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sav08:partial_congruences [2008/04/23 07:50]
sav08:partial_congruences [2008/04/26 17:42]
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 **Proof:​** ​ **Proof:​** ​
-Show $C^i(s) \cap T^2 = s$ for all $i \ge 0$.+Show $C^i(s) \cap T^2 = s$ for all $i \ge 0$, by induction. 
 +  * $i = 0$: s = s 
 +  * $i \rightarrow i+1$: 
 +$C^{i+1}(s) \cap T^2 = C(C^i(s) \cap T^2) \cap T^2 $, by induction hypothesis $C^i(s) \cap T^2 =s$. 
 +Thus $C(C^i(s) \cap T^2) \cap T^2 = C(s) \cap T^2$. 
 +As $C$ adds only needed term for congruence, the added term are either not in $T^2$ or $s$ in not a congruence. 
 +By hypothesis $s$ is a congruence, so $C(s) \cap T^2 = s$. 
 +Therefore $C^{i+1}(s) \cap T^2 = s$. 
 **Proof End.** **Proof End.**