set D = Int-Locations \/ FinSeq-Locations;
let I be InitHalting keepInt0_1 Program of SCM+FSA; :: thesis: for J being InitHalting Program of SCM+FSA
for s being State of SCM+FSA st Initialized (I ';' J) c= s holds
( IC (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = card I & DataPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = DataPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) & ProgramPart (Relocated (J,(card I))) c= Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1)) & (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) . (intloc 0) = 1 & ProgramPart s halts_on s & LifeSpan ((ProgramPart s),s) = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J))),((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J)))) & ( J is keeping_0 implies (Result ((ProgramPart s),s)) . (intloc 0) = 1 ) )
let J be InitHalting Program of SCM+FSA; :: thesis: for s being State of SCM+FSA st Initialized (I ';' J) c= s holds
( IC (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = card I & DataPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = DataPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) & ProgramPart (Relocated (J,(card I))) c= Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1)) & (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) . (intloc 0) = 1 & ProgramPart s halts_on s & LifeSpan ((ProgramPart s),s) = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J))),((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J)))) & ( J is keeping_0 implies (Result ((ProgramPart s),s)) . (intloc 0) = 1 ) )
let s be State of SCM+FSA; :: thesis: ( Initialized (I ';' J) c= s implies ( IC (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = card I & DataPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = DataPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) & ProgramPart (Relocated (J,(card I))) c= Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1)) & (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) . (intloc 0) = 1 & ProgramPart s halts_on s & LifeSpan ((ProgramPart s),s) = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J))),((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J)))) & ( J is keeping_0 implies (Result ((ProgramPart s),s)) . (intloc 0) = 1 ) ) )
set s1 = s +* I;
set s3 = (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J);
set m1 = LifeSpan ((ProgramPart (s +* I)),(s +* I));
set m3 = LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)));
A1: ((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)) c= (I ';' J) +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
assume A2: Initialized (I ';' J) c= s ; :: thesis: ( IC (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = card I & DataPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = DataPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) & ProgramPart (Relocated (J,(card I))) c= Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1)) & (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) . (intloc 0) = 1 & ProgramPart s halts_on s & LifeSpan ((ProgramPart s),s) = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J))),((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J)))) & ( J is keeping_0 implies (Result ((ProgramPart s),s)) . (intloc 0) = 1 ) )
then A3: s = s +* (Initialized (I ';' J)) by FUNCT_4:79;
set s4 = Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1));
T4: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) by AMI_1:123;
A4: ( Directed I c= I ';' J & I ';' J c= Initialized (I ';' J) ) by SCMFSA6A:26, SCMFSA6A:55;
reconsider m = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)))) as Element of NAT ;
A5: dom (Directed I) = dom I by FUNCT_4:105;
A6: (s +* I) +* (Directed I) = s +* (I +* (Directed I)) by FUNCT_4:15
.= s +* (Directed I) by A5, FUNCT_4:20
.= (s +* (Initialized (I ';' J))) +* (Directed I) by A2, LATTICE2:8
.= s +* ((Initialized (I ';' J)) +* (Directed I)) by FUNCT_4:15
.= s +* (Initialized (I ';' J)) by A4, LATTICE2:8, XBOOLE_1:1
.= s by A2, LATTICE2:8 ;
then A7: Directed I c= s by FUNCT_4:26;
Initialized J c= (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J) by FUNCT_4:26;
then dom (Initialized J) c= dom ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) by GRFUNC_1:8;
then A8: dom (Initialized J) c= the carrier of SCM+FSA by PARTFUN1:def 4;
A9: ProgramPart (Relocated (J,(card I))) c= I ';' J by FUNCT_4:26;
(I ';' J) +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA))) c= s by A2, FUNCT_4:15;
then A10: ((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)) c= s by A1, XBOOLE_1:1;
A11: Initialized I c= s +* I by A2, SCMFSA6A:52;
then A12: ProgramPart (s +* I) halts_on s +* I by Th5;
hence A13: IC (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = card I by A10, A6, Th21, FUNCT_4:26; :: thesis: ( DataPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = DataPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) & ProgramPart (Relocated (J,(card I))) c= Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1)) & (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) . (intloc 0) = 1 & ProgramPart s halts_on s & LifeSpan ((ProgramPart s),s) = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J))),((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J)))) & ( J is keeping_0 implies (Result ((ProgramPart s),s)) . (intloc 0) = 1 ) )

