let a be Int-Location ; :: thesis: for I being Program of
for s being State of
for k being Element of NAT st I is_closed_onInit s & I is_halting_onInit s & k < LifeSpan (s +* (Initialized I)) & IC (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = (IC (Computation (s +* (Initialized I)),k)) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = DataPart (Computation (s +* (Initialized I)),k) holds
( IC (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = (IC (Computation (s +* (Initialized I)),(k + 1))) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1)) )
let I be Program of ; :: thesis: for s being State of
for k being Element of NAT st I is_closed_onInit s & I is_halting_onInit s & k < LifeSpan (s +* (Initialized I)) & IC (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = (IC (Computation (s +* (Initialized I)),k)) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = DataPart (Computation (s +* (Initialized I)),k) holds
( IC (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = (IC (Computation (s +* (Initialized I)),(k + 1))) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1)) )
set D = Int-Locations \/ FinSeq-Locations ;
let s be State of ; :: thesis: for k being Element of NAT st I is_closed_onInit s & I is_halting_onInit s & k < LifeSpan (s +* (Initialized I)) & IC (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = (IC (Computation (s +* (Initialized I)),k)) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = DataPart (Computation (s +* (Initialized I)),k) holds
( IC (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = (IC (Computation (s +* (Initialized I)),(k + 1))) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1)) )
let k be Element of NAT ; :: thesis: ( I is_closed_onInit s & I is_halting_onInit s & k < LifeSpan (s +* (Initialized I)) & IC (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = (IC (Computation (s +* (Initialized I)),k)) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = DataPart (Computation (s +* (Initialized I)),k) implies ( IC (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = (IC (Computation (s +* (Initialized I)),(k + 1))) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1)) ) )
set s0 = Initialize s;
set sw = s +* (Initialized (while>0 a,I));
set sI = s +* (Initialized I);
set s0I = (Initialize s) +* (I +* (Start-At (insloc 0 )));
set s0w = (Initialize s) +* ((while>0 a,I) +* (Start-At (insloc 0 )));
set sK1 = Computation (s +* (Initialized (while>0 a,I))),(1 + k);
set sK2 = Computation (s +* (Initialized I)),k;
set l3 = IC (Computation (s +* (Initialized I)),k);
A1: s +* (Initialized I) = (Initialize s) +* (I +* (Start-At (insloc 0 ))) by SCMFSA8A:13;
A2: s +* (Initialized (while>0 a,I)) = (Initialize s) +* ((while>0 a,I) +* (Start-At (insloc 0 ))) by SCMFSA8A:13;
assume I is_closed_onInit s ; :: thesis: ( not I is_halting_onInit s or not k < LifeSpan (s +* (Initialized I)) or not IC (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = (IC (Computation (s +* (Initialized I)),k)) + 4 or not DataPart (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = DataPart (Computation (s +* (Initialized I)),k) or ( IC (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = (IC (Computation (s +* (Initialized I)),(k + 1))) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1)) ) )
then A3: I is_closed_on Initialize s by SCM_HALT:40;
assume I is_halting_onInit s ; :: thesis: ( not k < LifeSpan (s +* (Initialized I)) or not IC (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = (IC (Computation (s +* (Initialized I)),k)) + 4 or not DataPart (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = DataPart (Computation (s +* (Initialized I)),k) or ( IC (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = (IC (Computation (s +* (Initialized I)),(k + 1))) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1)) ) )
then A4: I is_halting_on Initialize s by SCM_HALT:41;
assume A5: k < LifeSpan (s +* (Initialized I)) ; :: thesis: ( not IC (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = (IC (Computation (s +* (Initialized I)),k)) + 4 or not DataPart (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = DataPart (Computation (s +* (Initialized I)),k) or ( IC (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = (IC (Computation (s +* (Initialized I)),(k + 1))) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1)) ) )
assume A6: IC (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = (IC (Computation (s +* (Initialized I)),k)) + 4 ; :: thesis: ( not DataPart (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = DataPart (Computation (s +* (Initialized I)),k) or ( IC (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = (IC (Computation (s +* (Initialized I)),(k + 1))) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1)) ) )
assume A7: DataPart (Computation (s +* (Initialized (while>0 a,I))),(1 + k)) = DataPart (Computation (s +* (Initialized I)),k) ; :: thesis: ( IC (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = (IC (Computation (s +* (Initialized I)),(k + 1))) + 4 & DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1)) )
hence IC (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = (IC (Computation (s +* (Initialized I)),(k + 1))) + 4 by A5, A6, A3, A4, A1, A2, SCMFSA_9:44; :: thesis: DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1))
thus DataPart (Computation (s +* (Initialized (while>0 a,I))),((1 + k) + 1)) = DataPart (Computation (s +* (Initialized I)),(k + 1)) by A5, A6, A7, A3, A4, A1, A2, SCMFSA_9:44; :: thesis: verum