Dependently typed programming advocates the use of various indexed versions of the same shape of data, but the formal relationship amongst these structurally similar datatypes usually needs to be established manually and tediously. Ornaments have been proposed as a formal mechanism to manage the relationships between such datatype variants. In this paper, we conduct a case study under an ornament framework; the case study concerns programming binomial heaps and their operations — including insertion and minimum extraction — by viewing them as lifted versions of binary numbers and numeric operations. We show how current dependently typed programming technology can lead to a clean treatment of the binomial heap constraints when implementing heap operations. We also identify some gaps between the current technology and an ideal dependently typed programming language that we would wish to have for our development.

We develop a methodology for writing interactive and object-based programs (in the sense of Wegner) in dependently typed functional programming languages. The methodology is implemented in the ooAgda library. ooAgda provides a syntax similar to the one used in object-oriented programming languages, thanks to Agda's copattern matching facility. The library allows for the development of graphical user interfaces (GUIs), including the use of action listeners.

Our notion of interactive programs is based on the IO monad defined by Hancock and Setzer, which is a coinductive data type. We use a sized coinductive type which allows us to write corecursive programs in a modular way. Objects are server-side interactive programs that respond to method calls by giving answers and changing their state. We introduce two kinds of objects: simple objects and IO objects. Methods in simple objects are pure, while method calls in IO objects allow for interactions before returning their result. Our approach also allows us to extend interfaces and objects by additional methods.

We refine our approach to state-dependent interactive programs and objects through which we can avoid exceptions. For example, with a state-dependent stack object, we can statically disable the pop method for empty stacks. As an example, we develop the implementation of recursive functions using a safe stack. Using a coinductive notion of object bisimilarity, we verify basic correctness properties of stack objects and show the equivalence of different stack implementations. Finally, we give a proof of concept that our interaction model allows to write GUI programs in a natural way: we present a simple drawing program, and a program which allows the users to move a small spaceship using a button.

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 Ornamental Algebras, Algebraic Ornaments. Unpublished). 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.

Unification is a core component of every proof assistant or programming language featuring dependent types. In many cases, it must deal with higher order problems up to conversion. Since unification in such conditions is undecidable, unification algorithms may include several heuristics to solve common problems. However, when the stack of heuristics grows large, the result and complexity of the algorithm can become unpredictable. Our contributions are twofold: (1) We present a full description of a new unification algorithm for the Calculus of Inductive Constructions (the base logic of COQ), building it up from a basic calculus to the full Calculus of Inductive Constructions as it is implemented in COQ, including universe polymorphism, canonical structures (the overloading mechanism baked into COQ's unification), and a small set of useful heuristics. (2) We implemented our algorithm, and tested it on several libraries, providing evidence that the selected set of heuristics suffices for large developments.

Rational sequences are possibly infinite sequences with a finite number of distinct suffixes. In this paper, we present different implementations of rational sequences in Martin–Löf type theory. First, we literally translate the above definition of rational sequence into the language of type theory, i.e., we construct predicates on possibly infinite sequences expressing the finiteness of the set of suffixes. In type theory, there exist several inequivalent notions of finiteness. We consider two of them, listability and Noetherianness, and show that in the implementation of rational sequences the two notions are interchangeable. Then we introduce the type of lists with backpointers, which is an inductive implementation of rational sequences. Lists with backpointers can be unwound into rational sequences, and rational sequences can be truncated into lists with backpointers. As an example, we see how to convert the fractional representation of a rational number into its decimal representation and vice versa.

Modern constructive type theory is based on pure dependently typed lambda calculus, augmented with user-defined datatypes. This paper presents an alternative called the Calculus of Dependent Lambda Eliminations, based on pure lambda encodings with no auxiliary datatype system. New typing constructs are defined that enable induction, as well as large eliminations with lambda encodings. These constructs are constructor-constrained recursive types, and a lifting operation to lift simply typed terms to the type level. Using a lattice-theoretic denotational semantics for types, the language is proved logically consistent. The power of CDLE is demonstrated through several examples, which have been checked with a prototype implementation called Cedille.

There has been sustained interest in functional programming languages with dependent types in recent years. The foundations of dependently typed programming can be traced back to Martin–Löf's work in the 1970s. In the past decades, this vision has given rise to the development of proof assistants and functional programming languages based on dependent types. The increased popularity of systems such as Agda, Coq, Idris, and many others, reflects the growing momentum in this research area. After sending out our first call for papers in October 2015, we are happy to accept six articles in this special issue covering a wide spectrum of topics.