let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for a being read-write Int-Location
for Ig being good Program of SCM+FSA st ( ProperBodyWhile=0 a,Ig,s,p or Ig is parahalting ) & s . (intloc 0) = 1 holds
for k being Element of NAT holds ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1
let s be State of SCM+FSA; :: thesis: for a being read-write Int-Location
for Ig being good Program of SCM+FSA st ( ProperBodyWhile=0 a,Ig,s,p or Ig is parahalting ) & s . (intloc 0) = 1 holds
for k being Element of NAT holds ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1
let a be read-write Int-Location ; :: thesis: for Ig being good Program of SCM+FSA st ( ProperBodyWhile=0 a,Ig,s,p or Ig is parahalting ) & s . (intloc 0) = 1 holds
for k being Element of NAT holds ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1
let Ig be good Program of SCM+FSA; :: thesis: ( ( ProperBodyWhile=0 a,Ig,s,p or Ig is parahalting ) & s . (intloc 0) = 1 implies for k being Element of NAT holds ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1 )
set I = Ig;
assume that
A1: ( ProperBodyWhile=0 a,Ig,s,p or Ig is parahalting ) and
A2: s . (intloc 0) = 1 ; :: thesis: for k being Element of NAT holds ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1
set SW = StepWhile=0 (a,Ig,p,s);
set PW = p +* (while=0 (a,Ig));
defpred S₁[ Nat] means ((StepWhile=0 (a,Ig,p,s)) . $1) . (intloc 0) = 1;
A4: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1 ; :: thesis: S₁[k + 1]

per cases ( ((StepWhile=0 (a,Ig,p,s)) . k) . a <> 0 or ((StepWhile=0 (a,Ig,p,s)) . k) . a = 0 ) ;

suppose ((StepWhile=0 (a,Ig,p,s)) . k) . a <> 0 ; :: thesis: S₁[k + 1]

then DataPart ((StepWhile=0 (a,Ig,p,s)) . (k + 1)) = DataPart ((StepWhile=0 (a,Ig,p,s)) . k) by Th24;
hence S₁[k + 1] by A5, SCMFSA6A:7; :: thesis: verum

end;

suppose A6: ((StepWhile=0 (a,Ig,p,s)) . k) . a = 0 ; :: thesis: S₁[k + 1]

set Ins = NAT ;
set SWkIS = Initialize ((StepWhile=0 (a,Ig,p,s)) . k);
set PWIS = (p +* (while=0 (a,Ig))) +* Ig;
set SWkI = Initialized ((StepWhile=0 (a,Ig,p,s)) . k);
set PWI = (p +* (while=0 (a,Ig))) +* Ig;
set ISWk = Initialized ((StepWhile=0 (a,Ig,p,s)) . k);
A9: DataPart ((StepWhile=0 (a,Ig,p,s)) . k) = DataPart (Initialized ((StepWhile=0 (a,Ig,p,s)) . k)) by A5, SCMFSA8C:7;
A10: ProperBodyWhile=0 a,Ig,s,p by A1, Th19;
then A11: Ig is_closed_on (StepWhile=0 (a,Ig,p,s)) . k,p +* (while=0 (a,Ig)) by A6, Def1;
Ig is_halting_on (StepWhile=0 (a,Ig,p,s)) . k,p +* (while=0 (a,Ig)) by A6, A10, Def1;
then A13: Ig is_halting_on Initialized ((StepWhile=0 (a,Ig,p,s)) . k),p +* (while=0 (a,Ig)) by A11, A9, SCMFSA8B:5;
Initialized ((StepWhile=0 (a,Ig,p,s)) . k) = Initialize (Initialized ((StepWhile=0 (a,Ig,p,s)) . k)) by MEMSTR_0:44;
then B14: (p +* (while=0 (a,Ig))) +* Ig halts_on Initialized ((StepWhile=0 (a,Ig,p,s)) . k) by A13, SCMFSA7B:def 7;
A15: (p +* (while=0 (a,Ig))) +* Ig halts_on Initialized ((StepWhile=0 (a,Ig,p,s)) . k) by B14;
A16: Initialized ((StepWhile=0 (a,Ig,p,s)) . k) = Initialize ((StepWhile=0 (a,Ig,p,s)) . k) by A5, SCMFSA8C:4;
A17: DataPart (IExec (Ig,(p +* (while=0 (a,Ig))),((StepWhile=0 (a,Ig,p,s)) . k))) = DataPart (Result (((p +* (while=0 (a,Ig))) +* Ig),(Initialized ((StepWhile=0 (a,Ig,p,s)) . k)))) by SCMFSA6B:def 1
.= DataPart (Result (((p +* (while=0 (a,Ig))) +* Ig),(Initialized ((StepWhile=0 (a,Ig,p,s)) . k))))
.= DataPart (Comput (((p +* (while=0 (a,Ig))) +* Ig),(Initialize ((StepWhile=0 (a,Ig,p,s)) . k)),(LifeSpan (((p +* (while=0 (a,Ig))) +* Ig),(Initialize ((StepWhile=0 (a,Ig,p,s)) . k)))))) by A16, A15, EXTPRO_1:23 ;
Ig is_closed_on Initialized ((StepWhile=0 (a,Ig,p,s)) . k),p +* (while=0 (a,Ig)) by A11, A9, SCMFSA8B:3;
then DataPart ((StepWhile=0 (a,Ig,p,s)) . (k + 1)) = DataPart (IExec (Ig,(p +* (while=0 (a,Ig))),((StepWhile=0 (a,Ig,p,s)) . k))) by A5, A6, A13, Th25;
hence ((StepWhile=0 (a,Ig,p,s)) . (k + 1)) . (intloc 0) = (Comput (((p +* (while=0 (a,Ig))) +* Ig),(Initialize ((StepWhile=0 (a,Ig,p,s)) . k)),(LifeSpan (((p +* (while=0 (a,Ig))) +* Ig),(Initialize ((StepWhile=0 (a,Ig,p,s)) . k)))))) . (intloc 0) by A17, SCMFSA6A:7
.= 1 by A5, A11, SCMFSA8C:68 ;
:: thesis: verum

end;

end;

end;

A18: S₁[ 0 ] by A2, SCMFSA_9:def 4;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A18, A4); :: thesis: verum