Program verification in synthetic domain theory

Synthetic Domain Theory provides a setting to consider domains as sets with certain closure properties for computing suprema of ascending chains. As a consequence the notion of domain can be internalized which allows one to construct and reason about solutions of recursive domain equations. Moreover, one can derive that all functions are continuous. In this thesis such a synthetic theory of domains (Σ-domains) is developed based on a few axioms formulated in an adequate intuitionistic higher-order logic. This leads to an elegant theory of domains. It integrates the positive features of several approaches in the literature. In contrast to those, however, it is model independent and can therefore be formalized. A complete formalization of the whole theory of Σ-domains has been coded into a proof-checker (Lego) for impredicative type theory. There one can exploit dependent types in order to express program modules and modular specifications. As an application of this theory an entirely formal correctness proof of the Sieve of Eratosthenes, a recursive function on recursively defined streams, is presented. This demonstrates that Synthetic Domain Theory is amenable to formal program verification. A realizability model is defined which ensures that the theory is consistent. Suggestions for the formalization of two other approaches for Synthetic Domain Theory (Σ-replete objects and well-completes) are presented. Putting all this together, one gets a new Logic of Computable Functions more expressive, more comfortable, and more powerful than the LCF-language.