Inspired by Plotkin and Power’s algebraic treatment of computational effects and the principle of notions of computations as monoids, we propose a categorical framework for equational theories and models of monoids equipped with operations. This framework generalises Plotkin and Power’s algebraic treatment of effectful operations taking or returning values as input or output to operations that may take or return computations as input or output. Additionally, to give semantic models of computational effects in a modular way, we introduce a formal theory of modular constructions of algebraic structures based on the framework of lifting functors along fibrations.

Functional programmers have many things for which to thank the late David Turner: design decisions he made in his languages SASL, KRC, and Miranda over the last 50 years are still influential and inspirational now. In particular, Turner was a strong advocate of lazy evaluation and of list comprehensions. As an illustration of these techniques, he popularized a one-line recursive “sieve” to generate the infinite list of prime numbers. Turner called this algorithm The Sieve of Eratosthenes. In a lovely paper called “The Genuine Sieve of Eratosthenes”, Melissa O’Neill argued that Turner’s program is not in fact a faithful implementation of the algorithm, and gave a detailed presentation using priority queues of the real thing. She included a variation by Richard Bird, which uses only lists but makes clever use of circular programming. Bird describes his circular program again in his textbook “Thinking Functionally with Haskell”, and sets its proof of correctness as an exercise. In particular, why is this circular program productive? Unfortunately, Bird’s hint for a solution is incorrect. So what should a proof look like? One of the last projects Turner worked on was the notion of “Total Functional Programming”. He observed that most programs are already structurally recursive or corecursive, therefore guaranteed respectively terminating or productive; he conjectured that “with more practice we will find this is always true”. We explore Bird’s circular Sieve of Eratosthenes as a challenge problem for Turner’s Total Functional Programming.

The formal theory of monads shows that much of the theory of monads can be developed in the abstract at the level of 2-categories. This means that results about monads can be established once and for all and simply instantiated in settings such as enriched category theory. Unfortunately, these results can be hard to reason about as they involve more abstract machinery. In this paper, we present the formal theory of monads in terms of string diagrams — a graphical language for 2-categorical calculations. Using this perspective, we show that many aspects of the theory of monads, such as the Eilenberg–Moore and Kleisli resolutions of monads, liftings, and distributive laws, can be understood in terms of systematic graphical calculational reasoning. This paper will serve as an introduction both to the formal theory of monads and to the use of string diagrams, in particular, their application to calculations in monad theory.

We propose a notion of syntax with metavariables that generalises Miller’s decidable pattern fragment of second-order unification for simply typed $𝜆$ -calculus. Using categorical semantics, we show that, under some conditions, a generalisation of Miller’s unification algorithm applies. To illustrate our semantic analysis, we implemented our generic unification algorithm in Agda. The syntax with metavariables given as input of the algorithm is specified by a notion of signature generalising binding signatures, covering a wide range of examples, including ordered $𝜆$ -calculus and (intrinsic) polymorphic syntax such as System F. Although we do not explicitly handle equations, we also tackle simply typed $𝜆$ -calculus modulo $𝛽$ - and $𝜂$ -equations (Miller’s original setting) by working on the syntax of normal forms.

Contemporary proof assistants such as Coq require that recursive functions be terminating and corecursive functions be productive to maintain logical consistency of their type theories, and some ensure these properties using syntactic checks. However, being syntactic, they are inherently delicate and restrictive, preventing users from easily writing obviously terminating or productive functions at their whim. Meanwhile, there exist many sized type theories that perform type-based termination and productivity checking, including theories based on the Calculus of (Co)Inductive Constructions (CIC), the core calculus underlying Coq. These theories are more robust and compositional in comparison. So why haven’t they been adapted to Coq? In this paper, we venture to answer this question with CIC $\hat{\ast }$ , a sized type theory based on CIC. It extends past work on sized types in CIC with additional Coq features such as global and local definitions. We also present a corresponding size inference algorithm and implement it within Coq’s kernel; for maximal backward compatibility with existing Coq developments, it requires no additional annotations from the user. In our evaluation of the implementation, we find a severe performance degradation when compiling parts of the Coq standard library, inherent to the algorithm itself. We conclude that if we wish to maintain backward compatibility, using size inference as a replacement for syntactic checking is impractical in terms of performance.

