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, ) ?? ?(x 2 , u 2 ) iff for any x 1 ? ?(x 1 , u 1 ), there exists x 2 ? ?(x 2 , u 2 ) with x 1 ? X x 2 . Hence, (i) ?? (ii). (ii) =? (iii): Let x ? X, u ? U , x 1 ? (? x) and u 1 ? (? u), Proof: (i) ?? (ii): Let x 1 , x 2 ? X and u 1 , u 2 ? U Lemma 1, we have that ?
, Hence, from (iii), we have that ?
, To prove the second inclusion, it is sufficient to show that C is a safety controller for the transition system S and the safety specification X S . We have that dom(C) =? dom(C * ) ?? X S = X S , where the first equality comes from (i), the second inclusion comes from the fact that C * is a safety controller and the last equality comes from the lower closedness of X S . Hence, the first condition of Definition 3 is satisfied. Now let x ? dom(C) and u ? C(x). From construction of the controller C, we have the existence of x ? X such that x ? X x , x ? dom(C * ), From construction of the controller C, it follows immediately that ? dom(C * ) = dom(C)
, Then, ? dom(C * ) =? dom(C) = dom(C * ), where the last equality comes from (i) in Lemma 2. Hence, dom(C * ) is a lower closed set. (ii) Let x 1 , x 2 ? X with x 1 ? X x 2, Proof: (i) We have from (ii) in Lemma 2 that dom(C * ) = dom(C)
where the first inclusion comes from the fact that S is a monotone transition system and the last equality comes from (i). Hence, by maximality of C * , we have that u ? C * (x 1 ). Then, C * (x 2 ) ? C * (x 1 ). (iii) Let x ? X, u ? C * (x), Then, ?(x 2 , u) ? dom(C * ). Hence, we have that ?(x 1 , u) ?? ?(x 2 , u) ?? dom(C * ) = dom(C * ) ,