Flat theory with sequence variables and flexible arity symbols has a decidable infinitary matching and unification. We briefly describe a minimal complete at matching procedure and discuss its relations with the at matching implemented in the Mathematica system.

We describe the deductive and proof presentation capabilities of a rule-based system implemented in Mathematica. The system can compute proof objects, which are internal representations of deduction derivations which respect a specification given by the user. It can also visualize such deductions in human readable format, at various levels of detail. The presentation of the computed proof objects is done in a natural-language style which is derived and simplified for our needs from the proof presentation styles of Theorema.

We describe a matching algorithm for terms built over flexible arity function symbols and context, function, sequence, and individual variables. The algorithm is called a context sequence matching algorithm. Context variables allow matching to descend in term-trees to arbitrary depth. Sequence variables allow matching to move in term-trees in arbitrary breadth. The ability to explore terms in two orthogonal directions in a uniform way may be useful for querying data available as a large term, like XML documents. We extend the algorithm to process regular constraints and discuss its possible application in XML querying.