Combining programming with theorem proving

Applied Type System (ATS) is recently proposed as a framework for designing and formalizing (advanced) type systems in support Of practical programming. In ATS, the definition of type equality involves a constraint relation, which may or may not be algorithmically decidable. To support practical programming, we adopted a design in the past that imposes certain restrictions on the syntactic form of constraints so that some effective means can be found for solving constraints automatically. Evidently, this is a rather ad hoc design in its nature. In this paper, we rectify the situation by presenting a fundamentally different design, which we claim to be both novel and practical. Instead of imposing syntactical restrictions on constraints, we provide a means for the programmer to construct proofs that attest to the validity of constraints. In particular, we are to accommodate a programming paradigm that enables the programmer to combine programming with theorem proving. Also we present some concrete examples in support of the practicality of this design.