Abstract. In this paper, we present a formalisation of elementary group theory done in Coq. This work is the first milestone of a long-term effort to formalise Feit-Thompson theorem. As our further developments will heavily rely on this initial base, we took special care to articulate it in the most compositional way. 1

We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a first-class value in a datatype of descriptions. Moreover, the latter itself has a description. Datatype-generic programming thus becomes ordinary programming. We show some of the resulting generic operations and deploy them in particular, useful ways on the datatype of datatype descriptions itself. Surprisingly this apparently self-supporting setup is achievable without paradox or infinite regress. 1.

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Interactive Theorem Provers (ITPs) are tools meant to assist the user during the formal development of mathematics. Automatic proof searching procedures are a desirable aid, and most ITPs supply the user with an extensive set of facilities to improve automation. However, the black-box nature of most automatic procedure conflicts with the interactive nature of these tools: a newcomer running an automatic procedure learns nothing by its execution (especially in case of failure), and a trained user has no opportunities to interactively guide the procedure towards the solution, e.g. pruning wrong or not promising branches of the search tree. In this paper we discuss the implementation of the resolution based automatic procedure of the Matita ITP, explicitly conceived to be interactively driven by the user through a suitable, simple graphical interface. Keywords: Interactive theorem proving, SLD resolution, automation

This paper presents GMETA: a generic framework for first-order representations of variable binding that provides once and for all many of the so-called infrastructure lemmas and definitions required in mechanizations of formal metatheory. The framework employs datatype-generic programming and modular programming techniques to provide a universe representing a family of datatypes. This universe is generic in two different ways: it is language-generic in the sense that several object languages can be represented within the universe; and it is representation-generic, meaning that it is parameterizable over the particular choice of firstorder representations for binders (for example, locally nameless or de Bruijn). Using this universe, several libraries providing generic infrastructure lemmas and definitions are implemented. These libraries are used in case studies based on the POPLmark challenge, showing that dealing with challenging binding constructs, like the ones found in System F<:, is possible with GMETA. All of GMETA’s generic infrastructure is implemented in the Coq theorem prover, ensuring the soundness of that infrastructure. Furthermore, due to GMETA’s modular design, the libraries can be easily used, extended and customized by end users. 1.

We describe some formal topological results, formalized in Matita 1/2, presented in predicative intuitionistic logic and in terms of Overlap Algebras. Overlap Algebras are new algebraic structures designed to ease reasoning about subsets in an algebraic way within intuitionistic logic. We find that they also ease the formalization of formal topological results in an interactive theorem prover. Our main result is the existence of a functor between two categories of ‘generalized topological spaces’, one with points (Basic Pairs) and the other point-free (Basic Topologies). The reported formalization is part as a wider scientific collaboration with the inventor of the theory, Giovanni Sambin. His goal is to verify in what sense, and with what difficulties, his theory is ‘implementable’. We check that all intermediate constructions respect the stringent size requirements imposed by predicative logic. The formalization is quite unusual, since it has to make explicit size information that is often hidden. We found that the version of Matita used for the formalization was largely inappropriate. The formalization drove several major improvements of Matita that will be integrated in the next major release (Matita 1.0). We show some motivating examples for these improvements, taken directly from the formalization. We also describe a possibly sub-optimal solution in Matita 1/2, exploitable in other similar systems. We briefly discuss a better solution available in Matita 1.0. 1.

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