ATS/LF: A type system for construct- ing proofs as total functional programs

The development of Applied Type System (ATS) [36, 31] stems from an earlier attempt to introduce dependent types into practical programming [38, 37]. While there is already a framework Pure Type System [4] (PTS) that offers a simple and general approach to designing and formalizing type systems, it has been understood that there are some acute problems with PTS that make it difficult to support, especially, in the presence of dependent types various common programming features such as general recursion [10], recursive types [19], effects [17], exceptions [15] and input/output, etc. To address such limitations of PTS, ATS is proposed to allow for designing and formalizing type systems that can readily accommodate common realistic programming features. The key salient feature of ATS lies in a complete separation of the statics, where types are formed and reasoned about, from the dynamics, where programs are constructed and evaluated. With this separation, it is no longer possible for a program to occur in a type as is otherwise allowed in PTS. We have now designed and implemented ATS, a programming language with its type system rooted in ATS. The work we report here is primarily motivated by a need for combining programs with proofs in ATS. Before going into further details, we would like to present an example to clearly illustrate the motivation. In ATS, we can declare a function append (through a form of syntax rather similar to that of Standard ML [20]) in Figure 1. We use list as a type constructor. When applied to a type T and an integer I, list(T, I) forms a type for lists of length I in which each element is of type T . The two list constructors nil and cons are assigned the following