Formalization of Formal Topology by Means of the Interactive Theorem Prover Matita

The project entitled " Formalization of Formal Topology by means of the interactive theorem prover Matita " is an official bilateral project between the Universities of Padova and Bologna, funded by the former, active ¡¡¡¡¡¡¡ .mine from march 2008 until august 2010. The project brought together and exploited the synergic collaboration of two communities of researchers, both centered around constructive type theory: ======= from March 2008 until August 2010. The project aimed to bring together and exploit the synergic collaboration of two communities of researchers, both centered around constructive type theory: ¿¿¿¿¿¿¿ .r3432 on one side the Logic Group at the University of Padova, focused on developing formal, pointfree topology within a constructive and predicative framework; on the other side, the Helm group at the University of Bologna, developing the Matita Interactive Theorem Prover [2], a young proof assistant based on the Calculus of Inductive Constructions as its logical foundation. The idea of the project was to formalize and check the new approach to formal topology being developed in Padova by means of Matita, with the aim on one side to assess the truly foundational nature of the theoretical framework (i.e. its reduction to notions so elementary to be easily understood by an automatic device), and on the other to drive the development of Matita, testing the tool on a non trivial set of mathematical results, and addressing from an original theoretical perspective some key problems of constructive interactive proving The project is a rare example of a significant collaboration between mathematicians and computer scientists in handling mathematical knowledge. It is worth to emphasize that the interest in the formalization from the mathematical perspective is not in the automatic verification of the results (on which mathematicians are already largely confident with) but in the phenomenological goal to investigate the most natural way to organize a new foundational framework in a coherent set of interconnected components, their mutual relations and dependencies, our interaction with these representations, and their influence on the concrete mathematical experience (see [1]). Formalization is neither a goal nor a technique, but first and foremost a methodology. The formalization work was mainly focused on Overlap Algebras [4, 3], new algebraic structures designed to ease reasoning about subsets within intuition-istic logic. The main result checked in Matita [8] is the embedding of a suitable