August 6, 2007 I am curious about the possibility of developing Haskell programs spontaneously with proofs about their properties and have the type checker verify the proofs for us, in a way one would do in a dependently typed language. In the exercise below, I tried to redo part of the merge-sort example in Altenkirch, McBride, and McKinna’s introduction to Epigram: deal the input list into a binary tree, and fold the tree by the function merging two sorted lists into one. The property I am going to show is merely that the length of the input list is preserved. To begin with, we define the usual type-level representation of natural numbers and lists indexed by their lengths.
1. What is a dependent type 1. ADT -- the simplest dependent-type 2. Singleton types 3. Branding: type proxies 2. Lightweight static capabilities 1. Abstract and Introduction 2. Formalization and proofs 3. Accompanying source code 3. The question of verification 4. Genuine Dependent-type systems This is a joint work with Chung-chieh Shan. We describe several approaches to lightweight dependent-type programming, letting us gain experience with dependent types on existing programming language systems All these lightweight approaches rely on type-level proxies for values, so we can statically express properties (e.g., equality, inequality) of the values that are not generally known until the run time. ``This much is clear: many programmers are already finding practical uses for the approximants to dependent types which mainstream functional languages (especially Haskell) admit, by hook or by crook.''