Proof Planning Coinduction

Abstract

Coinduction is a proof rule which is the dual of induction. It allows reasoning about non-well-founded sets and is of particular use for reasoning about equivalences.In this thesis I present an automation of coinductive theorem proving. This automation is based on the ideas of proof planning [Bundy 88]. Proof planning as the name suggests, plans the higher level steps in a proof without performing the formal checking which is also required for a verification. The automation has focused on the use of coinduction to prove the equivalence of programs in a small lazy functional language which is similar to Haskell.One of the hardest parts in a coinductive proof is the choice of a relation, called a bisimulation. The automation here described makes an initial simplified guess at a bisimulation and then uses critics, revisions based on failure, and generalisation techniques to refine this guess.The proof plan for coinduction and the critic have been implemented in CLAM [Bundy et al 90b] with encouraging results. The planner has been successfully tested on a number of theorems. Comparison of the proof planner for coinduction with the proof plan for induction implemented in CLAM has gighlighted a number of equivalences and dualities in the process of these proofs and has also suggested improvements to both systems.This work has demonstrated not only the possibility of fully automated theorem provers for coinduction but has also demonstrated the uses of proof planning for comparison of proof techniques.This work has demonstrated not only the possibility of fully automated theorem provers for coinduction but has also demonstrated the uses of proof planning for comparison of proof techniques.