for the few of us.

S-C. Mu, H-S. Ko, and P. Jansson. Algebra of programming using dependent types. In Mathematics of Program Construction 2008, LNCS 5133, pp 268-283. July 2008.
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It seems that reductivity of Doornbos and Backhouse is in fact accessibility, which is often taken by type theorists to be an alternative definition of well-foundedness.

Given a relation < that has been shown to be well-founded, we know that a recursively defined function is terminating if its arguments becomes “smaller” with respect to < at each recursive call. We can encode such well-founded recursion in Agda’s style of structural recusion.

After seeing our code, Nils Anders Danielsson suggested two improvements. Firstly, to wrap the bound in the seed. Secondly, to define unfoldT↓ using well-founded recursion.

I hope to come up with a variation of unfoldr which, given a proof of well-foundedness, somehow passes the termination check.

The AoPA library allows one to encode Algebra of Programming style program derivation, both functional and relational, in Agda, accompanying the paper Algebra of Programming Using Dependent Types (MPC 2008) developed in co-operation with Hsiang-Shang Ko and Patrik Jansson.

This time I am implementing the example in another section of Why Dependent Types Matter by Altenkirch, McBride, and McKinna in Agda: proving that mergesort returns an ordered list.

Continuing with my Adga programming practice. Part of the mergesort example became easy once Max showed me how I should perform the pattern matching in insert.

During the WG 2.1 meeting, my attention was brought to Agda. So, here is my first Agda program. I am merely repeating some simple exercises on lists, which I tried using Haskell in a previous entry Developing Programs and Proofs Spontaneously using GADT.

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. 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.