A14: now

let x be set ; :: thesis: ( x in dom (DataPart (Initialized J)) implies (DataPart (Initialized J)) . b₁ = (DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) . b₁ )
assume x in dom (DataPart (Initialized J)) ; :: thesis: (DataPart (Initialized J)) . b₁ = (DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) . b₁
then A15: x in (dom (Initialized J)) /\ (Int-Locations \/ FinSeq-Locations) by RELAT_1:90, SCMFSA_2:127;
then A16: x in dom (Initialized J) by XBOOLE_0:def 4;
A17: x in Int-Locations \/ FinSeq-Locations by A15, XBOOLE_0:def 4;

per cases ( x in dom J or x = intloc 0 or x = IC SCM+FSA ) by A16, SCMFSA6A:44;

suppose A18: x in dom J ; :: thesis: (DataPart (Initialized J)) . b₁ = (DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) . b₁

dom J c= NAT by RELAT_1:def 18;
then reconsider l = x as Element of NAT by A18;
(DataPart (Initialized J)) . l = (DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) . l by A17, SCMFSA6A:37;
hence (DataPart (Initialized J)) . x = (DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) . x ; :: thesis: verum

end;

suppose A19: x = intloc 0 ; :: thesis: (DataPart (Initialized J)) . b₁ = (DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) . b₁

thus (DataPart (Initialized J)) . x = (Initialized J) . x by A17, FUNCT_1:72, SCMFSA_2:127
.= 1 by A19, SCMFSA6A:46
.= (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) . x by A11, A19, Def3
.= (DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) . x by A17, FUNCT_1:72, SCMFSA_2:127 ; :: thesis: verum

end;

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

hence (DataPart (Initialized J)) . x = (DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) . x by A15, SCMFSA6A:37, XBOOLE_0:def 4; :: thesis: verum

end;

end;

end;

A20: J +* (Start-At (0,SCM+FSA)) c= (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J) by FUNCT_4:26, SCMFSA6B:8;
A21: ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) halts_on (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J) by Th5, 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 A8, XBOOLE_1:26;
then dom (DataPart (Initialized J)) c= (dom (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) /\ (Int-Locations \/ FinSeq-Locations) by PARTFUN1:def 4;
then dom (DataPart (Initialized J)) c= dom (DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) by RELAT_1:90, SCMFSA_2:127;
then ( DataPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) = (DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))))) +* (DataPart (Initialized J)) & DataPart (Initialized J) c= DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) ) by A14, FUNCT_4:75, GRFUNC_1:8;
then A22: DataPart (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) = DataPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) by LATTICE2:8;
s +* I = (s +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) +* I by A10, FUNCT_4:79
.= (s +* I) +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA))) by Th19
.= s +* (I +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) by FUNCT_4:15
.= s +* (Initialized I) by FUNCT_4:15 ;
then X: DataPart (Comput ((ProgramPart s),s,(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) = DataPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) by A3, A12, A22, Th24, SCMFSA6A:39;
ProgramPart (s +* I) halts_on s +* I by A11, Th5;
hence A23: DataPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) = DataPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) by A10, A7, Th22, X; :: thesis: ( ProgramPart (Relocated (J,(card I))) c= Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1)) & (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) . (intloc 0) = 1 & ProgramPart s halts_on s & LifeSpan ((ProgramPart s),s) = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J))),((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J)))) & ( J is keeping_0 implies (Result ((ProgramPart s),s)) . (intloc 0) = 1 ) )
I ';' J c= Initialized (I ';' J) by SCMFSA6A:26;
then I ';' J c= s by A2, XBOOLE_1:1;
then ProgramPart (Relocated (J,(card I))) c= s by A9, XBOOLE_1:1;
then A24: ProgramPart (Relocated (J,(card I))) c= Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1)) by AMI_1:81;
hence ProgramPart (Relocated (J,(card I))) c= Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1)) ; :: thesis: ( (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) . (intloc 0) = 1 & ProgramPart s halts_on s & LifeSpan ((ProgramPart s),s) = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J))),((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J)))) & ( J is keeping_0 implies (Result ((ProgramPart s),s)) . (intloc 0) = 1 ) )
A25: intloc 0 in dom (Initialized J) by SCMFSA6A:45;
intloc 0 in Int-Locations by SCMFSA_2:9;
then A26: intloc 0 in Int-Locations \/ FinSeq-Locations by XBOOLE_0:def 3;
hence (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) . (intloc 0) = (DataPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))) . (intloc 0) by A23, FUNCT_1:72, SCMFSA_2:127
.= ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) . (intloc 0) by A26, FUNCT_1:72, SCMFSA_2:127
.= (Initialized J) . (intloc 0) by A25, FUNCT_4:14
.= 1 by SCMFSA6A:46 ;
:: thesis: ( ProgramPart s halts_on s & LifeSpan ((ProgramPart s),s) = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J))),((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J)))) & ( J is keeping_0 implies (Result ((ProgramPart s),s)) . (intloc 0) = 1 ) )
T: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) by AMI_1:123;
x: Comput ((ProgramPart s),s,(((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)))))) = Comput ((ProgramPart s),(Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))),(LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))))) by EXTPRO_1:5;
A27: Initialized J c= (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J) by FUNCT_4:26;
then IncAddr ((CurInstr ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),(Comput ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)),(LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)))))))),(card I)) = CurInstr ((ProgramPart s),(Comput ((ProgramPart s),(Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))),(LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))))))) by A13, A23, A24, Th12, T4;
then IncAddr ((CurInstr ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),(Comput ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)),(LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)))))))),(card I)) = CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,(((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)))))))) by x;
then A28: CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,m))) = IncAddr ((halt SCM+FSA),(card I)) by A21, EXTPRO_1:def 14
.= halt SCM+FSA by SCMFSA_4:8 ;
hence A29: ProgramPart s halts_on s by EXTPRO_1:30; :: thesis: ( LifeSpan ((ProgramPart s),s) = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J))),((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J)))) & ( J is keeping_0 implies (Result ((ProgramPart s),s)) . (intloc 0) = 1 ) )

