. ?r, {i} ¬?[i in j]) ?R.¬{j} ?U.({j} ?[i in j]) ? ( (n ? 1) R ?

. ?r, ({i} ?[i in j]) ?R.{j} ? ( (n + 1) R ?

. ?r, {i} ¬?[i in j]) ?R.¬{j} ?U.({j} ¬?[i in j])) ? ( n R ?

R. If, its valuation is left unmodified. Otherwise, as the valuation of R becomes equal to the one of Q, there will be an R-neighbour satisfying ? if there is such a Q-neighbour

. ?r, {i} ¬?[i in j]) ?R.¬{j} ?U.({j} ¬?

?. Pre, (?works in ? .{ph 1 }) ?U.(?works in ? .{ph 2 }) then P H := P H -ph 1 ; while ?U.(?treats ? .{ph 1 }) do select p with ?U.({p} treats ? .{ph 1 }); treats := treats -(ph 1 ,p); treats := treats +

=. ?u, Let us now apply rule 15 to get-ph 1 ](¬{ph 2 })[P H:= P H-ph 1 ])). Then, by applying the rules 2, 10 and 13, one obtains ?U.({ph 1 } (¬ P H{ph 1 }) ¬{ph 2 }) One can now observe that this is equivalent to ?U.({ ph 1 } ¬{ph 2 }). The same can be done with ?U.({ph 2 }}MS) to prove that it is left unchanged by the substitution, Thus inv 1 [P H:= P H-ph 1 ] = ?U.({ph 1 } ¬{ph 2 }) ?U.({ph 2 }}MS) ?U.(?treats. ?MS). This is trivially implied by P re = ?U.({ph 1 } PH{ph 2 }) ?U.({ph 2 }}MS) ?U.(?treats. ?MS)

?. Wp, vc(s 10 , inv 1 ) vc(s 2 , P ost), let us split the study of the validity of vc(s 1 , P ost) in four parts. Let us start with vc(s 10 , inv 1 ) As s 10 = s 11 ; s 12 ; s 13 , vc(s 10 , inv 1 ) = vc(s 11 , wp(s 12 ; s 13 , inv 1 )) vc(s 12 , wp(s 13 , inv 1 )) vc(s 13 , inv 1 )?; ? ?1

. Lemma-4, Let R be one of rules ? 1 -? 11 , A r be the right-hand side and A l be the left-hand side of R. Given any

. Proof, The proof uses the model M = (M, R , ?, V ) obtained form M by using the definitions in Section 3

. Lemma-5, Let R be one of rules ? 1 -? 13 , A r be the right-hand side and A l be the left-hand side of R. Given any

. Proof, The proof uses the model M = (M, R , ?, V ) obtained form M by using the definitions in Section 3

. Proof, Rule S 1 : As i 2 is added to ? 1 , ?(? 1 [add(i 2 )]) = ?

. Lemma-6, Let R be one of rules ? 12 -? 14 , A r be the righthand side and A l be the lefthand side of R. Given any model

?. Otherwise, Then, by maximality of L * , ?[d], (¬ < ? ? > (c? < ? > d) ? L * or ¬ < ? ? > (d ? A) ? L * that is ?)) ? L * or, by (? ? 2) and (? S 3), [? ? ; c?; ?](d ? ¬A)) ? L * that is [? ? ; c?; ?; A?]? ? L * that is, ¬d ? L * . As L * is stable under (Cov)¬A) ? L * . By (?2), < ? ? > (c ? [?]¬A) ? L * and thus [c] ? V ([?]¬A) that is [c] ? V (< ? > A). Thus V (< ? > A) ? {s|?[d] ? M.((s, [d]) ? R(?) ? [d] ? V (A)}

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