Weight-biased leftist heaps verified in Haskell using dependent types
This repo contains implementation of weight-biased leftist heap data
structure verified in Haskell using dependent types. This package is
intended to be a tutorial and technology demonstration. It is not
intended to be used in real-world applications (but if you find such a
use please let me know).
Weight-biased leftist heap is a binary tree that satisfies two
Priority invariant: priority of every node is higher than
priority of its children. (This property is true for heaps in
Rank invariant: for every node size of its left child is not
smaller than the size of its right child.
These two invariants give us a data structure that provides O(1)
access to element with the highest priority and O(log2 n) insert and
merge operations. See chapter 3 of Chris Okasaki's "Purely Functional
Data Structures" for more discussion. Note that my implementation
represents priorities using natural numbers where 0 is the highest
The main purpose of this implementation is to explain how proofs of
the two above invariants are constructed in Haskell. (The ideas
convey to other languages with dependent types.) You'll find lots of
comments in the source code. I assume that you already have been
exposed to basics of proofs with dependent types. In particular
you should be familiar with:
the concept of data-as-evidence as described in "Why Dependent
Types Matter" paper. All the ideas here are taken from that
singleton types as described in "Dependently Typed Programming
with Singletons" paper. I don't make heavy use of singletons but
you should understand why do we need encodings like singleton
types in Haskell
basics of formal reasoning (including refl). See online lecture
notes for the Computer Aided Formal Reasoning course by Thorsten
You should begin studying of this repo by getting familiar with
Basics directory. Then go to
TwoPassMerge directory and
NoProofs module followed by
PriorityProof (in any order) and finish with
SinglePassMerge and study the modules in the same order as
earlier. Alternatively, you might want to look at the single-pass
merge variant right after studying the two-pass implementation.
Requirements and conventions
This code has been tested with GHC 7.6.3 and GHC 7.8.3. It only
baselibrary. No other dependencies are required.
I'm not relying on anything from
I'm using GADT syntax for all data types, even if they are
Whenever I say "See #XYZV" I'm referring to GHC bug report number
located under address
See LICENSE file in the root of the repository.
I originally implemented verified weight-biased leftist heap in
Agda. This implementation is available
I also wrote a companion blog post.