Making a productive use of failure to generate witnesses for coinduction from divergent proof attempts

  • Authors:

  • L. A. Dennis;A. Bundy;I. Green

  • Affiliations:

  • Division of Informatics, University of Edinburgh, 80 South Bridge, Edinburgh, UK E-mail: louised@dai.ed.ac.uk;Division of Informatics, University of Edinburgh, 80 South Bridge, Edinburgh, UK;Division of Informatics, University of Edinburgh, 80 South Bridge, Edinburgh, UK

  • Venue:
  • Annals of Mathematics and Artificial Intelligence
  • Year:
  • 2001

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Abstract

Coinduction is a proof rule. It is the dual of induction. It allows reasoning about non-well-founded structures such as lazy lists or streams and is of particular use for reasoning about equivalences. A central difficulty in the automation of coinductive proof is the choice of a relation (called a bisimulation).We present an automation of coinductive theorem proving. This automation is based on the idea of proof planning [7]. Proof planning constructs the higher level steps in a proof, using knowledge of the general structure of a family of proofs and exploiting this knowledge to control the proof search. Part of proof planning involves the use of failure information to modify the plan by the use of a proof critic [23] which exploits the information gained from the failed proof attempt.Our approach to the problem was to develop a strategy that makes an initial simple guess at a bisimulation and then uses generalisation techniques, motivated by a critic, to refine this guess, so that a larger class of coinductive problems can be automatically verified.The implementation of this strategy has focused on the use of coinduction to prove the equivalence of programs in a small lazy functional language which is similar to Haskell [22].We have developed a proof plan for coinduction and a critic associated with this proof plan. These have been implemented in iCoCLAM, an extended version of iCLAM [9], with encouraging results. The planner has been successfully tested on a number of theorems.