Hereditarily finite (HF) set theory provides a standard universe of sets, but with no infinite sets. Its utility is demonstrated through a formalisation of the theory of regular languages and finite automata, including the Myhill-Nerode theorem and Brzozowski's minimisation algorithm. The states of an automaton are HF sets, possibly constructed by product, sum, powerset and similar operations.
We present a reflexive tactic for deciding the equational theory of Kleene algebras in the Coq proof assistant. This tactic relies on a careful implementation of efficient finite automata algorithms, so that it solves casual equations instantaneously and properly scales to larger expressions. The decision procedure is proved correct and complete: correctness is established w.r.t. any model by formalising Kozen's initiality theorem; a counter-example is returned when the given equation does not hold. The correctness proof is challenging: it involves both a precise analysis of the underlying automata algorithms and a lot of algebraic reasoning. In particular, we have to formalise the theory of matrices over a Kleene algebra. We build on the recent addition of firstorder typeclasses in Coq in order to work efficiently with the involved algebraic structures. Adhesive dressing compositions are disclosed which are useful in restoring and improving adhesive joint surfaces such as grout surfaces in ceramic tile installations by easy, efficient methods. Such compositions form stain resistant, water repellent, washable coverings which adhere to most adhesive surfaces and further have properties of preferential adherability to certain adhesive surfaces compared with adjacent adherend surfaces. The compositions comprise a polymer in the form of an emulsion, an alkali-thickenable polymer, an alkaline material and water with other components including pigments, plasticizers and solvents. 