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. ?c, Since w b ? G(Reach(?s, t ), ??(nobs(? b , t )).?c), w b satisfies ??

?. and ?. ?u, By construction, ? d satisfies ??(? d ) ??(nobs(? b , t )).?c and ? d = =? date(? d (1)) ? t . ?Projection on ?u: Since a ? ?u, ? t|?u = (?s.(t , a). obs(? d , t)) |?u . ? d ? tw(?c), thus ? t|?u = ? s|?u

?. and ?. ?c, By construction, ? d satisfies ??(? d ) ??(? b ).?c.a, and ? d = =? date(? d (1)) ? t . ?Projection on ?u: ? t|?u = (?s . obs(? d , t)) |?u . Since ? d ? tw(?c)