Universita` degli Studi di Udine Dipartimento di Matematica e Informatica Dottorato di Ricerca in Informatica Institut National Polytechnique de Lorraine Ecole´ Nationale Superieure´ des Mines, Nancy Laboratoire Lorraine de Recherche en Informatique et ses Applications Doctorat de Recherche en Informatique Ph.D. Thesis Certified Reasoning on Real Numbers and Objects in Co-inductive Type Theory Candidate: Supervisors: Alberto Ciaffaglione Furio Honsell Pietro Di Gianantonio Claude Kirchner Luigi Liquori °c 2003, Alberto Ciaffaglione Author’s e-mail: ciaff
[email protected], Alberto.Ciaff
[email protected] Author’s address: Dipartimento di Matematica e Informatica Universit`adegli Studi di Udine via delle Scienze, 206 33100 Udine (Italia) Abstract In this thesis we adopt Formal Methods based on Type Theory for reasoning on the semantics of computer programs. The ultimate goal is their certification, that is, to prove that a fragment of software meets its formal specification. Application areas of our research are the representation of and computation on the Real Numbers and the Object-oriented Languages based on Objects. In our investigation, Coinductive specification and proof principles play an essential role. In the first part of the thesis we deal with Constructive Real Numbers. We construct the reals using streams, i.e. infinite sequences, of signed digits. We work with constructive logic, that is, we develop “algorithmic” mathematics. We implement the Real Numbers in Coq using streams, which are managed using coinductive judgments and corecursive algorithms. Then we introduce a constructive axiomatization and we use it for proving the adequacy of our construction. In the second part of the thesis we approach Object-based Calculi with side-effects, in order to address important properties, such as Type Soundness, formally.