set D = Int-Locations \/ FinSeq-Locations ;
let I be parahalting keeping_0 Program of SCM+FSA ; :: thesis: for J being parahalting Program of SCM+FSA
for s being State of SCM+FSA st Initialized (I ';' J) c= s holds
( IC (Computation s,((LifeSpan (s +* I)) + 1)) = insloc (card I) & DataPart (Computation s,((LifeSpan (s +* I)) + 1)) = DataPart ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) & ProgramPart (Relocated J,(card I)) c= Computation s,((LifeSpan (s +* I)) + 1) & (Computation s,((LifeSpan (s +* I)) + 1)) . (intloc 0 ) = 1 & s is halting & LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) & ( J is keeping_0 implies (Result s) . (intloc 0 ) = 1 ) )
set SA0 = Start-At (insloc 0 );
let J be parahalting Program of SCM+FSA ; :: thesis: for s being State of SCM+FSA st Initialized (I ';' J) c= s holds
( IC (Computation s,((LifeSpan (s +* I)) + 1)) = insloc (card I) & DataPart (Computation s,((LifeSpan (s +* I)) + 1)) = DataPart ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) & ProgramPart (Relocated J,(card I)) c= Computation s,((LifeSpan (s +* I)) + 1) & (Computation s,((LifeSpan (s +* I)) + 1)) . (intloc 0 ) = 1 & s is halting & LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) & ( J is keeping_0 implies (Result s) . (intloc 0 ) = 1 ) )
let s be State of SCM+FSA ; :: thesis: ( Initialized (I ';' J) c= s implies ( IC (Computation s,((LifeSpan (s +* I)) + 1)) = insloc (card I) & DataPart (Computation s,((LifeSpan (s +* I)) + 1)) = DataPart ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) & ProgramPart (Relocated J,(card I)) c= Computation s,((LifeSpan (s +* I)) + 1) & (Computation s,((LifeSpan (s +* I)) + 1)) . (intloc 0 ) = 1 & s is halting & LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) & ( J is keeping_0 implies (Result s) . (intloc 0 ) = 1 ) ) )
set s1 = s +* I;
set s3 = (Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J);
set m1 = LifeSpan (s +* I);
set m3 = LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J));
A1: dom (Directed I) = dom I by FUNCT_4:105;
A2: ( Directed I c= I ';' J & I ';' J c= Initialized (I ';' J) ) by SCMFSA6A:26, SCMFSA6A:55;
assume A3: Initialized (I ';' J) c= s ; :: thesis: ( IC (Computation s,((LifeSpan (s +* I)) + 1)) = insloc (card I) & DataPart (Computation s,((LifeSpan (s +* I)) + 1)) = DataPart ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) & ProgramPart (Relocated J,(card I)) c= Computation s,((LifeSpan (s +* I)) + 1) & (Computation s,((LifeSpan (s +* I)) + 1)) . (intloc 0 ) = 1 & s is halting & LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) & ( J is keeping_0 implies (Result s) . (intloc 0 ) = 1 ) )
then ( Start-At (insloc 0 ) c= (I ';' J) +* (Start-At (insloc 0 )) & (I ';' J) +* (Start-At (insloc 0 )) c= s ) by Th8, FUNCT_4:26;
then A4: Start-At (insloc 0 ) c= s by XBOOLE_1:1;
A5: (s +* I) +* (Directed I) = s +* (I +* (Directed I)) by FUNCT_4:15
.= s +* (Directed I) by A1, FUNCT_4:20
.= (s +* (Initialized (I ';' J))) +* (Directed I) by A3, LATTICE2:8
.= s +* ((Initialized (I ';' J)) +* (Directed I)) by FUNCT_4:15
.= s +* (Initialized (I ';' J)) by A2, LATTICE2:8, XBOOLE_1:1
.= s by A3, LATTICE2:8 ;
then A6: Directed I c= s by FUNCT_4:26;
(I ';' J) +* (Start-At (insloc 0 )) c= s by A3, Th8;
then A7: s = s +* ((I ';' J) +* (Start-At (insloc 0 ))) by FUNCT_4:79;
A8: Initialized I c= s +* I by A3, SCMFSA6A:52;
then A9: I +* (Start-At (insloc 0 )) c= s +* I by Th8;
then A10: s +* I is halting by Th18;
hence A11: IC (Computation s,((LifeSpan (s +* I)) + 1)) = insloc (card I) by A4, A5, Th37, FUNCT_4:26; :: thesis: ( DataPart (Computation s,((LifeSpan (s +* I)) + 1)) = DataPart ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) & ProgramPart (Relocated J,(card I)) c= Computation s,((LifeSpan (s +* I)) + 1) & (Computation s,((LifeSpan (s +* I)) + 1)) . (intloc 0 ) = 1 & s is halting & LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) & ( J is keeping_0 implies (Result s) . (intloc 0 ) = 1 ) )
A12: intloc 0 in dom (Initialized I) by SCMFSA6A:45;

