The Productive use of Failure to Generate Witnesses from Divergent Proof Attempts for Coinduction

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. 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 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. We have developed a proof plan for coinduction and a critic associated with this proof plan. These have been implemented in CoClam, an extended version of Clam with encouraging results. The planner has been successfully tested on a number of theorems.

Are we teaching our students what they need to know about ageing? Results from the National Survey of Undergraduate Teaching in Ageing and Geriatric Medicine

Introduction - Learning about ageing and the appropriate management of older patients is important for all doctors. This survey set out to evaluate what medical undergraduates in the UK are taught about ageing and geriatric medicine and how this teaching is delivered. Methods – An electronic questionnaire was developed and sent to the 28/31 UK medical schools which agreed to participate. Results – Full responses were received from 17 schools. 8/21 learning objectives were recorded as taught, and none were examined, across every school surveyed. Elder abuse and terminology and classification of health were taught in only 8/17 and 2/17 schools respectively. Pressure ulcers were taught about in 14/17 schools but taught formally in only 7 of these and examined in only 9. With regard to bio- and socio- gerontology, only 9/17 schools reported teaching in social ageing, 7/17 in cellular ageing and 9/17 in the physiology of ageing. Discussion – Even allowing for the suboptimal response rate, this study presents significant cause for concern with UK undergraduate education related to ageing. The failure to teach comprehensively on elder abuse and pressure sores, in particular, may be significantly to the detriment of older patients.