Many programs process lists by recursing in a wide variety of sequential and/or divide-and-conquer patterns. Reasoning about the correctness and completeness of these programs requires reasoning about the lengths of the lists, techniques for which are typically undecidable or at least NP-complete. In this paper we show how introducing a relatively simple (sub-)language for expressions describing list lengths, whilst not completely general, covers a great number of these patterns. It includes not only doubling but also exponentiation (iterated doubling), and moreover admits a simple length-checking algorithm that is complete over a predictable problem domain. We prove termination of the algorithm via category-theoretic pullbacks, formalized in Agda, as well as providing a more realistic implementation in Rocq, and a toy language Fulbourn with interpreter in Haskell.

Abstract The syntax of almost every programming language includes a notion of binder and corresponding bound occurrences, along with the accompanying notions of α-equivalence, capture-avoiding substitution, typing contexts, runtime environments, and so on. In the past, implementing and reasoning about programming languages required careful handling to maintain the correct behaviour of bound variables. Modern programming languages include features that enable constraints like scope safety to be expressed in types. Nevertheless, the programmer is still forced to write the same boilerplate over again for each new implementation of a scope-safe operation (e.g., renaming, substitution, desugaring, printing), and then again for correctness proofs. We present an expressive universe of syntaxes with binding and demonstrate how to (1) implement scope-safe traversals once and for all by generic programming; and (2) how to derive properties of these traversals by generic proving. Our universe description, generic traversals and proofs, and our examples have all been formalised in Agda and are available in the accompanying material available online at https://github.com/gallais/generic-syntax .

Abstract We explore the design and implementation of Frank, a strict functional programming language with a bidirectional effect type system designed from the ground up around a novel variant of Plotkin and Pretnar’s effect handler abstraction. Effect handlers provide an abstraction for modular effectful programming: a handler acts as an interpreter for a collection of commands whose interfaces are statically tracked by the type system. However, Frank eliminates the need for an additional effect handling construct by generalising the basic mechanism of functional abstraction itself. A function is but the special case of a Frank operator that interprets no commands. Moreover, Frank’s operators can be multihandlers which simultaneously interpret commands from several sources at once, without disturbing the direct style of functional programming with values. Effect typing in Frank employs a novel form of effect polymorphism which avoids mentioning effect variables in source code. This is achieved by propagating an ambient ability inwards, rather than accumulating unions of potential effects outwards. With the ambient ability describing the effects that are available at a certain point in the code, it can become necessary to reconfigure access to the ambient ability. A primary goal is to be able to encapsulate internal effects, eliminating a phenomenon we call effect pollution . Moreover, it is sometimes desirable to rewire the effect flow between effectful library components. We propose adaptors as a means for supporting both effect encapsulation and more general rewiring. Programming with effects and handlers is in its infancy. We contribute an exploration of future possibilities, particularly in combination with other forms of rich type systems.

Almost every programming language’s syntax includes a notion of binder and corresponding bound occurrences, along with the accompanying notions of α-equivalence, capture avoiding substitution, typing contexts, runtime environments, and so on. In the past, implementing and reasoning about programming languages required careful handling to maintain the correct behaviour of bound variables. Modern programming languages include features that enable constraints like scope safety to be expressed in types. Nevertheless, the programmer is still forced to write the same boilerplate over again for each new implementation of a scope safe operation (e.g., renaming, substitution, desugaring, printing, etc.), and then again for correctness proofs. We present an expressive universe of syntaxes with binding and demonstrate how to (1) implement scope safe traversals once and for all by generic programming; and (2) how to derive properties of these traversals by generic proving. Our universe description, generic traversals and proofs, and our examples have all been formalised in Agda and are available in the accompanying material. NB. we recommend printing the paper in colour to benefit from syntax highlighting in code fragments.

We abstract the common type-and-scope safe structure from computations on λ-terms that deliver, e.g., renaming, substitution, evaluation, CPS-transformation, and printing with a name supply. By exposing this structure, we can prove generic simulation and fusion lemmas relating operations built this way. This work has been fully formalised in Agda.

