let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for a being read-write Int-Location
for I being Program of SCM+FSA
for k being Element of NAT st ( ( I is_halting_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) & I is_closed_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) ) or I is parahalting ) & ((StepWhile=0 (a,I,p,s)) . k) . a = 0 & ((StepWhile=0 (a,I,p,s)) . k) . (intloc 0) = 1 holds
DataPart ((StepWhile=0 (a,I,p,s)) . (k + 1)) = DataPart (IExec (I,(p +* (while=0 (a,I))),((StepWhile=0 (a,I,p,s)) . k)))
let s be State of SCM+FSA; :: thesis: for a being read-write Int-Location
for I being Program of SCM+FSA
for k being Element of NAT st ( ( I is_halting_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) & I is_closed_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) ) or I is parahalting ) & ((StepWhile=0 (a,I,p,s)) . k) . a = 0 & ((StepWhile=0 (a,I,p,s)) . k) . (intloc 0) = 1 holds
DataPart ((StepWhile=0 (a,I,p,s)) . (k + 1)) = DataPart (IExec (I,(p +* (while=0 (a,I))),((StepWhile=0 (a,I,p,s)) . k)))
let a be read-write Int-Location ; :: thesis: for I being Program of SCM+FSA
for k being Element of NAT st ( ( I is_halting_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) & I is_closed_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) ) or I is parahalting ) & ((StepWhile=0 (a,I,p,s)) . k) . a = 0 & ((StepWhile=0 (a,I,p,s)) . k) . (intloc 0) = 1 holds
DataPart ((StepWhile=0 (a,I,p,s)) . (k + 1)) = DataPart (IExec (I,(p +* (while=0 (a,I))),((StepWhile=0 (a,I,p,s)) . k)))
let I be Program of SCM+FSA; :: thesis: for k being Element of NAT st ( ( I is_halting_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) & I is_closed_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) ) or I is parahalting ) & ((StepWhile=0 (a,I,p,s)) . k) . a = 0 & ((StepWhile=0 (a,I,p,s)) . k) . (intloc 0) = 1 holds
DataPart ((StepWhile=0 (a,I,p,s)) . (k + 1)) = DataPart (IExec (I,(p +* (while=0 (a,I))),((StepWhile=0 (a,I,p,s)) . k)))
let k be Element of NAT ; :: thesis: ( ( ( I is_halting_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) & I is_closed_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) ) or I is parahalting ) & ((StepWhile=0 (a,I,p,s)) . k) . a = 0 & ((StepWhile=0 (a,I,p,s)) . k) . (intloc 0) = 1 implies DataPart ((StepWhile=0 (a,I,p,s)) . (k + 1)) = DataPart (IExec (I,(p +* (while=0 (a,I))),((StepWhile=0 (a,I,p,s)) . k))) )
set Ins = NAT ;
assume that
A1: ( ( I is_halting_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) & I is_closed_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) ) or I is parahalting ) and
A2: ((StepWhile=0 (a,I,p,s)) . k) . a = 0 and
A3: ((StepWhile=0 (a,I,p,s)) . k) . (intloc 0) = 1 ; :: thesis: DataPart ((StepWhile=0 (a,I,p,s)) . (k + 1)) = DataPart (IExec (I,(p +* (while=0 (a,I))),((StepWhile=0 (a,I,p,s)) . k)))
set ISWk = Initialized ((StepWhile=0 (a,I,p,s)) . k);
set SW = StepWhile=0 (a,I,p,s);
set PW = p +* (while=0 (a,I));
set SWkI = ((StepWhile=0 (a,I,p,s)) . k) +* (Initialize ((intloc 0) .--> 1));
