A Machine-Checked Proof of the Average-Case Complexity of Quicksort in Coq

Eelis Weegen, James McKinna

Research output: Chapter in Book/Report/Conference proceedingConference contribution


As a case-study in machine-checked reasoning about the complexity of algorithms in type theory, we describe a proof of the average-case complexity of Quicksort in Coq. The proof attempts to follow a textbook development, at the heart of which lies a technical lemma about the behaviour of the algorithm for which the original proof only gives an intuitive justification.

We introduce a general framework for algorithmic complexity in type theory, combining some existing and novel techniques: algorithms are given a shallow embedding as monadically expressed functional programs; we introduce a variety of operation-counting monads to capture worst- and average-case complexity of deterministic and nondeterministic programs, including the generalization to count in an arbitrary monoid; and we give a small theory of expectation for such non-deterministic computations, featuring both general map-fusion like results, and specific counting arguments for computing bounds.

Our formalization of the average-case complexity of Quicksort includes a fully formal treatment of the ‘tricky’ textbook lemma, exploiting the generality of our monadic framework to support a key step in the proof, where the expected comparison count is translated into the expected length of a recorded list of all comparisons.
Original languageEnglish
Title of host publicationTypes for Proofs and Programs
Subtitle of host publicationInternational Conference, TYPES 2008 Torino, Italy, March 26-29, 2008 Revised Selected Papers
EditorsStefano Berardi, Ferruccio Damiani, Ugo de'Liguoro
PublisherSpringer-Verlag GmbH
Number of pages16
ISBN (Electronic)978-3-642-02444-3
ISBN (Print)978-3-642-02443-6
Publication statusPublished - 2009

Publication series

NameLecture Notes in Computer Science
PublisherSpringer Berlin / Heidelberg
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


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