Dybjer and Setzer introduced the definitional principle of inductive-recursively defined families - i.e. of families (U : Set, T : U -> D) such that the inductive definition of U may depend on the recursively defined T --- by defining a type DS D E of codes. Each c : DS D E defines a functor [c] : Fam D -> Fam E, and (U, T) = \mu [c] : Fam D is exhibited as the initial algebra of [c]. This paper considers the composition of DS-definable functors: Given F : Fam C -> Fam D and G : Fam D -> Fam E, is G \circ F : Fam C -> Fam E DS-definable, if F and G are? We show that this is the case if and only if powers of families are DS-definable, which seems unlikely. To construct composition, we present two new systems UF and PN of codes for inductive-recursive definitions, with UF a subsytem of DS a subsystem of PN. Both UF and PN are closed under composition. Since PN defines a potentially larger class of functors, we show that there is a model where initial algebras of PN-functors exist by adapting Dybjer-Setzer's proof for DS.

We explore the design and implementation of Frank, a strict functional programming language with a bidirectional effect type system designed from the ground up around a novel variant of Plotkin and Pretnar's effect handler abstraction.

This special issue comprises selected papers which were presented at the workshop on dependently typed programming (DTP 10) in Edinburgh in July 2010 – affiliated with Federated Logic conferences (FLOC 10). Earlier workshops on dependently typed programming took place in Nottingham in 2008 (DTP 2008) and there also has been a Dagstuhl seminar (04381) on this subject in 2004. After DTP 2010, in 2011 a workshop on dependently typed programming (DTP 11) took place in Nijmegen affiliated with Interactive Theorem Proving 2011 (ITP 11). In September 2011 there also was a DTP workshop in Shonan, Japan.

Abstract We show that the syntactically rich notion of strictly positive families can be reduced to a core type theory with a fixed number of type constructors exploiting the novel notion of indexed containers. As a result, we show indexed containers provide normal forms for strictly positive families in much the same way that containers provide normal forms for strictly positive types. Interestingly, this step from containers to indexed containers is achieved without having to extend the core type theory. Most of the construction presented here has been formalized using the Agda system.

I present a datatype-generic treatment of recursive container types whose elements are guaranteed to be stored in increasing order, with the ordering invariant rolled out systematically. Intervals, lists and binary search trees are instances of the generic treatment. On the journey to this treatment, I report a variety of failed experiments and the transferable learning experiences they triggered. I demonstrate that a total element ordering is enough to deliver insertion and flattening algorithms, and show that (with care about the formulation of the types) the implementations remain as usual. Agda's instance arguments and pattern synonyms maximize the proof search done by the typechecker and minimize the appearance of proofs in program text, often eradicating them entirely. Generalizing to indexed recursive container types, invariants such as size and balance can be expressed in addition to ordering. By way of example, I implement insertion and deletion for 2-3 trees, ensuring both order and balance by the discipline of type checking.

Abstract Programming with dependent types is a blessing and a curse. It is a blessing to be able to bake invariants into the definition of datatypes: We can finally write correct-by-construction software. However, this extreme accuracy is also a curse: A datatype is the combination of a structuring medium together with a special purpose logic. These domain-specific logics hamper any attempt to reuse code across similarly structured data. In this paper, we capitalise on the structural invariants of datatypes. To do so, we first adapt the notion of ornament to our universe of inductive families. We then show how code reuse can be achieved by ornamenting functions. Using these functional ornaments, we capture the relationship between functions such as the addition of natural numbers and the concatenation of lists. With this knowledge, we demonstrate how the implementation of the former informs the implementation of the latter: The users can ask the definition of addition to be lifted to lists and they will only be asked the details necessary to carry on adding lists rather than numbers. Our presentation is formalised in the type theory with a universe of datatypes and all our constructions have been implemented as generic programs, requiring no extension to the type theory.

Haskell's type system has outgrown its Hindley-Milner roots to the extent that it now stretches to the basics of dependently typed programming. In this paper, we collate and classify techniques for programming with dependent types in Haskell, and contribute some new ones. In particular, through extended examples---merge-sort and rectangular tilings---we show how to exploit Haskell's constraint solver as a theorem prover, delivering code which, as Agda programmers, we envy. We explore the compromises involved in simulating variations on the theme of the dependent function space in an attempt to help programmers put dependent types to work, and to inform the evolving language design both of Haskell and of dependently typed languages more broadly.

