from a S PECTRL

F, for all j {0,..., k}, we have s From Lemma A.2, we get that ζ From Lemma A.2 we conclude that ζ These definitions are a standard extension of Boolean logic to real values. In particular, they preserve (1)--i.e., b |= s if and only if b q This stronger property follows from a straightforward induction on φ.