A30: now

let k be Element of NAT ; :: thesis: ( ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + k < m implies not CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,(((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + k)))) = halt SCM+FSA )
assume ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + k < m ; :: thesis: not CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,(((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + k)))) = halt SCM+FSA
then A31: k < LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))) by XREAL_1:8;
assume A32: CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,(((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + k)))) = halt SCM+FSA ; :: thesis: contradiction
T: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) by AMI_1:123;
x: Comput ((ProgramPart s),s,(((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + k)) = Comput ((ProgramPart s),(Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))),k) by EXTPRO_1:5;
IncAddr ((CurInstr ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),(Comput ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)),k)))),(card I)) = CurInstr ((ProgramPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1)))),(Comput ((ProgramPart s),(Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))),k))) by A13, A23, A27, A24, Th12, T4
.= halt SCM+FSA by A32, x, T ;
then InsCode (CurInstr ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),(Comput ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)),k)))) = 0 by SCMFSA_2:124, SCMFSA_4:22;
then CurInstr ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),(Comput ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)),k))) = halt SCM+FSA by SCMFSA_2:122;
hence contradiction by A21, A31, EXTPRO_1:def 14; :: thesis: verum

end;

now

let k be Element of NAT ; :: thesis: ( k < m implies CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,b₁))) <> halt SCM+FSA )
assume A33: k < m ; :: thesis: CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,b₁))) <> halt SCM+FSA

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

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

hence CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,k))) <> halt SCM+FSA by A11, A6, Th23; :: thesis: verum

end;

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

then (LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1 <= k by NAT_1:13;
then consider kk being Nat such that
A34: ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + kk = k by NAT_1:10;
reconsider kk = kk as Element of NAT by ORDINAL1:def 13;
((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + kk = k by A34;
hence CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,k))) <> halt SCM+FSA by A30, A33; :: thesis: verum

end;

end;

end;

then for k being Element of NAT st CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,k))) = halt SCM+FSA holds
m <= k ;
then A35: LifeSpan ((ProgramPart s),s) = m by A28, A29, EXTPRO_1:def 14;
s +* I = s +* (Initialized I) by A2, SCMFSA6A:51;
then Initialized I c= s +* I by FUNCT_4:26;
then X: ProgramPart (s +* I) halts_on s +* I by Th5;
Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I)))) = Result ((ProgramPart (s +* I)),(s +* I)) by X, EXTPRO_1:23;
hence LifeSpan ((ProgramPart s),s) = ((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1) + (LifeSpan ((ProgramPart ((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J))),((Result ((ProgramPart (s +* I)),(s +* I))) +* (Initialized J)))) by A35; :: thesis: ( J is keeping_0 implies (Result ((ProgramPart s),s)) . (intloc 0) = 1 )
A36: Initialized J c= (Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J) by FUNCT_4:26;

hereby :: thesis: verum

A37: DataPart (Comput ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)),(LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)))))) = DataPart (Comput ((ProgramPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1)))),(Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))),(LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)))))) by A13, A23, A27, A24, Th12;
assume A38: J is keeping_0 ; :: thesis: (Result ((ProgramPart s),s)) . (intloc 0) = 1
thus (Result ((ProgramPart s),s)) . (intloc 0) = (Comput ((ProgramPart s),s,m)) . (intloc 0) by A29, A35, EXTPRO_1:23
.= (Comput ((ProgramPart s),(Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))),(LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)))))) . (intloc 0) by EXTPRO_1:5
.= (Comput ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)),(LifeSpan ((ProgramPart ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J))),((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)))))) . (intloc 0) by A37, T, SCMFSA6A:38
.= ((Comput ((ProgramPart (s +* I)),(s +* I),(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) +* (Initialized J)) . (intloc 0) by A20, A38, SCMFSA6B:def 4
.= (Initialized J) . (intloc 0) by A25, A36, GRFUNC_1:8
.= 1 by SCMFSA6A:46 ; :: thesis: verum

end;