Total functional programming offers the beguiling vision that, just by virtue of the compiler accepting a program, we are guaranteed that it will always terminate. In the case of programs that are not intended to terminate, e.g., servers, we are guaranteed that programs will always be productive. Productivity means that, even if a program generates an infinite amount of data, each piece will be generated in finite time. The theoretical underpinning for productive programming with infinite output is provided by the category theoretic notion of final coalgebras. Hence, we speak of coprogramming with non-well-founded codata, as a dual to programming with well-founded data like finite lists and trees.

Ornaments aim at taming the multiplication of special-purpose data types in dependently typed programming languages. In type theory, purpose is logic. By presenting data types as the combination of a structure and a logic, ornaments relate these special-purpose data types through their common structure. In the original presentation, the concept of ornament was introduced concretely for an example universe of inductive families in type theory, but it was clear that the notion was more general. This paper digs out the abstract notion of ornaments in the form of a categorical model. As a necessary first step, we abstract the universe of data types using the theory of polynomial functors. We are then able to characterise ornaments as cartesian morphisms between polynomial functors. We thus gain access to powerful mathematical tools that shall help us understand and develop ornaments. We shall also illustrate the adequacy of our model. Firstly, we rephrase the standard ornamental constructions into our framework. Thanks to its conciseness, we gain a deeper understanding of the structures at play. Secondly, we develop new ornamental constructions, by translating categorical structures into type theoretic artefacts.

Programming with dependent types is a blessing and a curse. It is a blessing to be able to bake invariants into the definition of datatypes: we can finally write correct-by-construction software. However, this extreme accuracy is also a curse: a datatype is the combination of a structuring medium together with a special purpose logic. These domain-specific logics hamper any effort of code reuse among similarly structured data. In this paper, we exorcise our datatypes by adapting the notion of ornament to our universe of inductive families. We then show how code reuse can be achieved by ornamenting functions. Using these functional ornaments, we capture the relationship between functions such as the addition of natural numbers and the concatenation of lists. With this knowledge, we demonstrate how the implementation of the former informs the implementation of the latter: the user can ask the definition of addition to be lifted to lists and she will only be asked the details necessary to carry on adding lists rather than numbers. Our presentation is formalised in a type theory with a universe of datatypes and all our constructions have been implemented as generic programs, requiring no extension to the type theory.

I explore programming with the dependently typed functional language, AGDA. I present the progress which AGDA has made, demonstrate its usage in a small development, reflect critically on the state of the art, and speculate about the way ahead. I do not seek to persuade you to adopt AGDA as your primary tool for systems development, but argue that AGDA stimulates new useful ways to think about programming problems and deserves not just curiosity but interest, support and contribution.

We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description - a first-class value in a datatype of descriptions. Moreover, the latter itself has a description. Datatype-generic programming thus becomes ordinary programming. We show some of the resulting generic operations and deploy them in particular, useful ways on the datatype of datatype descriptions itself. Simulations in existing systems suggest that this apparently self-supporting setup is achievable without paradox or infinite regress.

We present the type rules for a dependently typed core calculus together with a straight-forward implementation in Haskell. We explicitly highlight the changes necessary to shift from a simply-typed lambda calculus to the dependently typed lambda calculus. We also describe how to extend our core language with data types and write several small example programs. The article is accompanied by an executable interpreter and example code that allows immediate experimentation with the system we describe.

This special issue of the Journal of Functional Programming collects revised selected articles arising from the inaugural meeting of the Workshop on Mathematically Structured Functional Programming, MSFP 2006, held in Kuressaare, Estonia, on 2 July 2006, with support from the European Union's FP6 IST Coordination Action TYPES. This workshop raised the curtain for the Eighth International Conference on Mathematics of Program Construction, MPC 2006, but where MPC is concerned primarily with extrinsic mathematics supporting the programming process, MSFP has a complementary focus on the mathematics intrinsic to programs themselves. MSFP is about the extraction of functionality from structure.

This paper introduces a small but useful generalisation to the 'derivative' operation on datatypes underlying Huet's notion of 'zipper', giving a concrete representation to one-hole contexts in data which is undergoing transformation. This operator, 'dissection', turns a container-like functor into a bifunctor representing a one-hole context in which elements to the left of the hole are distinguished in type from elements to its right.

