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THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



It is clear that any 2n-pole N belongs in C or in exactly one of the six- 

 teen elementary categories whose union is (0 + 1 + 2 + 3, 0+1 + 

 2 + 3). 



Table 4.04 shows for each category of Nr , except (0, 0), which opera- 

 tions may be applied, and the possible categories of the resulting N^+i . 



A 2n-pole N not in (0, 0) has at least one matrix, and if it has two these 

 are of the same degree (2.07, 2.13). We may then denote the degree of 

 whatever matrix N has simply by 5(N). The fourth and fifth columns 

 of Table 4.04 show the relations of 5(Nr) to 5(Nr+i), and of Ur to w^+i . 



4,05 Table 4.04 summarizes facts to be proved in 4.1 et seq. Assuming 

 now that the assertions in this table are true, we can show that the 

 inductive procedure is effective. 



We observe first that the category C and every possible elementary 

 category (/, A) except (0, 0) is contained in at least one of the categories 

 listed in the first column. Hence to any 2n-pole not in (0, 0) there is at 

 least one operation applicable. Further we note that the category (0, 0) 

 does not appear in the third column. Since by hypothesis No is not in 

 (0, 0), it follows by induction that no N^ will be. Therefore the process 

 can stop only by the operation F: completion. 



Second, we notice that if N^ is not in the category (1, 1), then an 

 applicable operation can be found which actually reduces one of the 

 two numbers 5(Nr), n^ . Furthermore, if Nr is in (1, 1), a sequence of two 

 operations can be found which reduces one of 5(Nr), ih . Therefore 

 before the realization process terminates (with F), 



(i) There are not more operations chosen from the list IP, AP, IB, 



AB, than the integer S(No); 

 (ii) There are not more operations chosen from the list ID, AD, than 

 the integer no — 1 (since after these, still ?ir+i > 0) ; 



Table 4.04 



But rir+i > 0. 



