We describe rippling: a tactic for the heuristic control of the key part of proofs by mathematical induction. This tactic significantly reduces the search for a proof of a wide variety of inductive theorems. We first present a basic version of rippling, followed by various extensions which are necessary to capture larger classes of inductive proofs. Finally, we present a generalised form of rippling which embodies these extensions as special cases. We prove that generalised rippling always terminates, and we discuss the implementation of the tactic and its relation with other inductive proof search heuristics.

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The technique of proof plans, is outlined. This technique is used to guide automatic inference in order to avoid a combinatorial explosion. Empirical research to test this technique in the domain of theorem proving by mathematical induction is described. Heuristics, adapted from the work of Boyer and Moore, have been implemented as Prolog programs, called tactics, and used to guide an inductive proof checker, Oyster. These tactics have been partially specified in a meta-logic, and plan formation has been used to reason with these specifications and form plans. These plans are then executed by running their associated tactics and, hence, performing an Oyster proof. Results are presented of the use of this technique on a number of standard theorems from the literature. Searching in the planning space is shown to be considerably cheaper than searching directly in Oyster's search space. The success rate on the standard theorems is high. These preliminary results are very encouraging.

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