. If-?, then, since ? ? Pre(?, time(? following lemma A.14, since obs(?, time(?.(?, a))) = ?, obs(? s0 , time(? . (?, a))) = ? s = obs(? |?u , time(? . (?, a))) = ? |?u and ? ? (nobs(? s0 , time(? . (?, a)))) . ? c = ? ? (? |?c ) Since ? . (?, a) ? Pre(?, t), and ? ? Pre(?, time(? . (?, a))), following the definition of Pre(?, t) and Pre(?, time(? . (?, a))) (Definition 4.20), there exists t ? R ?0 such that time(?, ? ? (obs(? . (?, a), t ) |?c )) = ?. Let us consider the minimum such t . Since t ? time This means that: G(Reach((? . (?, a)) |?u , t ), ? ? ((? . (?, a)) |?c )) = ?. (A.2), p.then obs(?

?. For-any-t and . ?0, after (0, a) after (? , t ) ? v, then v, ? ? (nobs(? , t )) ?1 . buf c , 1 ? W 0 , because ? is winning for player 0. By construction of ? , and because of the different constraints required on G s , this implies that all states v ? V s such that Reach(? s0 , time(? . (?, a))) after (0, a) after (? , t ) ? v are in W 0 , for any t ? R ?0 . We know by construction of ? that this holds for some t

F. Now and A. , then: (? E ) * that are minimal in the number of rules delay(), i.e. the word obtained by merging two consecutive rules delay() into one with the sum of delays of the two rules, until stabilisation. This allows to define Rules(?, t) correctly, without " cheating " by slicing time to increase the length of the word. Note that the words obtained by merging or adding delay() rules this way reach exactly the same configurations in the end

. Now, let P(?, t) be the predicate " (?, ? s0 , ? c ) ? store ? =? (output(?, t) = obs(? s0 , t) ? Reach E (?, t) = nobs(? s0 , t), ? c , Reach(? s0 , t), t ? time(obs(? s0 , t))) " , and P(?) be the predicate, Let us then show by induction that P(?) holds for any ? ? tw(?)

?. and ?. ?0, P(?) holds. Let us consider?, ? s0 , ? c ) ? store ? , (? . (?, a), ? t0 , ? d ) ? store ? , ? s = obs(? s0 , time(? . (?, a))), and c = Reach E (?, time(? . (?, a))). Then, by induction hypothesis, c = nobs(? s0 , time(?, Reach(? s0 time(? . (?, a)) ? time, p.time(?

<. If-t and . Time, ?, a), t) = Reach E (?, t), and obs(? t0 , t) = obs(? s0 , t), meaning that P(? . (?, a), t) holds. Then

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