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The Stern-Brocot Tree
| Title: | The Stern-Brocot Tree |
| Authors: | Peter Gammie and Andreas Lochbihler |
| Submission date: | 2015-12-22 |
| Abstract: | The Stern-Brocot tree contains all rational numbers exactly once and in their lowest terms. We formalise the Stern-Brocot tree as a coinductive tree using recursive and iterative specifications, which we have proven equivalent, and show that it indeed contains all the numbers as stated. Following Hinze, we prove that the Stern-Brocot tree can be linearised looplessly into Stern's diatonic sequence (also known as Dijkstra's fusc function) and that it is a permutation of the Bird tree.
The reasoning stays at an abstract level by appealing to the uniqueness of solutions of guarded recursive equations and lifting algebraic laws point-wise to trees and streams using applicative functors. |
| BibTeX: |
@article{Stern_Brocot-AFP,
author = {Peter Gammie and Andreas Lochbihler},
title = {The Stern-Brocot Tree},
journal = {Archive of Formal Proofs},
month = dec,
year = 2015,
note = {\url{http://isa-afp.org/entries/Stern_Brocot.shtml},
Formal proof development},
ISSN = {2150-914x},
}
|
| License: | BSD License |
| Depends on: | Applicative_Lifting |
| Status: [skipped] | This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations. |
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