Labelled State Transition Systems

For simplicity, we adopt the following convention: x, x1, x2, y, y1, y2, z, z1, z2, X, X1, X2 are sets, E is a non empty set, e is an element of E, u, v, v1, v2, w, w1, w2 are elements of Eω, F , F1, F2 are subsets of Eω, and k, l are natural numbers. Next we state a number of propositions: (1) For every finite sequence p such that k ∈ dom p holds (〈x〉 a p)(k+ 1) = p(k). (2) For every finite sequence p such that p 6= ∅ there exists a finite sequence q and there exists x such that p = q a 〈x〉 and len p = len q + 1. (3) For every finite sequence p such that k ∈ dom p and k+1 / ∈ dom p holds len p = k. (4) Let R be a binary relation, P be a reduction sequence w.r.t. R, and q1, q2 be finite sequences. Suppose P = q1 a q2 and len q1 > 0 and len q2 > 0. Then q1 is a reduction sequence w.r.t. R and q2 is a reduction sequence w.r.t. R.