A well-known representation of monoids and its application to the function ‘vector reverse’

Abstract Vectors—or length-indexed lists—are classic example of a dependent type. Yet, most tutorials stay clear any function on vectors whose definition requires non-trivial equalities between natural numbers to type check. This pearl shows how write functions, such as vector reverse, that rely monoidal be correct without having additional proofs. These techniques can applied many other functions over types indexed by monoid, written using an accumulating parameter, and even used decide arbitrary monoids ‘for free.’