let s be State of SCM+FSA ; :: thesis: for a being read-write Int-Location
for I being Program of SCM+FSA st ( ProperBodyWhile>0 a,I, Initialize s or I is parahalting ) & WithVariantWhile>0 a,I, Initialize s holds
DataPart (IExec (while>0 a,I),s) = DataPart ((StepWhile>0 a,I,(Initialize s)) . (ExitsAtWhile>0 a,I,(Initialize s)))
let a be read-write Int-Location ; :: thesis: for I being Program of SCM+FSA st ( ProperBodyWhile>0 a,I, Initialize s or I is parahalting ) & WithVariantWhile>0 a,I, Initialize s holds
DataPart (IExec (while>0 a,I),s) = DataPart ((StepWhile>0 a,I,(Initialize s)) . (ExitsAtWhile>0 a,I,(Initialize s)))
let I be Program of SCM+FSA ; :: thesis: ( ( ProperBodyWhile>0 a,I, Initialize s or I is parahalting ) & WithVariantWhile>0 a,I, Initialize s implies DataPart (IExec (while>0 a,I),s) = DataPart ((StepWhile>0 a,I,(Initialize s)) . (ExitsAtWhile>0 a,I,(Initialize s))) )
assume that
A1: ( ProperBodyWhile>0 a,I, Initialize s or I is parahalting ) and
A2: WithVariantWhile>0 a,I, Initialize s ; :: thesis: DataPart (IExec (while>0 a,I),s) = DataPart ((StepWhile>0 a,I,(Initialize s)) . (ExitsAtWhile>0 a,I,(Initialize s)))
set WH = while>0 a,I;
set Ins = NAT ;
set Ids = s +* (Initialized (while>0 a,I));
set Is = Initialize s;
A3: s +* (Initialized (while>0 a,I)) = (Initialize s) +* ((while>0 a,I) +* (Start-At (insloc 0 ))) by SCMFSA8A:13;
while>0 a,I is_halting_on Initialize s by A1, A2, Th33, Th34;
then A4: s +* (Initialized (while>0 a,I)) is halting by A3, SCMFSA7B:def 8;
consider k being Element of NAT such that
A5: ExitsAtWhile>0 a,I,(Initialize s) = k and
( ((StepWhile>0 a,I,(Initialize s)) . k) . a <= 0 & ( for i being Element of NAT st ((StepWhile>0 a,I,(Initialize s)) . i) . a <= 0 holds
k <= i ) ) and
A6: DataPart (Computation ((Initialize s) +* ((while>0 a,I) +* (Start-At (insloc 0 )))),(LifeSpan ((Initialize s) +* ((while>0 a,I) +* (Start-At (insloc 0 )))))) = DataPart ((StepWhile>0 a,I,(Initialize s)) . k) by A1, A2, Def6;
NAT misses Int-Locations \/ FinSeq-Locations by SCMFSA_2:13, SCMFSA_2:14, XBOOLE_1:70;
then A7: dom (s | NAT ) misses Int-Locations \/ FinSeq-Locations by SCMFSA8A:3;
thus DataPart (IExec (while>0 a,I),s) = DataPart ((Result (s +* (Initialized (while>0 a,I)))) +* (s | NAT )) by SCMFSA6B:def 1
.= DataPart (Result (s +* (Initialized (while>0 a,I)))) by A7, FUNCT_4:94, SCMFSA_2:127
.= DataPart ((StepWhile>0 a,I,(Initialize s)) . (ExitsAtWhile>0 a,I,(Initialize s))) by A3, A4, A5, A6, AMI_1:122 ; :: thesis: verum