FORMAL REALIZABILITY THEORY — II 553 



(iii) There are not more operations chosen from the list Res, Con, 



than the integer 5 (No) -{- tio — 1. 

 Finally, then, the process must terminate after at most 25 (No) + 

 2a?o — 1 operations. 



4.06 Besides the data in 4.04, one other fact must be established about 

 each operation: that Nr is physically realizable if Nr+i is. This ^vill be 

 done as we discuss each operation. When it is established, we reason 

 back from the result of operation F, which pro\ddes a physical realiza- 

 tion of some N„. {m < 25(No) + no — 1), through N^-i to Nq = N, and 

 obtain a realization of N in finitely many steps. 



4.07 Finally, we shall prove about each step that: 



If Nr+i can be realized with .iv+i reactive elements, then Nr can be 

 realized with 



Xr+i 4- S(Nr) - 5(N.+i) 



reactive elements. This observation will pro\dde the basis for proving 

 the theorem of 2.18. For if N^ is the network with which the process 

 terminates, then by 3.21 N„ can be realized with 5(Nm) reactive elements. 

 Reading back through the construction, each increment of degree that 

 is encountered is balanced by an equal increment in the total number 

 of reactive elements, so that, finally, 5(N) is the total number of reactive 

 elements used. That no construction using fewer reactive elements can 

 succeed will be shown in Section 6. 



We now turn to IP, ID, Res, and IB, omitting the dual considera- 

 tions. In each case, notation is simplified by writing Z, Y, N, n respec- 

 tively for Zr , Yr ,'Nr , rir , and Zi , Fi , Ni , ni for Zr+i , Fr+i , N^+i , 



4.1 Given a 2n-pole N in any category for which / = 3, its impedance 

 matrLx Z(p) exists by hypothesis and has poles on p = tco. These can 

 be removed successively by 2.05 and 2.06, giving 



Zip) = pR„-\--Ro-i-J2 -2-^2 Rk + Z,{p). (1) 



p fc=i p -r (^k 



In this expansion, either of i?o , ^oo may of course be absent, and all the 

 Rk are real, symmetric, constant and semidefinite, for k = 0, 1, • • • , 

 K, 00 . Furthermore, Zi{p) is PR and has no poles on p = ioj, by 2.05, 

 2.06 and construction. 



Let Ni be the 2wi-pole whose impedance matrLx is Zi(p). We define 

 IP to be the operation giving Ni from N. Either Ni e C, or / = 1 or 2 



