Overview
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Weightbiased leftist heaps verified in Haskell using dependent types
This repo contains implementation of weightbiased 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 realworld applications (but if you find such a use please let me know).
Weightbiased leftist heap is a binary tree that satisfies two invariants:

Priority invariant: priority of every node is higher than priority of its children. (This property is true for heaps in general).

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
priority (See Basics
and Nat
modules).
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 dataasevidence as described in "Why Dependent Types Matter" paper. All the ideas here are taken from that paper

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 Altenkirch
You should begin studying of this repo by getting familiar with
modules in Basics
directory. Then go to TwoPassMerge
directory and
begin with NoProofs
module followed by RankProof
and
PriorityProof
(in any order) and finish with CombinedProofs
. Then
move to SinglePassMerge
and study the modules in the same order as
earlier. Alternatively, you might want to look at the singlepass
merge variant right after studying the twopass implementation.
Requirements and conventions

This code has been tested with GHC 7.6.3 and GHC 7.8.3. It only depends on
base
library. No other dependencies are required. 
I'm not relying on anything from
Prelude
except forundefined
function. 
I'm using GADT syntax for all data types, even if they are ordinary ADTs.

Whenever I say "See #XYZV" I'm referring to GHC bug report number located under address
http://hackage.haskell.org/trac/ghc/ticket/XYZV
.
License
See LICENSE file in the root of the repository.
See also
I originally implemented verified weightbiased leftist heap in Agda. This implementation is available here.
I also wrote a companion blog post.