When Is a Type Refinement an Inductive Type?

Abstract

Abstract. Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the Nindexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce purposespecific refinements on an ad hoc basis, developers must anticipate which refinements to include in libraries, and implementations often store redundant information about data and their refinements. This paper shows how to generically derive inductive characterisations of refinements of inductive types, and argues that these characterisations can alleviate some of the aforementioned difficulties associated with ad hoc refinements. These characterisations also ensure that standard techniques for programming with and reasoning about inductive types are applicable to refinements, and that refinements can themselves be further refined.

Transporting functions across ornaments

“…We shall not reiterate it here, the implementation being essentially the same for our universe. A categorical presentation was also given in Atkey et al [2011] in which the connection with refinement types was explored.…”

“…Our presentation of algebraic ornament has been greatly improved by the categorical model developed by Atkey et al [2011]: the authors gave a conceptually clear treatment of algebraic ornament in a Lawvere fibration. At the technical level, the authors connected the definition of algebraic ornament with truth-preserving liftings, which are also used in the construction of induction principles, and op-reindexing, which models Σ-types in type theory.…”

“…In particular, the idea that natural numbers realise lists of the corresponding length appears in this system under the guise of vectors, the reflection of the realisability predicate. The strength of the realisability interpretation is that it is naturally defined on functions: while McBride and Atkey et al [2011] only consider ornaments on datatypes, their work is the first, to our knowledge, to capture a general notion of functions realising -i.e. ornamentingother functions.…”

Fibrational Induction Meets Effects

“…The condition we use to ensure this is that the Lawvere fibration in which we work has very strong sums. The following important property of very strong sums is from [1]: If U : E → B is a Lawvere fibration with very strong sums, F : B → B is a functor, f : X → Y is a morphism, and P ∈ E X , thenF (Σ f P ) = Σ F fF P . Using this, we can prove that in a Lawvere fibration with very strong sums lifting is actually a strong monoidal functor, i.e., that lifting preserves functor composition.…”

“…1 We write T for the monad (T, η, μ) when no confusion may result. An Eilenberg-Moore algebra for T is a T -algebra h : T X → X such that h•μ X = h•T h and h•η X = id; we can think of such algebra as a T -algebra that respects the unit and multiplication of T .…”