To me as a programmer, to write mathematical proofs that are mechanically checked by a computer feels empowering. To have these proofs as executable programs feels even more empowering. Therefore, our proofs have a computational aspect and vice-versa: our programs have a logical aspect. To be able to get an instant feedback while proving a theorem is amazing. With Agda, a dependently typed functional programming language, one can interactively write a proof by getting guidance from Agda as to what is left to prove. Furthermore, Agda checks the correctness of proofs by following a set of rules. Unlike with pen and paper proofs, proofs in Agda are much more rigorous because there is no room for hand-waving nor unwarranted claims for something to be trivial. An uninformed mathematician will likely find this comparison to Agda hard to believe.

# Tag Archives: Agda

# Function Totality: Abstraction Tool in Programming

Edsger Dijkstra said the following of abstraction:

“The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise.”

Abstraction is a cornerstone of programming a complex software system. Without it, a complex software system is a complicated software system. In this article I will explore an important tool in achieving abstraction in programming: function totality.

# Hooked on Dependent Types

Just this morning these two books landed my mailbox:

The Little Typer is a new book on dependent type theory, written by Daniel P. Friedman, an author of The Little Schemer, and David Thrane Christiansen, an Idris contributor. Verified Functional Programming in Agda is a book by Aaron Stump, on using dependent types in Agda to prove various properties of programs. After having read the Type-driven Development with Idris book by Edwin Brady, I am hoping these two books will significantly expand my knowledge of and improve my skills in type theory, theorem proving and typed functional programming. Looking forward to reading the books! If you haven’t noticed by now, I got hooked on dependent types!

# Typed Functional Programming and Software Correctness

Software testing has been an important, if not prevalent way of checking software correctness. In this article I will tell how have my doctoral dissertation on testing and verification of imperative software as well as my work experience after the studies led me to typed functional programming, which consequently gave me a different perspective on automatic software testing. Furthermore, I’ll explain why functional programming and static type systems are important for software correctness.

## Typed Holes in Programming

### Aside

# Totality in Programming

You might think it’s only imperative programmers that don’t see value in function totality, but neither do many functional programmers. The ability to abstract in programming is severely impaired when totality is ignored. A programmer should strive to work in a programming language with a compiler that supports totality checking, such as Agda and Idris.

## Incomplete Pattern-matching in Haskell

### Aside

From Haskell’s safe-exceptions library: “[I]ncomplete pattern matches can generate impure exceptions.” This makes me think of Agda, its totality and absence of exceptions.