Relative Monads Formalised

Relative monads (Altenkirch, Chapman, and Uustalu 2010) are a recent generalisation of ordinary monads to cover similar structures where the underlying functor need not be an endofunctor. Our interest in this generalisation was triggered by some structures from programming theory that, in many ways, are strikingly similar to monads (even respecting the same laws), have similar programming applications, but nonetheless fail to be monads. Some examples of relative monads include untyped and typed λ-terms, finite-dimensional vector spaces, Hughes’s arrows. Our research effort on relative monads is ongoing. In this paper, we describe a formalisation of the basic theory of relative monads in the interactive theorem prover and dependently typed programming language Agda (Agda team 2010). This formalisation work is motivated by a number of basic observations. First of all, by moving from a conventional functional programming language like Haskell (The Haskell Team 2010) to a dependently typed language like Agda (Agda team 2010), we gain the opportunity to formally certify our programs. Indeed, we are able to move from programs to “deliverables”, which contain partial specifications of programs as types; and the correctness of the assertions can be automatically verified by a type checker.