We present a personal view and strategy for algorithm-supported mathematical theory exploration and draw some conclusions for the desirable functionality of future mathematical software systems. The main points of emphasis are: The use of schemes for bottom-up mathematical invention, the algorithmic generation of conjectures from failing proofs for top-down mathematical invention, and the possibility to program new reasoners within the logic on which the reasoners work ("meta-programming").

Term equations involving individual and sequence variables, and individual and sequence function symbols are studied. Function symbols can have either fixed or flexible arity. A new unification procedure for solving such equations is presented. Decidability of unification is proved. Completeness and almost minimality of the procedure is shown. 1

Term equations involving individual and sequence variables and sequence function symbols are studied. Function symbols can have either fixed or flexible arity. A sequence variable can be instantiated by any finite sequence of terms. A sequence function abbreviates a finite sequence of functions all having the same argument lists. It is proved that solvability of systems of equations of this form is decidable. A new unification procedure that enumerates a complete almost minimal set of solutions is presented, together with variations for special cases. The procedure terminates if the solution set is finite. Applications in various areas of artificial intelligence, symbolic computation, and programming are discussed. 1

Recently, as part of a general formal (i.e. logic based) methodology for mathematical knowledge management we also introduced a method for the automated synthesis of correct algorithms, which we called the lazy thinking method. For a given concrete problem specification (in predicate logic), the method tries out various algorithm schemes and derives specifications for the subalgorithms in the algorithm scheme.

Abstract. We describe a design for a system for mathematical theory exploration that can be extended by implementing new reasoners using the logical input language of the system. Such new reasoners can be applied like the built-in reasoners, and it is possible to reason about them, e.g. proving their soundness, within the system. This is achieved in a practical and attractive way by adding reflection, i.e. a representation mechanism for terms and formulae, to the system’s logical language, and some knowledge about these entities to the system’s basic reasoners. The approach has been evaluated using a prototypical implementation called Mini-Tma. It will be incorporated into the Theorema system. 1

Integro-differential polynomials are a novel generalization of the well-known differential polynomials extensively used in differential algebra [17]. They were introduced in [29] as a kind of universal extensions of integro-differential algebras and have recently been applied in a confluence proof [34] for the rewrite system