Abstract. The prototype of a content based search engine for mathematical knowledge supporting a small set of queries requiring matching and/or typing operations is described. The prototype — called Whelp — exploits a metadata approach for indexing the information that looks far more flexible than traditional indexing techniques for structured expressions like substitution, discrimination, or context trees. The prototype has been instantiated to the standard library of the Coq proof assistant extended with many user contributions. 1

When proving a theorem, one makes intermediate claims, leaving parts temporarily unspecified. These `open' parts may be proofs but also terms. In interactive theorem proving systems, one prominently deals with these `unfinished proofs' and `open terms'. We study these `open phenomena' from the point of view of logic. This amounts to finding a correctness criterion for `unfinished proofs' (where some parts may be left open, but the logical steps that have been made are still correct). Furthermore we want to capture the notion of `proof state'. Proof states are the objects that interactive theorem provers operate on and we want to understand them in terms of logic. In this paper we define `open higher order predicate logic', an extension of higher order logic with unfinished (open) proofs and open terms. Then we define a type theoretic variant of this open higher order logic together with a formulas-as-types embedding from open higher order logic to this type theory. We show how this type theory nicely captures the notion of `proof state', which is now a type-theoretic context.

Abstract. We address the problem of representing mathematical structures in a proof assistant which: 1) is based on a type theory with dependent types, telescopes and a computational version of Leibniz equality; 2) implements coercive subtyping, accepting multiple coherent paths between type families; 3) implements a restricted form of higher order unification and type reconstruction. We show how to exploit the previous quite common features to reduce the “syntactic ” gap between pen&paper and formalised algebra. However, to reach our goal we need to propose unification and type reconstruction heuristics that are slightly different from the ones usually implemented. We have implemented them in Matita. 1

Mathematical notation has the characteristic of being ambiguous: operators can be overloaded and information that can be deduced is often omitted. Mathematicians are used to this ambiguity and can easily disambiguate a formula making use of the context and of their ability to find the right interpretation. Software applications that have to deal with formulae usually avoid these issues by fixing an unambiguous input notation. This solution is annoying for mathematicians because of the resulting tricky syntaxes and becomes a show stopper to the simultaneous adoption of tools characterized by different input languages. In this paper we present an efficient algorithm suitable for ambiguous parsing of mathematical formulae. The only requirement of the algorithm is the existence of a “validity” predicate over abstract syntax trees of incomplete formulae with placeholders. This requirement can be easily fulfilled in the applicative area of interactive proof assistants, and in several other areas of Mathematical Knowledge Management.

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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

In this paper we make an overview of some existing systems of open (incomplete) terms including ALF, Typelab, OLEG, L, Automath, c and s e.

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