# (Programming Languages) in Agda = Programming (Languages in Agda)

YOW! Lambda Jam 2019

The most profound connection between logic and computation is a pun. The doctrine of Propositions as Types asserts that propositions correspond to types, proofs to programs, and simplification of proofs to evaluation of programs. The proof of a conjunction is a pair, the proof of a disjunction is a case expression, and the proof of an implication is a lambda expression. Proof by induction is just programming by recursion.

Dependently-typed programming languages, such as Agda, exploit this pun. To prove properties of programming languages in Agda, all we need do is program a description of those languages Agda. Finding an abstruse mathematical proof becomes as simple and as fun as hacking a program. This talk introduces *Programming Language Foundations in Agda*, a new textbook that is also an executable Agda script---and also explains the role Agda is playing in IOHK's new cryptocurrency.