Abstract In this article, we introduce Applicative functors – an abstract characterisation of an applicative style of effectful programming, weaker than Monads and hence more widespread. Indeed, it is the ubiquity of this programming pattern that drew us to the abstraction. We retrace our steps in this article, introducing the applicative pattern by diverse examples, then abstracting it to define the Applicative type class and introducing a bracket notation that interprets the normal application syntax in the idiom of an Applicative functor. Furthermore, we develop the properties of applicative functors and the generic operations they support. We close by identifying the categorical structure of applicative functors and examining their relationship both with Monads and with Arrow.

In this paper, we show how to manipulate syntax with binding using a mixed representation of names for free variables (with respect to the task in hand) and de Bruijn indices [5] for bound variables. By doing so, we retain the advantages of both representations: naming supports easy, arithmetic-free manipulation of terms; de Bruijn indices eliminate the need for α-conversion. Further, we have ensured that not only the user but also the implementation need never deal with de Bruijn indices, except within key basic operations.Moreover, we give a hierarchical representation for names which naturally reflects the structure of the operations we implement. Name choice is safe and straightforward. Our technology combines easily with an approach to syntax manipulation inspired by Huet's 'zippers'[10].Without the ideas in this paper, we would have struggled to implement EPIGRAM [19]. Our example-constructing inductive elimination operators for datatype families-is but one of many where it proves invaluable.

Pattern matching has proved an extremely powerful and durable notion in functional programming. This paper contributes a new programming notation for type theory which elaborates the notion in various ways. First, as is by now quite well-known in the type theory community, definition by pattern matching becomes a more discriminating tool in the presence of dependent types, since it refines the explanation of types as well as values. This becomes all the more true in the presence of the rich class of datatypes known as inductive families (Dybjer, 1991). Secondly, as proposed by Peyton Jones (1997) for Haskell, and independently rediscovered by us, subsidiary case analyses on the results of intermediate computations, which commonly take place on the right-hand side of definitions by pattern matching, should rather be handled on the left. In simply-typed languages, this subsumes the trivial case of Boolean guards; in our setting it becomes yet more powerful. Thirdly, elementary pattern matching decompositions have a well-defined interface given by a dependent type; they correspond to the statement of an induction principle for the datatype. More general, user-definable decompositions may be defined which also have types of the same general form. Elementary pattern matching may therefore be recast in abstract form, with a semantics given by translation. Such abstract decompositions of data generalize Wadler's (1987) notion of ‘view’. The programmer wishing to introduce a new view of a type $\mathit{T}$ , and exploit it directly in pattern matching, may do so via a standard programming idiom. The type theorist, looking through the Curry–Howard lens, may see this as proving a theorem , one which establishes the validity of a new induction principle for $\mathit{T}$ . We develop enough syntax and semantics to account for this high-level style of programming in dependent type theory. We close with the development of a typechecker for the simply-typed lambda calculus, which furnishes a view of raw terms as either being well-typed, or containing an error. The implementation of this view is ipso facto a proof that typechecking is decidable.

First-order unification algorithms (Robinson, 1965) are traditionally implemented via general recursion, with separate proofs for partial correctness and termination. The latter tends to involve counting the number of unsolved variables and showing that this total decreases each time a substitution enlarges the terms. There are many such proofs in the literature (Manna & Waldinger, 1981; Paulson, 1985; Coen, 1992; Rouyer, 1992; Jaume, 1997; Bove, 1999). This paper shows how a dependent type can relate terms to the set of variables over which they are constructed. As a consequence, first-order unification becomes a structurally recursive program, and a termination proof is no longer required. Both the program and its correctness proof have been checked using the proof assistant L EGO (Luo & Pollack, 1992; McBride, 1999).

Dependent types reflect the fact that validity of data is often a relative notion by allowing prior data to affect the types of subsequent data. Not only does this make for a precise type system, but also a highly generic one: both the type and the program for each instance of a family of operations can be computed from the data which codes for that instance. Recent experimental extensions to the Haskell type class mechanism give us strong tools to relativize types to other types . We may simulate some aspects of dependent typing by making counterfeit type-level copies of data, with type constructors simulating data constructors and type classes simulating datatypes. This paper gives examples of the technique and discusses its potential.