A wide range of computer programs, including compilers and theorem provers, manipulate data structures that involve names and binding. However, the design of programming idioms which allow performing these manipulations in a safe and natural style has, to a large extent, remained elusive.In this paper, we present a novel approach to the problem. Our proposal can be viewed either as a programming language design or as a library: in fact, it is currently implemented within Agda. It provides a safe and expressive means of programming with names and binders. It is abstract enough to support multiple concrete implementations: we present one in nominal style and one in de Bruijn style. We use logical relations to prove that "well-typed programs do not mix names with different scope". We exhibit an adequate encoding of Pitts-style nominal terms into our system.

A wide range of computer programs, including compilers and theorem provers, manipulate data structures that involve names and binding. However, the design of programming idioms which allow performing these manipulations in a safe and natural style has, to a large extent, remained elusive.In this paper, we present a novel approach to the problem. Our proposal can be viewed either as a programming language design or as a library: in fact, it is currently implemented within Agda. It provides a safe and expressive means of programming with names and binders. It is abstract enough to support multiple concrete implementations: we present one in nominal style and one in de Bruijn style. We use logical relations to prove that "well-typed programs do not mix names with different scope". We exhibit an adequate encoding of Pitts-style nominal terms into our system.

De Bruijn indices are a well known technique for programming with names and binders. They provide a representation that is both simple and canonical. However, programming errors tend to be really easy to make. We propose a safer programming interface implemented as a library. Whereas indexing the types of names and terms by a numerical bound is a famous technique, we index them by worlds, a different notion of index that is both finer and more abstract. While being more finely typed, our approach incurs no loss of expressiveness or efficiency. Via parametricity we obtain properties about functions polymorphic on worlds. For instance, well-typed world-polymorphic functions over open λ-terms commute with any renaming of the free variables. Our whole development is conducted within Agda, from the code of the library, to its soundness proof and the properties of external functions. The soundness of our library is demonstrated via the construction of a logical relations argument.

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