The other day, I was assembling lecture material for a course on Agda. Pursuing an application-driven approach, I was looking for correctness proofs of popular algorithms. One of my all-time favourites is Huffman data compression (Huffman, 1952). Even though it is probably safe to assume that you are familiar with this algorithmic gem, a brief reminder of the essential idea may not be amiss.

Many programmers use dependently typed languages such as Coq to machine-verify high-assurance software. However, existing compilers for these languages provide no guarantees after compiling, nor when linking after compilation. Type-preserving compilers preserve guarantees encoded in types and then use type checking to verify compiled code and ensure safe linking with external code. Unfortunately, standard compiler passes do not preserve the dependent typing of commonly used (intensional) type theories. This is because assumptions valid in simpler type systems no longer hold, and intensional dependent type systems are highly sensitive to syntactic changes, including compilation. We develop an A-normal form (ANF) translation with join-point optimization—a standard translation for making control flow explicit in functional languages—from the Extended Calculus of Constructions (ECC) with dependent elimination of booleans and natural numbers (a representative subset of Coq). Our dependently typed target language has equality reflection, allowing the type system to encode semantic equality of terms. This is key to proving type preservation and correctness of separate compilation for this translation. This is the first ANF translation for dependent types. Unlike related translations, it supports the universe hierarchy, and does not rely on parametricity or impredicativity.

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 .

The observation that program structure follows data structure is a key lesson in introductory programming: good hints for possible program designs can be found by considering the structure of the data concerned. In particular, this lesson is a core message of the influential textbook “How to Design Programs” by Felleisen, Findler, Flatt, and Krishnamurthi. However, that book discusses using only the structure of input data for guiding program design, typically leading towards structurally recursive programs. We argue that novice programmers should also be taught to consider the structure of output data, leading them also towards structurally corecursive programs.

Gradually typed languages are designed to support both dynamically typed and statically typed programming styles while preserving the benefits of each. Sound gradually typed languages dynamically check types at runtime at the boundary between statically typed and dynamically typed modules. However, there is much disagreement in the gradual typing literature over how to enforce complex types such as tuples, lists, functions and objects. In this paper, we propose a new perspective on the design of runtime gradual type enforcement: runtime type casts exist precisely to ensure the correctness of certain type-based refactorings and optimizations. For instance, for simple types, a language designer might desire that beta-eta equality is valid. We show that this perspective is useful by demonstrating that a cast semantics can be derived from beta-eta equality. We do this by providing an axiomatic account program equivalence in a gradual cast calculus in a logic we call gradual type theory (GTT). Based on Levy’s call-by-push-value, GTT allows us to axiomatize both call-by-value and call-by-name gradual languages. We then show that we can derive the behavior of casts for simple types from the corresponding eta equality principle and the assumption that the language satisfies a property called graduality, also known as the dynamic gradual guarantee. Since we can derive the semantics from the assumption of eta equality, we also receive a useful contrapositive: any observably different cast semantics that satisfies graduality must violate the eta equality. We show the consistency and applicability of our axiomatic theory by proving that a contract-based implementation using the lazy cast semantics gives a logical relations model of our type theory, where equivalence in GTT implies contextual equivalence of the programs. Since GTT also axiomatizes the dynamic gradual guarantee, our model also establishes this central theorem of gradual typing. The model is parameterized by the implementation of the dynamic types, and so gives a family of implementations that validate type-based optimization and the gradual guarantee.

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.

