let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis:
let a be Int-Location ; :: thesis:
let I be Program of SCM+FSA; :: thesis:
set D = Data-Locations SCM+FSA;
let s be State of SCM+FSA; :: thesis:
let k be Element of NAT ; :: thesis:
set s0 = Initialized s;
set sw = s +* (Initialized (while>0 (a,I)));
set Pw = P +* (while>0 (a,I));
set sI = s +* (Initialized I);
set PI = P +* I;
set s0I = (Initialized s) +* (Initialize I);
set s0w = (Initialized s) +* (Initialize (while>0 (a,I)));
set sK1 = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(1 + k));
set sK2 = Comput ((P +* I),(s +* (Initialized I)),k);
set l3 = IC (Comput ((P +* I),(s +* (Initialized I)),k));
A1: s +* (Initialized I) = (Initialized s) +* (Initialize I) by SCMFSA8A:13;
A2: s +* (Initialized (while>0 (a,I))) = (Initialized s) +* (Initialize (while>0 (a,I))) by SCMFSA8A:13;
assume I is_closed_onInit s,P ; :: thesis:
then A3: I is_closed_on Initialized s,P by SCM_HALT:40;
assume I is_halting_onInit s,P ; :: thesis:
then A4: I is_halting_on Initialized s,P by SCM_HALT:41;
assume A5: k < LifeSpan ((P +* I),(s +* (Initialized I))) ; :: thesis:
assume A6: IC (Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(1 + k))) = (IC (Comput ((P +* I),(s +* (Initialized I)),k))) + 4 ; :: thesis:
assume A7: DataPart (Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(1 + k))) = DataPart (Comput ((P +* I),(s +* (Initialized I)),k)) ; :: thesis:
hence IC (Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),((1 + k) + 1))) = (IC (Comput ((P +* I),(s +* (Initialized I)),(k + 1)))) + 4 by A5, A6, A3, A4, A1, A2, SCMFSA_9:44; :: thesis:
thus DataPart (Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),((1 + k) + 1))) = DataPart (Comput ((P +* I),(s +* (Initialized I)),(k + 1))) by A5, A6, A7, A3, A4, A1, A2, SCMFSA_9:44; :: thesis: