Full First-Order Free Variable Sequents and Tableaux in Implicit Induction

We show how to integrate implicit inductive theorem proving into free variable sequent and tableau calculi and compare the appropri-ateness of tableau calculi for this integration with that of sequent calculi. When rst-order validity is introduced to students it comes with some complete calculus. If this calculus happens to be an analytic calculus augmented with a Cut rule like a sequent or tableau calculus the students can compare the formal proofs with the informal ones they are hopefully acquainted with. This is because these calculi can mirror the human proof search process better than others. While knowing a complete calculus does not mean to know much about rst-order theorem proving, the interrelation of a human-oriented calculus and the informal proof search of the students will turn out to be fruitful for their later mathematical work. It is a pity that|while nearly all proofs of a working mathematician include induction|nothing comparable for inductive rst-order validity is ooered to the students. Some may argue that this is generally impossible because not even the theory of the Peano algebra of natural numbers is recursively enumerable, cf. e.g. Enderton (1973). Nevertheless, there really is some general way a working mathematician searches for an informal proof, may it be inductive or not. The inductive version of this proof search method goes back to the ancient Greeks and was rediscovered under the name \descente in-nie" by Pierre de Fermat (1601-1665). If you want to prove a conjecture, this method requires that you show, for each assumed counterexample of the conjecture , the existence of another counterexample of the conjecture that is strictly smaller in some wellfounded ordering. The working mathematician applies it in the following fashion. He (who may be female!) starts with the conjecture and simpliies it in case analyses which can be described as steps in a sequent or tableau calculus with Cut. When he realizes that the goals become similar to a diierent instance of the conjecture, he applies the conjecture just like a lemma, but keeps in mind that he actually has applied some induction hypothesis. Finally , he searches for some wellfounded ordering in which all the instances of the conjecture that he has applied as induction hypotheses are smaller than the original conjecture itself. Looking for a formal inductive calculus for mirroring this style of human inductive theorem proving (ITP), the \implicit induction" of Bachmair (1988) was a starting point …