This paper describes a technique for assembling both data types and functions from isolated individual components. We also explore how the same technology can be used to combine free monads and, as a result, structure Haskell's monolithic IO monad.

This paper has something new and positive to say about propositional equality in programming and proof systems based on the Curry-Howard correspondence between propositions and types. We have found a way to present a propositional equality type• which is substitutive, allowing us to reason by replacing equal for equal in propositions; • which reflects the observable behaviour of values rather than their construction: in particular, we have extensionalityfunctions are equal if they take equal inputs to equal outputs; • which retains strong normalisation, decidable typechecking and canonicity-the property that closed normal forms inhabiting datatypes have canonical constructors; • which allows inductive data structures to be expressed in terms of a standard characterisation of well-founded trees; • which is presented syntactically-you can implement it directly, and we are doing so-this approach stands at the core of Epigram 2; • which you can play with now: we have simulated our system by a shallow embedding in Agda 2, shipping as part of the standard examples package for that system [20].Until now, it has always been necessary to sacrifice some of these aspects. The closest attempt in the literature is Altenkirch's construction of a setoid-model for a system with canonicity and extensionality on top of an intensional type theory with proof-irrelevant propositions [4]. Our new proposal simplifies Altenkirch's construction by adopting McBride's heterogeneous approach to equality [18].

This paper exhibits the power of programming with dependent types by dint of embedding three domain-specific languages: Cryptol, a language for cryptographic protocols; a small data description language; and relational algebra. Each example demonstrates particular design patterns inherent to dependently-typed programming. Documenting these techniques paves the way for further research in domain-specific embedded type systems.

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This paper explores the recent addition to Agda enabling reflection, in the style of Lisp and Template Haskell. It gives a brief introduction to using reflection, and details the complexities encountered when automating certain proofs with proof by reflection. It presents a library that can be used for automatically quoting a class of concrete Agda terms to a non-dependent, user-defined inductive data type, alleviating some of the burden a programmer faces when using reflection in a practical setting.