A13: now

let x be set ; :: thesis: ( x in dom (DataPart (Initialized J)) implies (DataPart (Initialized J)) . b₁ = (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) . b₁ )
assume x in dom (DataPart (Initialized J)) ; :: thesis: (DataPart (Initialized J)) . b₁ = (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) . b₁
then A14: x in (dom (Initialized J)) /\ (Int-Locations \/ FinSeq-Locations ) by RELAT_1:90, SCMFSA_2:127;
then A15: x in dom (Initialized J) by XBOOLE_0:def 4;
A16: x in Int-Locations \/ FinSeq-Locations by A14, XBOOLE_0:def 4;

per cases ( x in dom J or x = intloc 0 or x = IC SCM+FSA ) by A15, SCMFSA6A:44;

suppose A17: x in dom J ; :: thesis: (DataPart (Initialized J)) . b₁ = (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) . b₁

dom J c= NAT by RELAT_1:def 18;
then reconsider l = x as Instruction-Location of SCM+FSA by A17, AMI_1:def 4;
(DataPart (Initialized J)) . l = (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) . l by A16, SCMFSA6A:37;
hence (DataPart (Initialized J)) . x = (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) . x ; :: thesis: verum

end;

suppose A18: x = intloc 0 ; :: thesis: (DataPart (Initialized J)) . b₁ = (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) . b₁

thus (DataPart (Initialized J)) . x = (Initialized J) . x by A16, FUNCT_1:72, SCMFSA_2:127
.= 1 by A18, SCMFSA6A:46
.= (Initialized I) . x by A18, SCMFSA6A:46
.= (s +* I) . x by A8, A12, A18, GRFUNC_1:8
.= (Computation (s +* I),(LifeSpan (s +* I))) . x by A9, A18, Def4
.= (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) . x by A16, FUNCT_1:72, SCMFSA_2:127 ; :: thesis: verum

end;

suppose x = IC SCM+FSA ; :: thesis: (DataPart (Initialized J)) . b₁ = (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) . b₁

hence (DataPart (Initialized J)) . x = (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) . x by A14, SCMFSA6A:37, XBOOLE_0:def 4; :: thesis: verum

end;

end;

end;

