 
Summary: The Critical Pair Lemma:
A Case Study for Induction Proofs With Partial Functions \Lambda
J¨urgen Giesl
TU Darmstadt, Alexanderstr. 10, 64283 Darmstadt, Germany,
Email: giesl@informatik.tudarmstadt.de
Abstract
In [9] we presented a calculus for automated induction proofs about partial functions. In contrast
to previous work, our approach also allows us to derive induction schemes from the recursions of partial
(and in particular, nonterminating) algorithms. In this way, existing induction theorem provers can be
directly extended to partial functions without changing their logical framework.
This report contains a large collection of theorems from the area of term rewriting systems which
were proved with our calculus (including Knuth and Bendix' wellknown critical pair lemma). These
examples demonstrate the power of our approach and they show that induction schemes based on partial
functions are indeed needed frequently.
1 Introduction
Induction is the essential proof method for the verification of functional programs. For that reason, several
techniques 1 have been developed to compute suitable induction relations and to perform induction proofs
automatically, cf. e.g. [2, 5, 12, 20, 21]. However, most of these techniques are only sound if all occurring
functions are total.
In [9] we showed that by slightly restricting the prerequisites of these techniques it is nevertheless possible