Plotkin and Pretnar’s effect handlers offer a versatile abstraction for modular programming with user-defined effects. This paper focuses on foundations for implementing effect handlers, for the three different kinds of effect handlers that have been proposed in the literature: deep, shallow, and parameterised. Traditional deep handlers are defined by folds over computation trees and are the original construct proposed by Plotkin and Pretnar. Shallow handlers are defined by case splits (rather than folds) over computation trees. Parameterised handlers are deep handlers extended with a state value that is threaded through the folds over computation trees. We formulate the extensions both directly and via encodings in terms of deep handlers and illustrate how the direct implementations avoid the generation of unnecessary closures. We give two distinct foundational implementations of all the kinds of handlers we consider: a continuation-passing style (CPS) transformation and a CEK-style abstract machine. In both cases, the key ingredient is a generalisation of the notion of continuation to accommodate stacks of effect handlers. We obtain our CPS translation through a series of refinements as follows. We begin with a first-order CPS translation into untyped lambda calculus which manages a stack of continuations and handlers as a curried sequence of arguments. We then refine the initial CPS translation by uncurrying it to yield a properly tail-recursive translation and then moving towards more and more intensional representations of continuations in order to support different kinds of effect handlers. Finally, we make the translation higher order in order to contract administrative redexes at translation time. Our abstract machine design then uses the same generalised continuation representation as the CPS translation. We have implemented both the abstract machine and the CPS transformation (plus extensions) as backends for the Links web programming language.

We compare the expressive power of three programming abstractions for user-defined computational effects: Plotkin and Pretnar’s effect handlers, Filinski’s monadic reflection, and delimited control. This comparison allows a precise discussion about the relative expressiveness of each programming abstraction. It also demonstrates the sensitivity of the relative expressiveness of user-defined effects to seemingly orthogonal language features. We present three calculi, one per abstraction, extending Levy’s call-by-push-value. For each calculus, we present syntax, operational semantics, a natural type-and-effect system, and, for effect handlers and monadic reflection, a set-theoretic denotational semantics. We establish their basic metatheoretic properties: safety, termination, and, where applicable, soundness and adequacy. Using Felleisen’s notion of a macro translation, we show that these abstractions can macro express each other, and show which translations preserve typeability. We use the adequate finitary set-theoretic denotational semantics for the monadic calculus to show that effect handlers cannot be macro expressed while preserving typeability either by monadic reflection or by delimited control. Our argument fails with simple changes to the type system such as polymorphism and inductive types. We supplement our development with a mechanised Abella formalisation.

A sound gradual type system ensures that untyped components of a program can never break the guarantees of statically typed components. This assurance relies on runtime checks, which in turn impose performance overhead in proportion to the frequency and nature of interaction between typed and untyped components. The literature on gradual typing lacks rigorous descriptions of methods for measuring the performance of gradual type systems. This gap has consequences for the implementors of gradual type systems and developers who use such systems. Without systematic evaluation of mixed-typed programs, implementors cannot precisely determine how improvements to a gradual type system affect performance. Developers cannot predict whether adding types to part of a program will significantly degrade (or improve) its performance. This paper presents the first method for evaluating the performance of sound gradual type systems. The method quantifies both the absolute performance of a gradual type system and the relative performance of two implementations of the same gradual type system. To validate the method, the paper reports on its application to 20 programs and 3 implementations of Typed Racket.

In higher order settings, control-flow analysis aims to model the propagation of both data and control by finitely approximating program behaviors across all possible executions. The polyvariance of an analysis describes the number of distinct abstract representations, or variants, for each syntactic entity (e.g., functions, variables, or intermediate expressions). Monovariance, one of the most basic forms of polyvariance, maintains only a single abstract representation for each variable or expression. Other polyvariant strategies allow a greater number of distinct abstractions and increase analysis complexity with the aim of increasing analysis precision. For example, k -call sensitivity distinguishes flows by the most recent k call sites, k -object sensitivity by a history of allocation points, and argument sensitivity by a tuple of dynamic argument types. From this perspective, even a concrete operational semantics may be thought of as an unboundedly polyvariant analysis. In this paper, we develop a unified methodology that fully captures this design space. It is easily tunable and guarantees soundness regardless of how tuned. We accomplish this by extending the method of abstracting abstract machines, a systematic approach to abstract interpretation of operational abstract-machine semantics. Our approach permits arbitrary instrumentation of the underlying analysis and arbitrary tuning of an abstract-allocation function. We show that the design space of abstract allocators both unifies and generalizes existing notions of polyvariance. Simple changes to the behavior of this function recapitulate classic styles of analysis and yield novel combinations and variants.