set s4 = Computation s,((LifeSpan (s +* I)) + 1);
reconsider m = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J))) as Element of NAT ;
A19: ProgramPart (Relocated J,(card I)) c= I ';' J by FUNCT_4:26;
A20: Initialized J c= (Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J) by FUNCT_4:26;
Initialized J c= (Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J) by FUNCT_4:26;
then dom (Initialized J) c= dom ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) by GRFUNC_1:8;
then A21: dom (Initialized J) c= the carrier of SCM+FSA by AMI_1:79;
A22: (Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J) is halting by Th19, FUNCT_4:26;
dom (DataPart (Initialized J)) = (dom (Initialized J)) /\ (Int-Locations \/ FinSeq-Locations ) by RELAT_1:90, SCMFSA_2:127;
then dom (DataPart (Initialized J)) c= the carrier of SCM+FSA /\ (Int-Locations \/ FinSeq-Locations ) by A21, XBOOLE_1:26;
then dom (DataPart (Initialized J)) c= (dom (Computation (s +* I),(LifeSpan (s +* I)))) /\ (Int-Locations \/ FinSeq-Locations ) by AMI_1:79;
then dom (DataPart (Initialized J)) c= dom (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) by RELAT_1:90, SCMFSA_2:127;
then ( DataPart ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) = (DataPart (Computation (s +* I),(LifeSpan (s +* I)))) +* (DataPart (Initialized J)) & DataPart (Initialized J) c= DataPart (Computation (s +* I),(LifeSpan (s +* I))) ) by A13, FUNCT_4:75, GRFUNC_1:8;
then A23: DataPart (Computation (s +* I),(LifeSpan (s +* I))) = DataPart ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) by LATTICE2:8;
s +* I = (s +* (Start-At (insloc 0 ))) +* I by A4, FUNCT_4:79
.= (s +* I) +* (Start-At (insloc 0 )) by Th14
.= s +* (I +* (Start-At (insloc 0 ))) by FUNCT_4:15 ;
then DataPart (Computation s,(LifeSpan (s +* I))) = DataPart ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) by A7, A10, A23, Th40, SCMFSA6A:39;
hence A24: DataPart (Computation s,((LifeSpan (s +* I)) + 1)) = DataPart ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) by A9, A4, A6, Th18, Th38; :: thesis: ( ProgramPart (Relocated J,(card I)) c= Computation s,((LifeSpan (s +* I)) + 1) & (Computation s,((LifeSpan (s +* I)) + 1)) . (intloc 0 ) = 1 & s is halting & LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) & ( J is keeping_0 implies (Result s) . (intloc 0 ) = 1 ) )
I ';' J c= Initialized (I ';' J) by SCMFSA6A:26;
then I ';' J c= s by A3, XBOOLE_1:1;
then ProgramPart (Relocated J,(card I)) c= s by A19, XBOOLE_1:1;
then A25: [(ProgramPart (Relocated J,(card I)))] c= Computation s,((LifeSpan (s +* I)) + 1) by AMI_1:81;
hence ProgramPart (Relocated J,(card I)) c= Computation s,((LifeSpan (s +* I)) + 1) ; :: thesis: ( (Computation s,((LifeSpan (s +* I)) + 1)) . (intloc 0 ) = 1 & s is halting & LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) & ( J is keeping_0 implies (Result s) . (intloc 0 ) = 1 ) )
A26: intloc 0 in dom (Initialized J) by SCMFSA6A:45;
intloc 0 in Int-Locations by SCMFSA_2:9;
then A27: intloc 0 in Int-Locations \/ FinSeq-Locations by XBOOLE_0:def 3;
hence (Computation s,((LifeSpan (s +* I)) + 1)) . (intloc 0 ) = (DataPart ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J))) . (intloc 0 ) by A24, FUNCT_1:72, SCMFSA_2:127
.= ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) . (intloc 0 ) by A27, FUNCT_1:72, SCMFSA_2:127
.= (Initialized J) . (intloc 0 ) by A26, FUNCT_4:14
.= 1 by SCMFSA6A:46 ;
:: thesis: ( s is halting & LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) & ( J is keeping_0 implies (Result s) . (intloc 0 ) = 1 ) )
A28: J +* (Start-At (insloc 0 )) c= (Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J) by Th8, FUNCT_4:26;
then IncAddr (CurInstr (Computation ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)),(LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J))))),(card I) = CurInstr (Computation (Computation s,((LifeSpan (s +* I)) + 1)),(LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)))) by A11, A24, A25, Th27;
then IncAddr (CurInstr (Computation ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)),(LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J))))),(card I) = CurInstr (Computation s,(((LifeSpan (s +* I)) + 1) + (LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J))))) by AMI_1:51;
then A29: CurInstr (Computation s,m) = IncAddr (halt SCM+FSA ),(card I) by A22, AMI_1:def 46
.= halt SCM+FSA by SCMFSA_4:8 ;
hence A30: s is halting by AMI_1:def 20; :: thesis: ( LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) & ( J is keeping_0 implies (Result s) . (intloc 0 ) = 1 ) )

