Functional and ( Constraint ) Logic Programming 18 th International Workshop , WFLP ’ 09 part of the Federated Conference on Rewriting , Deduction

In previous works, we have introduced the notion of abstract strategies for abstract reduction systems and de ned adequate properties of termination, con uence and normalization under strategies. Thanks to this abstract strategy concept, we draw a parallel between strategies for computation and strategies for deduction. Then, deduction rules can be viewed as rewrite rules, a deduction step as a rewriting step and a proof construction step as a narrowing step for an adequate abstract reduction system, possibly in constraint handling settings. The fundamental complementarity between deduction and computation, as emphasized in particular in deduction modulo [9], gives now rise to a completely new generation of proof assistants where customized deductions are performed modulo appropriate and user de nable computations [5,6]. This has in particular the advantage to allow for a uniform implementation of higher-order and rst-order logics [8,7] making possible the safe use of existing dedicated proof environments [16,10,4]. This generalizes classical approaches used in rst-order theorem proving [17], as well as higher-order ones like PVS [18], TPS [1,2], Omega [3,19], Coq [11] or Mizar [20], to mention just a few. Proof search in these environments goes back to the late sixties and then throught the design of ML as the metalanguage of LCF. It requires to guide proof discovery using so called strategies, tactics, tacticals or proof plans, terms widely used in arti cial intelligence, in automated or interactive reasoning, in semantics of programming languages as well as in every day life. The collusion of deduction and computation in next-generation proof assistants has inspired our recent attempt at providing an uniform (domain-agnostic, if you will) de nition for strategies, starting from a rule-based view point [13]. For term rewriting, reduction strategies study which expressions should be selected for evaluation and which rules should be applied. These choices are usualy made to increase the e ciency of evaluation but may a ect fundamental properties of computations such as con uence or (non-)termination. Programming languages like TOM, ELAN, Maude and Stratego allow for the explicit de nition of the evaluation strategy, whereas languages like Clean, Curry, and Haskell allow for its modi cation. In theorem proving environments, including automated theorem provers, proof checkers, and logical frameworks, strategies (also called tacticals in some 1 http://tom.loria.fr