Abstraction and invariance for algebraically indexed types

Abstract

Reynolds' relational parametricity provides a powerful way to reason about programs in terms of invariance under changes of data representation. A dazzling array of applications of Reynolds' theory exists, exploiting invariance to yield "free theorems", non-inhabitation results, and encodings of algebraic datatypes. Outside computer science, invariance is a common theme running through many areas of mathematics and physics. For example, the area of a triangle is unaltered by rotation or flipping. If we scale a triangle, then we scale its area, maintaining an invariant relationship between the two. The transformations under which properties are invariant are often organised into groups, with the algebraic structure reflecting the composability and invertibility of transformations.