Using reflection to build efficient and certified decision procedures

Boutin, Samuel

doi:10.1007/bfb0014565

Cited by 82 publications

(65 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…To facilitate reasoning about trees, we use the method known as the two-level approach (see, for example, [3] and [6]). It is a general technique to automate the proof of a class of goals by internalizing it as an inductive type.…”

Section: The Two-level Approach For Treesmentioning

confidence: 99%

See 1 more Smart Citation

Certifying the Fast Fourier Transform with Coq

Capretta

2001

Lecture Notes in Computer Science

View full text Add to dashboard Cite

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.

show abstract

Section: The Two-level Approach For Treesmentioning

confidence: 99%

“…We prove that it is equivalent to a more natural one. In Section 3 we apply the two-level approach (see [3] and [6]) to the tree representation to prove some basic facts about it. Section 4 contains the definition and proof of correctness of FFT.…”

Section: Introductionmentioning

confidence: 99%

Certifying the Fast Fourier Transform with Coq

Capretta

2001

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…With these definitions at hand, then if we have some b : B such that I b " a, we can replace the original proof-term ∆ with sound b prefl_equal trueq where refl_equal has type @x : bool, x " x. Typechecking that the proof-term above has the expected type (P a) effectively amounts to (i) executing the procedure D on b, (ii) checking that its result is equal to true, (iii) and checking that the interpretation of b is equal to a. 3 Previous works [11,4] have exposed several advantages and weaknesses of proof by reflection, especially in comparison with the traditional LCF proof style [9]. In a nutshell, the former is considered more robust to change, while the latter is easier to write.…”

Section: Introductionmentioning

confidence: 99%

Lightweight Proof by Reflection Using a Posteriori Simulation of Effectful Computation

Claret

Huesca

Régis-Gianas

et al. 2013

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. Proof-by-reflection is a well-established technique that employs decision procedures to reduce the size of proof-terms. Currently, decision procedures can be written either in Type Theory-in a purely functional way that also ensures termination-or in an effectful programming language, where they are used as oracles for the certified checker. The first option offers strong correctness guarantees, while the second one permits more efficient implementations. We propose a novel technique for proof-by-reflection that marries, in Type Theory, an effectful language with (partial) proofs of correctness. The key to our approach is to use simulable monads, where a monad is simulable if, for all terminating reduction sequences in its equivalent effectful computational model, there exists a witness from which the same reduction may be simulated a posteriori by the monad. We encode several examples using simulable monads and demonstrate the advantages of the technique over previous approaches.

show abstract

“…This is clearly one of the advantages of Coq [5] with respect to other proof assistants, like Isabelle/HOL [22]. This characteristic is the base of reflective tactics, pioneered by S. Boutin [6], and successfully used, for instance, in [14,20].…”

Section: Introductionmentioning

confidence: 99%