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Proof.
Since is a coarsening of i for each i N, K i( ( ) ).
Hence, K 1N ( ( ) ),
and since by definition K i( ( ) ) = { | i( ) ( ) } = ( ),
K 1N ( ( ) ) =
i NK i( ( ) ) = ( )
Applying the recursive definition of mutual knowledge, for any m 1,
K mN ( ( ) ) =
i NK i ( K m - 1N ( ( ) ) =
i NK i( ( ) ) = ( )
so, since ( ) , by definition we have K *N ( ( ) ).
Peter Vanderschraaf peterv@cyrus.andrew.cmu.edu |