We consider the task of constructing 2-3 trees. Given a sequence of elements we seek to build a 2-3 tree–in linear time–that contains the elements in symmetric order. We discuss three approaches: top-down, bottom-up, and incremental. The incremental approach is more flexible than the other two in that it allows us to interleave the construction work with other operations, for example, queries.

Iris is a framework for higher-order concurrent separation logic, which has been implemented in the Coq proof assistant and deployed very effectively in a wide variety of verification projects. Iris was designed with the express goal of simplifying and consolidating the foundations of modern separation logics, but it has evolved over time, and the design and semantic foundations of Iris itself have yet to be fully written down and explained together properly in one place. Here, we attempt to fill this gap, presenting a reasonably complete picture of the latest version of Iris (version 3.1), from first principles and in one coherent narrative.

Enumerations represented as bijections between the natural numbers and elements of some given type have recently garnered interest in property-based testing because of their efficiency and flexibility. There are, however, many ways of defining these bijections, some of which are better than others. This paper offers a new property of enumeration combinators called fairness that identifies enumeration combinators that are better suited to property-based testing. Intuitively, the result of a fair combinator indexes into its argument enumerations equally when constructing its result. For example, extracting the nth element from our enumeration of three-tuples indexes about $\sqrt[3]{𝑛}$ elements into each of its components instead of, say, indexing $\sqrt[2]{𝑛}$ into one and $\sqrt[4]{𝑛}$ into the other two, as you would if a three-tuple were built out of nested pairs. Similarly, extracting the nth element from our enumeration of a three-way union returns an element that is $\frac{𝑛}{3}$ into one of the argument enumerators. The paper presents a semantics of enumeration combinators, a theory of fairness, proofs establishing fairness of our new combinators and that some combinations of fair combinators are not fair. We also report on an evaluation of fairness for the purpose of finding bugs in programming-language models. We show that fair enumeration combinators have complementary strengths to an existing, well-tuned ad hoc random generator (better on short time scales and worse on long time scales) and that using unfair combinators is worse across the board.

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type is proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also for paths. In this paper, we consider a programming application of higher inductive types. Version control systems such as Darcs are based on the notion of patches—syntactic representations of edits to a repository. We show how patch theory can be developed in homotopy type theory. Our formulation separates formal theories of patches from their interpretation as edits to repositories. A patch theory is presented as a higher inductive type. Models of a patch theory are given by maps out of that type, which, being functors, automatically preserve the structure of patches. Several standard tools of homotopy theory come into play, demonstrating the use of these methods in a practical programming context.

The main aim of the paper is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between α-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact-oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the “easy” direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.

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.

