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Concurrent Systems/notes/10 - Consensus Implementation.md

@@ -4,5 +4,16 @@ The **consensus number** of an object of type T is the greatest number n such th
 
 For all T, CN(T) > 0; if there is no sup, we let CN(T) := +∞
 
-**Thm:** let CN(T1) < CN(T2), then there exists no wait free implementation of T2 that only uses objects of type T1 and atomic R/W registers for all n s.t. CN(T1) < n <= CN(T2).
+**Thm:** let CN(T1) < CN(T2), then there exists no wait free implementation of T2 that only uses objects of type T1 and atomic R/W registers, for all n s.t. CN(T1) < n <= CN(T2).
 
+*Proof:*
+- Fix such an n; by contr., there exists a wait free implementation of objects of type T2 in a system of n processes that only uses objects of type T1 and atomic RW reg.s.
+- Since n ≤ CN(T2), by def. of CN, there exists a wait free implementation of consensus in a system of n processes that only uses objects of type T2 and atomic RW reg.s.
+- Hence, there exists a wait free implementation of consensus in a system of n processes that only uses objects of type T1 and atomic RW reg.s.
+	- contradiction with CN(T1) < n
+
+### Schedules and Configurations
+**Schedule:** sequence of operation invocations issued by processes
+**Configuration:** the global state of a system at a given execution time (values of the shared memory + local state of every process).
+
+Given a configuration C and a schedule S, we denote with S(C) the configuration obtained starting from C and applying S.