SYNTFIESIS OF SWITCHING CIRCUITS 85 



We must now adjust the values to the right of L — L to the values 

 03 , 04 , • • • , o„ . Let us denote the sequence 



4, 8, • • • , 2«-S (2-^' + R), 3-2«, 3- 2"+', • • • (2" + S), 2"+\ • ■ • , 2" 



by n\ , jji-i , ■ • • , yin-2 ■ Now since all the rows in the above addition are 

 non-decreasing to the right oi L — L, and no I's appear, we will have proved 

 the lemma if we can show that 



S Mi < S '^. i = 1, 2, • • ■ , (n - 2) 



1 = 1 i=3 



since we have shown this to be a sufficient condition that 



03 , 04 , • • • , On = (mi , Mn , • • • , Mn-2) 



and the decomposition proof we used for the first part will work. For 

 i< q— 2, i.e., before the term (23-i -f R) 



E M. - 4(2'' - 1) 



and 



smce 



Hence 



i+3 



3 



q< P 



i i+3 



m "i < XI o» i < q — 2 



3 



Next, for {q - \) < i < {p - 3), i.e., before the term {2^ + S) 

 Z M. = -^(2'"' - \) + R + 3-2^(2' "^^ - 1) 



< 3-2'^' - 4 < 3-2*^' 



smce 



also again 



i? < 2« 



i+3 



Z ai > 12"-' 



3 



SO that in this interval we also have the desired inequality. Finally for the 

 last interval. 



