Datatype-generic programming makes it possible to define a construction once and apply it to a large class of datatypes. It is often used to avoid code duplication in languages that encourage the definition of custom datatypes, in particular state-of-the-art dependently typed languages where one can have many variants of the same datatype with different type-level invariants. In addition to giving access to familiar programming constructions for free, datatype-generic programming in the dependently typed setting also allows for the construction of generic proofs. However, the current interfaces available for this purpose are needlessly hard to use or are limited in the range of datatypes they handle. In this paper, we describe the design of a library for safe and user-friendly datatype-generic programming in the Agda language. Generic constructions in our library are regular Agda functions over a broad universe of datatypes, yet they can be specialized to native Agda datatypes with a simple one-liner. Furthermore, we provide building blocks so that library designers can too define their own datatype-generic constructions.

Most existing programming languages provide little support to formally state and prove properties about programs. Adding such capabilities is far from trivial, as it requires significant re-engineering of the existing compilers and tools. This paper proposes a novel technique to write correct-by-construction programs in languages without built-in verification capabilities, while maintaining the ability to use existing tools. This is achieved in three steps. Firstly, we give a shallow embedding of the language (or a subset) into a dependently typed language. Secondly, we write a program in that embedding, and we use dependent types to guarantee correctness properties of interest within the embedding. Thirdly, we extract a program written in the original language, so it can be used with existing compilers and tools. Our main insight is that it is possible to express all three steps in a single language that supports both dependent types and reflection. Essentially, this allows us to express a program, its formal properties, and a compiler for it hand-in-hand, offering a lot of flexibility to programmers. We demonstrate this three-step approach by embedding a subset of the PostScript language in Agda, and illustrating it with several short examples. Thus we use the power of reflection to bring the benefits of dependent types to languages that had to go without them so far.

Dependently typed programming languages and proof assistants such as Agda and Coq rely on computation to automatically simplify expressions during type checking. To overcome the lack of certain programming primitives or logical principles in those systems, it is common to appeal to axioms to postulate their existence. However, one can only postulate the bare existence of an axiom, not its computational behaviour. Instead, users are forced to postulate equality proofs and appeal to them explicitly to simplify expressions, making axioms dramatically more complicated to work with than built-in primitives. On the other hand, the equality reflection rule from extensional type theory solves these problems by collapsing computation and equality, at the cost of having no practical type checking algorithm. This paper introduces Rewriting Type Theory (RTT), a type theory where it is possible to add computational assumptions in the form of rewrite rules. Rewrite rules go beyond the computational capabilities of intensional type theory, but in contrast to extensional type theory, they are applied automatically so type checking does not require input from the user. To ensure type soundness of RTT-as well as effective type checking-we provide a framework where confluence of user-defined rewrite rules can be checked modularly and automatically, and where adding new rewrite rules is guaranteed to preserve subject reduction. The properties of RTT have been formally verified using the MetaCoq framework and an implementation of rewrite rules is already available in the Agda proof assistant.

Dependently typed languages such as Coq and Agda can statically guarantee the correctness of our proofs and programs. To provide this guarantee, they restrict users to certain schemes a- such as strictly positive datatypes, complete case analysis, and well-founded induction a- that are known to be safe. However, these restrictions can be too strict, making programs and proofs harder to write than necessary. On a higher level, they also prevent us from imagining the different ways the language could be extended. In this paper I show how to extend a dependently typed language with user-defined higher-order non-linear rewrite rules. Rewrite rules are a form of equality reflection that is applied automatically by the typechecker. I have implemented rewrite rules as an extension to Agda, and I give six examples how to use them both to make proofs easier and to experiment with extensions of type theory. I also show how to make rewrite rules interact well with other features of Agda such as-equality, implicit arguments, data and record types, irrelevance, and universe level polymorphism. Thus rewrite rules break the chains on computation and put its power back into the hands of its rightful owner: Yours.

In a dependently typed language, we can guarantee correctness of our programmes by providing formal proofs. To check them, the typechecker elaborates these programs and proofs into a low-level core language. However, this core language is by nature hard to understand by mere humans, so how can we know we proved the right thing? This question occurs in particular for dependent copattern matching, a powerful language construct for writing programmes and proofs by dependent case analysis and mixed induction/coinduction. A definition by copattern matching consists of a list of clauses that are elaborated to a case tree, which can be further translated to primitive eliminators. In previous work this second step has received a lot of attention, but the first step has been mostly ignored so far. We present an algorithm elaborating definitions by dependent copattern matching to a core language with inductive data types, coinductive record types, an identity type, and constants defined by well-typed case trees. To ensure correctness, we prove that elaboration preserves the first-match semantics of the user clauses. Based on this theoretical work, we reimplement the algorithm used by Agda to check left-hand sides of definitions by pattern matching. The new implementation is at the same time more general and less complex, and fixes a number of bugs and usability issues with the old version. Thus, we take another step towards the formally verified implementation of a practical dependently typed language.