The essence of ornaments

Functional programmers from all horizons strive to use, and sometimes abuse, their favorite type system in order to capture the invariants of their programs. A widely-used tool in that trade consists in defining finely-indexed datatypes. Operationally, these types classify the programmer’s data, following the ML tradition. Logically, these types enforce the program invariants in a novel manner. This new programming pattern, by which one programs over inductive definitions to account for some invariants, lead to the development of a theory of ornaments (McBride, 2011). However, ornaments originate as a dependently-typed object and may thus appear rather daunting to a functional programmer of the non-dependent kind. This article aims at presenting ornaments from first-principles and, in particular, to declutter their presentation from syntactic considerations. To do so, we shall give a sufficiently abstract model of indexed datatypes by means of many-sorted signatures. In this process, we formalize our intuition that an indexed datatype is the combination of a data-structure and a data-logic. Over this abstraction of datatypes, we shall recast the definition of ornaments, effectively giving a model of ornaments. Benefiting both from the operational and abstract nature of many-sorted signatures, ornaments should appear applicable and, one hopes, of interest beyond the type-theoretic circles, case in point being languages with generalized abstract datatypes or refinement types.