Proofs by Induction

Abstract

This chapter explores the fact that properties of inductively defined objects can often be verified by inductive proofs, which is of fundamental importance to computing. The induction principle is motivated by examples based on natural numbers and then expanded to other data types and generalised to principles of structural, strong and well-founded induction. As a motivating example an inductive argument for verifying the closed formula for adding up n consecutive numbers is considered. Based on this, the induction principle is developed from different points of view, and further basic examples are explored. The principle of strong induction, where the induction hypothesis for all predecessor numbers can be used, is then presented and applied. Next, the correspondence between inductive definitions and inductive proofs is examined in detail and various inductive proofs about Fibonacci numbers are presented. The two last sections are devoted to pseudo-proofs such as the Sorites paradox and inductive proofs in computer science applications.