AUTOMATED LOGIC FOR SEMI-AUTOMATED MATHEMATICS
UNCLASSIFIED In previous work with our colleagues, we have investigated some of the possibilities of proving mathematical theorems on a computer on a man-machine basis. At the intermediate stages in a proof we are, in general, trying to prove some formula from certain suppositions and previously proved heorems. If such intermediate steps have "trivial" pr. zfs, we might hope to have the machine verify this automatically. This report describes some algorithms which verify certain "trivial"
... . These algorithms can be read iff from the definition of proof in the formal syst, 1 through S 6 described in this report. Algorithms concerning the propositional connectives are explicated by systems S 1 and S 2 quantifiers by S 3 and S 4 ; V-order predicatefunction calculus by S 5 ; and many sorted variables and constants for t/-order predicate-function calculus by S 6 ACKNOWLEDGEMENTS The author wishes to acki owledge the contributions of his colleagues, James H. Bennett, William B. Easton, and Thomas H. Mott, to this report. As a team, we developed the SAM programs; and though the development of the automatic logic was the author's personal responsibility, he had many >eipful discussions with his colleagues which bear on this report. The author is affiliated with the Mathematics Department of Princeton University and wishes to acknowledge their encouragement in this study in bridging mathematics and computing. Finally, thanks to the staff at Applied Loqic for admlnistrative assistance. TABLE OF CONTENTS SUMMARY 0. SSys:enA S 5 is n :-oraer -re -fca n-ction cacuus ,atror.ncd after E S4. A method for handling types is 'Included in S5 which allows us tc -"ve theorems at the lowest pcssible type and yet have instances of these thexems involving higher types availabie by substitution. Function-like properties of predicates can also be obtained as instances of more general theorems concerning functions by substitution. System S 6 is an extension of S 5 to a many-sorted '" -order predtcatefunction calculus. It is believed that many f the problems attendent to mechanizing a many-sorted calculus are resolved by -S . A careful treatment of the matching process has not however as yet been carried out for S 6 . However no insurmountable obstacles are expected. 6