A31: now

let k be Element of NAT ; :: thesis: ( ((LifeSpan (s +* I)) + 1) + k < m implies not CurInstr (Computation s,(((LifeSpan (s +* I)) + 1) + k)) = halt SCM+FSA )
assume ((LifeSpan (s +* I)) + 1) + k < m ; :: thesis: not CurInstr (Computation s,(((LifeSpan (s +* I)) + 1) + k)) = halt SCM+FSA
then A32: k < LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) by XREAL_1:8;
assume A33: CurInstr (Computation s,(((LifeSpan (s +* I)) + 1) + k)) = halt SCM+FSA ; :: thesis: contradiction
IncAddr (CurInstr (Computation ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)),k)),(card I) = CurInstr (Computation (Computation s,((LifeSpan (s +* I)) + 1)),k) by A11, A24, A28, A25, Th27
.= halt SCM+FSA by A33, AMI_1:51 ;
then InsCode (CurInstr (Computation ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)),k)) = 0 by SCMFSA_2:124, SCMFSA_4:22;
then CurInstr (Computation ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)),k) = halt SCM+FSA by SCMFSA_2:122;
hence contradiction by A22, A32, AMI_1:def 46; :: thesis: verum

end;

now

let k be Element of NAT ; :: thesis: ( k < m implies CurInstr (Computation s,b₁) <> halt SCM+FSA )
assume A34: k < m ; :: thesis: CurInstr (Computation s,b₁) <> halt SCM+FSA

per cases ( k <= LifeSpan (s +* I) or LifeSpan (s +* I) < k ) ;

suppose k <= LifeSpan (s +* I) ; :: thesis: CurInstr (Computation s,b₁) <> halt SCM+FSA

hence CurInstr (Computation s,k) <> halt SCM+FSA by A8, A5, Th39; :: thesis: verum

end;

suppose LifeSpan (s +* I) < k ; :: thesis: CurInstr (Computation s,b₁) <> halt SCM+FSA

then (LifeSpan (s +* I)) + 1 <= k by NAT_1:13;
then consider kk being Nat such that
A35: ((LifeSpan (s +* I)) + 1) + kk = k by NAT_1:10;
kk in NAT by ORDINAL1:def 13;
hence CurInstr (Computation s,k) <> halt SCM+FSA by A31, A34, A35; :: thesis: verum

end;

end;

end;

then for k being Element of NAT st CurInstr (Computation s,k) = halt SCM+FSA holds
m <= k ;
then A36: LifeSpan s = m by A29, A30, AMI_1:def 46;
s +* I = s +* (Initialized I) by A3, SCMFSA6A:51;
then Initialized I c= s +* I by FUNCT_4:26;
hence LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) by A36, Th19, AMI_1:122; :: thesis: ( J is keeping_0 implies (Result s) . (intloc 0 ) = 1 )
A37: J +* (Start-At (insloc 0 )) c= (Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J) by Th8, FUNCT_4:26;

hereby :: thesis: verum

A38: DataPart (Computation ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)),(LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)))) = DataPart (Computation (Computation s,((LifeSpan (s +* I)) + 1)),(LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)))) by A11, A24, A28, A25, Th27;
assume A39: J is keeping_0 ; :: thesis: (Result s) . (intloc 0 ) = 1
thus (Result s) . (intloc 0 ) = (Computation s,m) . (intloc 0 ) by A30, A36, AMI_1:122
.= (Computation (Computation s,((LifeSpan (s +* I)) + 1)),(LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)))) . (intloc 0 ) by AMI_1:51
.= (Computation ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)),(LifeSpan ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)))) . (intloc 0 ) by A38, SCMFSA6A:38
.= ((Computation (s +* I),(LifeSpan (s +* I))) +* (Initialized J)) . (intloc 0 ) by A37, A39, Def4
.= (Initialized J) . (intloc 0 ) by A26, A20, GRFUNC_1:8
.= 1 by SCMFSA6A:46 ; :: thesis: verum

end;