Induction=I-Axiomatization+First-Order Consistency

In the early 80's, there was a number of papers on what should be called proofs by consistency. They describe how to perform inductive proofs, without using an explicit induction scheme, in the context of equational speciications and ground-convergent rewrite systems. The method was explicitly stated as a rst-order consistency proof in case of pure equational, constructor based, speciications. In this paper, we show how, in general, inductive proofs can be reduced to rst-order consistency and hence be performed by a rst-order theorem prover. Moreover, we extend previous methods, allowing non-equational speciications (even non-Horn speciications), designing some speciic strategies. Finally, we also show how to drop the ground convergence requirement (which is called saturatedness for general clauses). 1 Inductive proofs First-order speciications are ubiquitous in virtually all areas of computer science. In many cases, the intended meaning of a speciication E is not its standard rst-order semantics, i.e., the class of all its models, but rather a more speciic class M. Perhaps the best known example is the initial or minimal Herbrand model semantics, where M consists of the unique minimal Herbrand model of a set of Horn clauses with or without equality E (an algebraic speciication, a logic program, a deductive data base, ...). Other interesting semantics that are used in practice as well include