let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for I being parahalting good Program of SCM+FSA
for a being read-write Int-Location st s . (intloc 0) = 1 & s . a <= 0 holds
DataPart (IExec ((Times (a,I)),P,s)) = DataPart s
let s be State of SCM+FSA; :: thesis: for I being parahalting good Program of SCM+FSA
for a being read-write Int-Location st s . (intloc 0) = 1 & s . a <= 0 holds
DataPart (IExec ((Times (a,I)),P,s)) = DataPart s
set D = Data-Locations SCM+FSA;
let I be parahalting good Program of SCM+FSA; :: thesis: for a being read-write Int-Location st s . (intloc 0) = 1 & s . a <= 0 holds
DataPart (IExec ((Times (a,I)),P,s)) = DataPart s
let a be read-write Int-Location ; :: thesis: ( s . (intloc 0) = 1 & s . a <= 0 implies DataPart (IExec ((Times (a,I)),P,s)) = DataPart s )
assume A1: s . (intloc 0) = 1 ; :: thesis: ( not s . a <= 0 or DataPart (IExec ((Times (a,I)),P,s)) = DataPart s )
set s0 = Initialized s;
A2: Stop SCM+FSA is_halting_on Initialized s,P by SCMFSA7B:25;
assume s . a <= 0 ; :: thesis: DataPart (IExec ((Times (a,I)),P,s)) = DataPart s
then A3: (Initialized s) . a <= 0 by SCMFSA6C:3;
A4: IExec (((Stop SCM+FSA) ';' (Stop SCM+FSA)),P,(Initialized s)) = (IExec ((Stop SCM+FSA),P,(IExec ((Stop SCM+FSA),P,(Initialized s))))) +* (Start-At (((IC (IExec ((Stop SCM+FSA),P,(IExec ((Stop SCM+FSA),P,(Initialized s)))))) + (card (Stop SCM+FSA))),SCM+FSA)) by SCMFSA6B:44;
A5: for f being FinSeq-Location holds (Initialized s) . f = s . f by SCMFSA6C:3;
A6: for a being read-write Int-Location holds (Initialized s) . a = s . a by SCMFSA6C:3;
A7: (Initialized s) . (intloc 0) = 1 by SCMFSA6C:3;
A8: Stop SCM+FSA is_closed_on Initialized s,P by SCMFSA7B:24;
then A9: Times (a,I) is_halting_on Initialized s,P by A3, A2, SCMFSA8B:24;
A11: Stop SCM+FSA is_closed_on Initialized s,P by SCMFSA7B:24;
A12: DataPart (IExec ((Stop SCM+FSA),P,(Initialized s))) = DataPart (Initialized (Initialized s)) by Th38
.= DataPart (Initialized s) ;
A13: Directed (Stop SCM+FSA) is_pseudo-closed_on Initialized s,P by A11, SCMFSA7B:25, SCMFSA8A:37;
Times (a,I) is_closed_on Initialized s,P by A3, A8, A2, SCMFSA8B:24;
then DataPart (IExec ((Times (a,I)),P,s)) = DataPart (IExec ((Times (a,I)),P,(Initialized s))) by A9, A7, A6, A5, Th45
.= DataPart (IExec (((Stop SCM+FSA) ';' (Stop SCM+FSA)),P,(Initialized s))) by A3, A7, Th73, A13 ;
hence DataPart (IExec ((Times (a,I)),P,s)) = DataPart (IExec ((Stop SCM+FSA),P,(IExec ((Stop SCM+FSA),P,(Initialized s))))) by A4, SCMFSA8A:10
.= DataPart (IExec ((Stop SCM+FSA),P,(Initialized s))) by A8, A2, A7, A12, Th46
.= DataPart s by A1, A12, Th27 ;
:: thesis: verum