set PWI = (p +* (while=0 (a,I))) +* I;
A5: (p +* (while=0 (a,I))) +* (while=0 (a,I)) = p +* (while=0 (a,I)) by FUNCT_4:99;
DataPart (Initialized ((StepWhile=0 (a,I,p,s)) . k)) = DataPart ((StepWhile=0 (a,I,p,s)) . k) by A3, SCMFSA8C:27;
then A6: ( I is_closed_on (StepWhile=0 (a,I,p,s)) . k,p +* (while=0 (a,I)) & I is_halting_on (StepWhile=0 (a,I,p,s)) . k,p +* (while=0 (a,I)) ) by A1, SCMFSA7B:24, SCMFSA7B:25, SCMFSA8B:8;
I is_halting_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) by A1, SCMFSA7B:25;
then A7: I is_halting_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) ;
XX: Start-At (0,SCM+FSA) c= Initialized ((StepWhile=0 (a,I,p,s)) . k) by SCMFSA6B:4;
((StepWhile=0 (a,I,p,s)) . k) +* (Initialize ((intloc 0) .--> 1)) = ((StepWhile=0 (a,I,p,s)) . k) +* (Initialize ((intloc 0) .--> 1))
.= Initialized ((StepWhile=0 (a,I,p,s)) . k)
.= Initialize (Initialized ((StepWhile=0 (a,I,p,s)) . k)) by XX, FUNCT_4:104 ;
then A9: (p +* (while=0 (a,I))) +* I halts_on ((StepWhile=0 (a,I,p,s)) . k) +* (Initialize ((intloc 0) .--> 1)) by A7, SCMFSA7B:def 8;
A10: (p +* (while=0 (a,I))) +* I halts_on ((StepWhile=0 (a,I,p,s)) . k) +* (Initialize ((intloc 0) .--> 1)) by A9;
A11: dom (((StepWhile=0 (a,I,p,s)) . k) | NAT) c= NAT by RELAT_1:87;
NAT misses Int-Locations \/ FinSeq-Locations by SCMFSA_2:13, SCMFSA_2:14, XBOOLE_1:70;
then A12: dom (((StepWhile=0 (a,I,p,s)) . k) | NAT) misses Int-Locations \/ FinSeq-Locations by A11, XBOOLE_1:63;
set IS = Initialize I;
set SWkIS = Initialize ((StepWhile=0 (a,I,p,s)) . k);
set PWIS = (p +* (while=0 (a,I))) +* I;
A13: ((StepWhile=0 (a,I,p,s)) . k) +* (Initialize ((intloc 0) .--> 1)) = Initialize ((StepWhile=0 (a,I,p,s)) . k) by A3, SCMFSA8C:18;
set WHS = Initialize (while=0 (a,I));
A14: (StepWhile=0 (a,I,p,s)) . (k + 1) = Comput (((p +* (while=0 (a,I))) +* (while=0 (a,I))),(Initialize ((StepWhile=0 (a,I,p,s)) . k)),((LifeSpan (((p +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,p,s)) . k)))) + 3)) by A5, SCMFSA_9:def 4;
A15: DataPart (IExec (I,(p +* (while=0 (a,I))),((StepWhile=0 (a,I,p,s)) . k))) = DataPart ((Result (((p +* (while=0 (a,I))) +* I),(((StepWhile=0 (a,I,p,s)) . k) +* (Initialize ((intloc 0) .--> 1))))) +* (((StepWhile=0 (a,I,p,s)) . k) | NAT)) by SCMFSA6B:def 1
.= DataPart (Result (((p +* (while=0 (a,I))) +* I),(((StepWhile=0 (a,I,p,s)) . k) +* (Initialize ((intloc 0) .--> 1))))) by A12, FUNCT_4:76, SCMFSA_2:127
.= DataPart (Comput (((p +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,p,s)) . k)),(LifeSpan (((p +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,p,s)) . k)))))) by A13, A10, EXTPRO_1:23 ;
thus DataPart ((StepWhile=0 (a,I,p,s)) . (k + 1)) = DataPart (Comput (((p +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,p,s)) . k)),(LifeSpan (((p +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,p,s)) . k)))))) by A2, A6, Th23, A14
.= DataPart (IExec (I,(p +* (while=0 (a,I))),((StepWhile=0 (a,I,p,s)) . k))) by A15 ; :: thesis: verum