The study of programming with and reasoning about inductive datatypes such as lists and trees has benefited from the simple categorical principle of initial algebras. In initial algebra semantics, each inductive datatype is represented by an initial f -algebra for an appropriate functor f . The initial algebra principle then supports the straightforward derivation of definitional principles and proof principles for these datatypes. This technique has been expanded to a whole methodology of structured functional programming, often called origami programming. In this article we show how to extend initial algebra semantics from pure inductive datatypes to inductive datatypes interleaved with computational effects. Inductive datatypes interleaved with effects arise naturally in many computational settings. For example, incrementally reading characters from a file generates a list of characters interleaved with input/output actions, and lazily constructed infinite values can be represented by pure data interleaved with the possibility of non-terminating computation. Straightforward application of initial algebra techniques to effectful datatypes leads either to unsound conclusions if we ignore the possibility of effects, or to unnecessarily complicated reasoning because the pure and effectful concerns must be considered simultaneously. We show how pure and effectful concerns can be separated using the abstraction of initial f -and- m -algebras, where the functor f describes the pure part of a datatype and the monad m describes the interleaved effects. Because initial f -and- m -algebras are the analogue for the effectful setting of initial f -algebras, they support the extension of the standard definitional and proof principles to the effectful setting. Initial f -and- m -algebras are originally due to Filinski and Støvring, who studied them in the category Cpo. They were subsequently generalised to arbitrary categories by Atkey, Ghani, Jacobs, and Johann in a FoSSaCS 2012 paper. In this article we aim to introduce the general concept of initial f -and- m -algebras to a general functional programming audience.

Moggi’s Computational Monads and Power et al .‘s equivalent notion of Freyd category have captured a large range of computational effects present in programming languages. Examples include non-termination, non-determinism, exceptions, continuations, side effects and input/output. We present generalisations of both computational monads and Freyd categories, which we call parameterised monads and parameterised Freyd categories, that also capture computational effects with parameters. Examples of such are composable continuations, side effects where the type of the state varies and input/output where the range of inputs and outputs varies. By considering structured parameterisation also, we extend the range of effects to cover separated side effects and multiple independent streams of I/O. We also present two typed λ-calculi that soundly and completely model our categorical definitions – with and without symmetric monoidal parameterisation – and act as prototypical languages with parameterised effects.

The Iterator pattern gives a clean interface for element-by-element access to a collection, independent of the collection’s shape. Imperative iterations using the pattern have two simultaneous aspects: mapping and accumulating . Various existing functional models of iteration capture one or other of these aspects, but not both simultaneously. We argue that C. McBride and R. Paterson’s applicative functors (Applicative programming with effects, J. Funct. Program. , 18 (1): 1–13, 2008), and in particular the corresponding traverse operator, do exactly this, and therefore capture the essence of the Iterator pattern. Moreover, they do so in a way that nicely supports modular programming. We present some axioms for traversal, discuss modularity concerns and illustrate with a simple example, the wordcount problem.

We have built the first family of tagless interpretations for a higher-order typed object language in a typed metalanguage (Haskell or ML) that require no dependent types, generalized algebraic data types, or postprocessing to eliminate tags. The statically type-preserving interpretations include an evaluator, a compiler (or staged evaluator), a partial evaluator, and call-by-name and call-by-value continuation-passing style (CPS) transformers. Our principal technique is to encode de Bruijn or higher-order abstract syntax using combinator functions rather than data constructors. In other words, we represent object terms not in an initial algebra but using the coalgebraic structure of the λ-calculus. Our representation also simulates inductive maps from types to types, which are required for typed partial evaluation and CPS transformations. Our encoding of an object term abstracts uniformly over the family of ways to interpret it, yet statically assures that the interpreters never get stuck. This family of interpreters thus demonstrates again that it is useful to abstract over higher-kinded types.

Abstract Regular-expression derivatives are an old, but elegant, technique for compiling regular expressions to deterministic finite-state machines. It easily supports extending the regular-expression operators with boolean operations, such as intersection and complement. Unfortunately, this technique has been lost in the sands of time and few computer scientists are aware of it. In this paper, we reexamine regular-expression derivatives and report on our experiences in the context of two different functional-language implementations. The basic implementation is simple and we show how to extend it to handle large character sets (e.g., Unicode). We also show that the derivatives approach leads to smaller state machines than the traditional algorithm given by McNaughton and Yamada.

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.

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 $\text{𝑇}$ , 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 $\text{𝑇}$ . 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 LEGO (Luo & Pollack, 1992; McBride, 1999).