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 sav08:partial_congruences [2008/04/23 07:50]vkuncak sav08:partial_congruences [2008/04/26 17:42]damien Both sides previous revision Previous revision 2008/04/26 17:42 damien 2008/04/23 07:50 vkuncak 2008/04/23 07:50 vkuncak 2008/04/23 07:49 vkuncak 2008/04/23 07:45 vkuncak 2008/04/23 07:43 vkuncak created 2008/04/26 17:42 damien 2008/04/23 07:50 vkuncak 2008/04/23 07:50 vkuncak 2008/04/23 07:49 vkuncak 2008/04/23 07:45 vkuncak 2008/04/23 07:43 vkuncak created Line 7: Line 7: **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.**