let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for a being read-write Int-Location
for k being Nat
for I being really-closed MacroInstruction of SCM+FSA st ( I is_halting_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 k being Nat
for I being really-closed MacroInstruction of SCM+FSA st ( I is_halting_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 k being Nat
for I being really-closed MacroInstruction of SCM+FSA st ( I is_halting_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 Nat; :: thesis: for I being really-closed MacroInstruction of SCM+FSA st ( I is_halting_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 really-closed MacroInstruction of SCM+FSA ; :: thesis: ( ( I is_halting_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)) 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 = Initialized ((StepWhile=0 (a,I,p,s)) . k);
set PWI = (p +* (while=0 (a,I))) +* I;
DataPart (Initialized ((StepWhile=0 (a,I,p,s)) . k)) = DataPart ((StepWhile=0 (a,I,p,s)) . k) by A3, SCMFSA_M:19;
then A4: I is_halting_on (StepWhile=0 (a,I,p,s)) . k,p +* (while=0 (a,I)) by A1, SCMFSA7B:19, SCMFSA8B:5;
I is_halting_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) by A1, SCMFSA7B:19;
then A5: I is_halting_on Initialized ((StepWhile=0 (a,I,p,s)) . k),p +* (while=0 (a,I)) ;
Initialized ((StepWhile=0 (a,I,p,s)) . k) = Initialize (Initialized ((StepWhile=0 (a,I,p,s)) . k)) by MEMSTR_0:44;
then A6: (p +* (while=0 (a,I))) +* I halts_on Initialized ((StepWhile=0 (a,I,p,s)) . k) by A5, SCMFSA7B:def 7;
A7: (p +* (while=0 (a,I))) +* I halts_on Initialized ((StepWhile=0 (a,I,p,s)) . k) by A6;
set SWkIS = Initialize ((StepWhile=0 (a,I,p,s)) . k);
set PWIS = (p +* (while=0 (a,I))) +* I;
A8: Initialized ((StepWhile=0 (a,I,p,s)) . k) = Initialize ((StepWhile=0 (a,I,p,s)) . k) by A3, SCMFSA_M:18;
A9: (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)))) + 2)) by SCMFSA_9:def 4;
A10: DataPart (IExec (I,(p +* (while=0 (a,I))),((StepWhile=0 (a,I,p,s)) . k))) = DataPart (Result (((p +* (while=0 (a,I))) +* I),(Initialized ((StepWhile=0 (a,I,p,s)) . k)))) by SCMFSA6B:def 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 A8, A7, 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, A4, Th17, A9
.= DataPart (IExec (I,(p +* (while=0 (a,I))),((StepWhile=0 (a,I,p,s)) . k))) by A10 ; :: thesis: verum