Fix-Point Equations for Well-Founded Recursion in Type Theory

Balaa, Antonia; Bertot, Yves

doi:10.1007/3-540-44659-1_1

Cited by 25 publications

(32 citation statements)

References 7 publications

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“…The "converging iterations" approach In [1], a recursive function f is defined as the solution of a fixpoint equation involving a functional F . The definition involves proving that the fixpoint equation terminates in a finite number of iterations.…”

Section: Other Approaches In Coqmentioning

confidence: 99%

“…The approach has been implemented in a prototype tool that automatically generates some of the definitions, proof obligations, and corresponding proofs. The tool of Balaa and Bertot [1], which is currently the sole tool for Coq based on this approach, does not satisfy the criteria (DT) and (ST) and (X), and only partially satisfies the criterion (SR), as it only provides the fixpoint equation for the function.…”

Section: Other Approaches In Coqmentioning

confidence: 99%

“…Combining [9] and [1] In [6], the "converging iterations" and "ad-hoc predicate" approaches are merged into a powerful technique for recursive function definition, allowing for partial functions with nested recursive calls. The method is implementable in Coq.…”

Section: Other Approaches In Coqmentioning

confidence: 99%

See 2 more Smart Citations

Defining and Reasoning About Recursive Functions: A Practical Tool for the Coq Proof Assistant

Barthe

Forest

Pichardie

et al. 2006

Functional and Logic Programming

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Abstract. We present a practical tool for defining and proving properties of recursive functions in the Coq proof assistant. The tool generates from pseudo-code the graph of the intended function as an inductive relation. Then it proves that the relation actually represents a function, which is by construction the function that we are trying to define. Then, we generate induction and inversion principles, and a fixpoint equation for proving other properties of the function. Our tool builds upon stateof-the-art techniques for defining recursive functions, and can also be used to generate executable functions from inductive descriptions of their graph. We illustrate the benefits of our tool on two case studies.

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Section: Other Approaches In Coqmentioning

confidence: 99%

Section: Other Approaches In Coqmentioning

confidence: 99%

See 1 more Smart Citation

Defining and Reasoning About Recursive Functions: A Practical Tool for the Coq Proof Assistant

Barthe

Forest

Pichardie

et al. 2006

Functional and Logic Programming

View full text Add to dashboard Cite

show abstract

“…An alternative way, adopted by Balaa and Bertot in [1], sees the partial function as a fixed point of an operator F that maps total functions to total functions. It can be approximated by a finite number of iterations of F on an arbitrary base function.…”

Section: [P]]([[a]]) = [[R]] Where [[−]] Is the Interpretationmentioning

confidence: 99%

Type-Theoretic Functional Semantics

Bertot¹,

Capretta²,

Barman³

2002

Lecture Notes in Computer Science

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Abstract. We describe the operational and denotational semantics of a small imperative language in type theory with inductive and recursive definitions. The operational semantics is given by natural inference rules, implemented as an inductive relation. The realization of the denotational semantics is more delicate: The nature of the language imposes a few difficulties on us. First, the language is Turing-complete, and therefore the interpretation function we consider is necessarily partial. Second, the language contains strict sequential operators, and therefore the function necessarily exhibits nested recursion. Our solution combines and extends recent work by the authors and others on the treatment of general recursive functions and partial and nested recursive functions. The first new result is a technique to encode the approach of Bove and Capretta for partial and nested recursive functions in type theories that do not provide simultaneous induction-recursion. A second result is a clear understanding of the characterization of the definition domain for general recursive functions, a key aspect in the approach by iteration of Balaa and Bertot. In this respect, the work on operational semantics is a meaningful example, but the applicability of the technique should extend to other circumstances where complex recursive functions need to be described formally.

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“…The recursion is wellfounded because f e and f o have smaller degree than f . This requires the use of course-of-value recursion on the degree of the polynomial, which can be realized in type theory using a general method for wellfounded recursion (see [1]). The formalization of the algorithm that we obtain is more complex than necessary, because it contains the proofs of the fact that the degrees decrease.…”

Section: Data Representationmentioning

confidence: 99%

Certifying the Fast Fourier Transform with Coq

Capretta

2001

Lecture Notes in Computer Science

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Abstract. We program the Fast Fourier Transform in type theory, using the tool Coq. We prove its correctness and the correctness of the Inverse Fourier Transform. A type of trees representing vectors with interleaved elements is defined to facilitate the definition of the transform by structural recursion. We define several operations and proof tools for this data structure, leading to a simple proof of correctness of the algorithm. The inverse transform, on the other hand, is implemented on a different representation of the data, that makes reasoning about summations easier. The link between the two data types is given by an isomorphism. This work is an illustration of the two-level approach to proof development and of the principle of adapting the data representation to the specific algorithm under study. CtCoq, a graphical user interface of Coq, helped in the development. We discuss the characteristics and usefulness of this tool.

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Fix-Point Equations for Well-Founded Recursion in Type Theory

Cited by 25 publications

References 7 publications

Defining and Reasoning About Recursive Functions: A Practical Tool for the Coq Proof Assistant

Defining and Reasoning About Recursive Functions: A Practical Tool for the Coq Proof Assistant

Type-Theoretic Functional Semantics

Certifying the Fast Fourier Transform with Coq

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