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Extra info for Applications of Nonlinear Partial Differential Equations in Mathematical Physics. Proceedings of Symposia in Applied Mathematics Volume XVII
V L m , —L v Ki v K2 v . . v K n Prom such a pair, the clause Li v L2 v . . v L m v Ki v K2 v . . K n can be inferred. This clause (called the resolvent) is then added to the set of clauses which have been accumulated from previous inferences. Furthermore, a substitution is considered only if it is the most general that could be made, thus maintaining the maximum degree of generality in the result. Another closely related method of inference is factoring: A substitution is sought such that (a) two or more of the literals of a clause will collapse into a single literal, and (b) no more general substitution would have the same effect.
The constraint imposed on 3 is: if the resolvent of C and D is a non-unit whose level is a specified bound ko, or a unit whose level exceeds ko, then it is not added to the corresponding list, and the pair is treated as if no resolvent were generated. To illustrate the difficulty thus avoided consider the set consisting of the clauses P(a), — P(x) v P(f(x)), which correspond to a subset of the Peano axioms, and some clause of length 3. , ad infinitum. We would be caught in an infinite loop which would continually present and execute the task of resolving a new unit with the same 2-clause instead of either passing to the proof recovery or to the non-unit section.
6. 7. 8. 9. 10. 11. 12. 13. 14. P(x,e,x) P(e,x,x) - P ( x , y , u ) v -P(y,z,v) - P ( x , y , u ) v -P(y,z,v) P(x,x,e) P(a,b,c) -P(b,a,c) —P(y,z,v) v —P(e,z,w) - P ( y , w , v ) v P(y,v,w) —P(x,z,u) v —P(x,e,w) —P(w,z,u) v P(u,z,w) P(c,b,a) P(c,a,b) -P(c,a,b) v -P(u,z,w) v P(x,v,w) v -P(x,v,w) v P(u,z,w) v P(y,v,w) v P(u,z,w) (Al) (A2) (A3) (A4) (A5) (A6) (A7) (from (from (from (from (from (from (from 5 and 3) 2 and 8) 5 and 4) 1 and 10) 6 and 11) 12 and 9) 7 and 11) Since 13 and 14 are manifestly contradictory, the proof is complete.