repeatedly during proof for testing suitable subgoals. Our tool uses testdata generators which are de ned inside Agda/Alfa. We can therefore use the type system to prove properties about them, in particular surjectivity stating that all possible test cases can indeed be generated.
Partial recursive functions in higherorder logic
 Int. Joint Conference on Automated Reasoning (IJCAR 2006), LNCS
, 2006
"... Abstract. Based on inductive definitions, we develop an automated tool for defining partial recursive functions in HigherOrder Logic and providing appropriate reasoning tools for them. Our method expresses termination in a uniform manner and includes a very general form of pattern matching, where p ..."
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Cited by 12 (2 self)
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Abstract. Based on inductive definitions, we develop an automated tool for defining partial recursive functions in HigherOrder Logic and providing appropriate reasoning tools for them. Our method expresses termination in a uniform manner and includes a very general form of pattern matching, where patterns can be arbitrary expressions. Termination proofs can be deferred, restricted to subsets of arguments and are interchangeable with other proofs about the function. We show that this approach can also facilitate termination arguments for total functions, in particular for nested recursions. We implemented our tool as a definitional specification mechanism for Isabelle/HOL. 1
P.: Dependent types at work
 LerNet 2008. LNCS
, 2009
"... Abstract. In these lecture notes we give an introduction to functional programming with dependent types. We use the dependently typed programming language Agda which is an extension of MartinLöf type theory. First we show how to do simply typed functional programming in the style of Haskell and ML. ..."
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Abstract. In these lecture notes we give an introduction to functional programming with dependent types. We use the dependently typed programming language Agda which is an extension of MartinLöf type theory. First we show how to do simply typed functional programming in the style of Haskell and ML. Some differences between Agda’s type system and the HindleyMilner type system of Haskell and ML are also discussed. Then we show how to use dependent types for programming and we explain the basic ideas behind typechecking dependent types. We go on to explain the CurryHoward identification of propositions and types. This is what makes Agda a programming logic and not only a programming language. According to CurryHoward, we identify programs and proofs, something which is possible only by requiring that all program terminate. However, at the end of these notes we present a method for encoding partial and general recursive functions as total functions using dependent types.
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 9 (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.
A Unifying Approach to Recursive and Corecursive Definitions
 IN [5
, 2002
"... In type theory based logical frameworks, recursive and corecursive definitions are subject to syntactic restrictions that ensure their termination and productivity. These restrictions however greately decrease the expressive power of the language. In this work we propose a general approach for s ..."
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Cited by 9 (1 self)
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In type theory based logical frameworks, recursive and corecursive definitions are subject to syntactic restrictions that ensure their termination and productivity. These restrictions however greately decrease the expressive power of the language. In this work we propose a general approach for systematically defining fixed points for a broad class of well given recursive definition. This approach unifies the ones based on wellfounded order to the ones based on complete metrics and contractive functions, thus allowing for mixed recursive/corecursive definitions.
An overview of the programatica toolset
 High Confidence Software and Systems Conference, HCSS04, http://www.cse.ogi.edu/~hallgren/ Programatica/HCSS04
, 2004
"... With ever increasing use of computers in safety and security critical applications, the need for trustworthy computer systems has never been greater. But how can such trust be established? For example, how can we be sure that our computer systems will not destroy or corrupt valuable data, compromise ..."
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Cited by 7 (0 self)
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With ever increasing use of computers in safety and security critical applications, the need for trustworthy computer systems has never been greater. But how can such trust be established? For example, how can we be sure that our computer systems will not destroy or corrupt valuable data, compromise privacy, or trigger
Verifying haskell programs by combining testing and proving
 In Proceedings of the Third International Conference on Quality Software
"... We propose a method for improving confidence in the correctness of Haskell programs by combining testing and proving. Testing is used for debugging programs and specification before a costly proof attempt. During a proof development, testing also quickly eliminates wrong conjectures. Proving helps u ..."
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Cited by 4 (0 self)
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We propose a method for improving confidence in the correctness of Haskell programs by combining testing and proving. Testing is used for debugging programs and specification before a costly proof attempt. During a proof development, testing also quickly eliminates wrong conjectures. Proving helps us to decompose a testing task in a way that is guaranteed to be correct. To demonstrate the method we have extended the Agda/Alfa proof assistant for dependent type theory with a tool for random testing. As an example we show how the correctness of a BDDalgorithm written in Haskell is verified by testing properties of component functions. We also discuss faithful translations from Haskell to type theory.
Recursive definitions of monadic functions
 In Proc. of PAR 2010
, 2010
"... Using standard domaintheoretic fixedpoints, we present an approach for defining recursive functions that are formulated in monadic style. The method works both in the simple option monad and the stateexception monad of Isabelle/HOL’s imperative programming extension, which results in a convenient ..."
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Cited by 3 (0 self)
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Using standard domaintheoretic fixedpoints, we present an approach for defining recursive functions that are formulated in monadic style. The method works both in the simple option monad and the stateexception monad of Isabelle/HOL’s imperative programming extension, which results in a convenient definition principle for imperative programs, which were previously hard to define. For such monadic functions, the recursion equation can always be derived without preconditions, even if the function is partial. The construction is easy to automate, and convenient induction principles can be derived automatically. 1
Corecursive Algebras: A Study of General Structured Corecursion (Extended Abstract)
"... Abstract. We study general structured corecursion, dualizing the work of Osius, Taylor, and others on general structured recursion. We call an algebra of a functor corecursive if it supports general structured corecursion: there is a unique map to it from any coalgebra of the same functor. The conce ..."
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Abstract. We study general structured corecursion, dualizing the work of Osius, Taylor, and others on general structured recursion. We call an algebra of a functor corecursive if it supports general structured corecursion: there is a unique map to it from any coalgebra of the same functor. The concept of antifounded algebra is a statement of the bisimulation principle. We show that it is independent from corecursiveness: Neither condition implies the other. Finally, we call an algebra focusing if its codomain can be reconstructed by iterating structural refinement. This is the strongest condition and implies all the others. 1
Using structural recursion for corecursion
 In Types for Proofs and Programs, International Conference, TYPES 2008, volume 5497 of LNCS
, 2009
"... Abstract. We propose a (limited) solution to the problem of constructing stream values defined by recursive equations that do not respect the guardedness condition. The guardedness condition is imposed on definitions of corecursive functions in Coq, AGDA, and other higherorder proof assistants. In ..."
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Abstract. We propose a (limited) solution to the problem of constructing stream values defined by recursive equations that do not respect the guardedness condition. The guardedness condition is imposed on definitions of corecursive functions in Coq, AGDA, and other higherorder proof assistants. In this paper, we concentrate in particular on those nonguarded equations where recursive calls appear under functions. We use a correspondence between streams and functions over natural numbers to show that some classes of nonguarded definitions can be modelled through the encoding as structural recursive functions. In practice, this work extends the class of stream values that can be defined in a constructive type theorybased theorem prover with inductive and coinductive types, structural recursion and guarded corecursion.