Filters on coinductive streams, an application to eratosthenes’ sieve (2005)
| Venue: | Typed Lambda Calculi and Applications, 7th International Conference, TLCA 2005 |
| Citations: | 15 - 4 self |
BibTeX
@INPROCEEDINGS{Bertot05filterson,
author = {Yves Bertot},
title = {Filters on coinductive streams, an application to eratosthenes’ sieve},
booktitle = {Typed Lambda Calculi and Applications, 7th International Conference, TLCA 2005},
year = {2005},
pages = {102--115},
publisher = {Springer-Verlag}
}
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Abstract
Our objective is to describe a formal proof of correctness for the following Haskell [13] program in a type theory-based proof verification system, such as the Coq system [10, 1]. sieve (p:rest) = p:sieve [r | r <- rest, r ‘rem ‘ p / = 0] primes = sieve [2..] This program is a functional implementation of Eratosthenes ’ sieve that consists in removing all multiples of previously found primes from the sequence of natural numbers. We want to prove that the expression primes is the stream containing all the prime numbers in increasing order. This work relies on co-inductive types [5, 11, 12] because the program manipulates infinite lists, also known as streams. It first uses the infinite list of natural numbers larger than 2, then the infinite list of numbers larger than 3 and containing no multiples of 2, then the infinite list of numbers larger than 4 and containing no multiples of prime numbers smaller than 4, and so on. This example was initially proposed as a challenge by G. Kahn and used as an illustration of a program and its proof of correctness in a
