set D = Data-Locations ;
let s be State of SCM+FSA; :: according to SCM_HALT:def 2 :: thesis: ( Initialize ((intloc 0) .--> 1) c= s implies for P being Instruction-Sequence of SCM+FSA st I ';' J c= P holds
P halts_on s )
assume A1: Initialize ((intloc 0) .--> 1) c= s ; :: thesis: for P being Instruction-Sequence of SCM+FSA st I ';' J c= P holds
P halts_on s
then A2: s = Initialized s by FUNCT_4:98;
let p be Instruction-Sequence of SCM+FSA; :: thesis: ( I ';' J c= p implies p halts_on s )
assume A3: I ';' J c= p ; :: thesis: p halts_on s
A4: p = p +* (I ';' J) by A3, FUNCT_4:98;
set p1 = p +* I;
set s3 = (Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1));
set p3 = (p +* I) +* J;
A7: J c= (p +* I) +* J by FUNCT_4:25;
set m1 = LifeSpan ((p +* I),s);
set s4 = Comput (p,s,((LifeSpan ((p +* I),s)) + 1));
A8: I c= p +* I by FUNCT_4:25;
A9: Reloc (J,(card I)) c= I ';' J by FUNCT_4:25;
set m3 = LifeSpan (((p +* I) +* J),((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))));
A11: dom (DataPart (Initialize ((intloc 0) .--> 1))) = (dom (Initialize ((intloc 0) .--> 1))) /\ (Data-Locations ) by RELAT_1:61;
reconsider m = ((LifeSpan ((p +* I),s)) + 1) + (LifeSpan (((p +* I) +* J),((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))))) as Element of NAT ;
A17: Initialize ((intloc 0) .--> 1) c= Initialized s by FUNCT_4:25;
I c= p +* I by FUNCT_4:25;
then A18: p +* I halts_on Initialized s by Def2, A17;

A19: now

let x be set ; :: thesis: ( x in dom (DataPart (Initialize ((intloc 0) .--> 1))) implies (DataPart (Comput ((p +* I),s,(LifeSpan ((p +* I),s))))) . b₁ = (DataPart (Initialize ((intloc 0) .--> 1))) . b₁ )
DataPart (Initialize ((intloc 0) .--> 1)) c= Initialize ((intloc 0) .--> 1) by RELAT_1:59;
then AA: dom (DataPart (Initialize ((intloc 0) .--> 1))) c= dom (Initialize ((intloc 0) .--> 1)) by RELAT_1:11;
assume ZZ: x in dom (DataPart (Initialize ((intloc 0) .--> 1))) ; :: thesis: (DataPart (Comput ((p +* I),s,(LifeSpan ((p +* I),s))))) . b₁ = (DataPart (Initialize ((intloc 0) .--> 1))) . b₁
then x in dom (Initialize ((intloc 0) .--> 1)) by AA;
then pc: x in {(intloc 0),(IC )} by diS, ENUMSET1:1;

per cases ( x = intloc 0 or x = IC ) by pc, TARSKI:def 2;

suppose A24: x = intloc 0 ; :: thesis: (DataPart (Comput ((p +* I),s,(LifeSpan ((p +* I),s))))) . b₁ = (DataPart (Initialize ((intloc 0) .--> 1))) . b₁

then x in Int-Locations by SCMFSA_2:2;
then A25: x in Data-Locations by SCMFSA_2:100, XBOOLE_0:def 3;
hence (DataPart (Comput ((p +* I),s,(LifeSpan ((p +* I),s))))) . x = (Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) . x by FUNCT_1:49
.= 1 by A1, A24, Def3, A8
.= (DataPart (Initialize ((intloc 0) .--> 1))) . x by A25, A24, FUNCT_1:49, SCMFSA6A:43 ;
:: thesis: verum

end;

suppose S: x = IC ; :: thesis: (DataPart (Initialize ((intloc 0) .--> 1))) . b₁ = (DataPart (Comput ((p +* I),s,(LifeSpan ((p +* I),s))))) . b₁

dom (DataPart (Initialize ((intloc 0) .--> 1))) c= Data-Locations by RELAT_1:58;
hence (DataPart (Initialize ((intloc 0) .--> 1))) . x = (DataPart (Comput ((p +* I),s,(LifeSpan ((p +* I),s))))) . x by S, ZZ, STRUCT_0:3; :: thesis: verum

end;

end;

end;

take m ; :: according to EXTPRO_1:def 8 :: thesis: ( IC (Comput (p,s,m)) in proj1 p & CurInstr (p,(Comput (p,s,m))) = halt SCM+FSA )
IC (Comput (p,s,m)) in NAT ;
hence IC (Comput (p,s,m)) in dom p by PARTFUN1:def 2; :: thesis: CurInstr (p,(Comput (p,s,m))) = halt SCM+FSA
Directed I c= I ';' J by SCMFSA6A:16;
then A27: Directed I c= p by A3, XBOOLE_1:1;
Initialize ((intloc 0) .--> 1) c= (Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1)) by FUNCT_4:25;
then dom (Initialize ((intloc 0) .--> 1)) c= dom ((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))) by GRFUNC_1:2;
then dom (Initialize ((intloc 0) .--> 1)) c= the carrier of SCM+FSA by PARTFUN1:def 2;
then dom (DataPart (Initialize ((intloc 0) .--> 1))) c= the carrier of SCM+FSA /\ (Data-Locations ) by A11, XBOOLE_1:26;
then dom (DataPart (Initialize ((intloc 0) .--> 1))) c= (dom (Comput ((p +* I),s,(LifeSpan ((p +* I),s))))) /\ (Data-Locations ) by PARTFUN1:def 2;
then dom (DataPart (Initialize ((intloc 0) .--> 1))) c= dom (DataPart (Comput ((p +* I),s,(LifeSpan ((p +* I),s))))) by RELAT_1:61;
then DataPart (Initialize ((intloc 0) .--> 1)) c= DataPart (Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) by A19, GRFUNC_1:2;
then A30: DataPart (Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) = (DataPart (Comput ((p +* I),s,(LifeSpan ((p +* I),s))))) +* (DataPart (Initialize ((intloc 0) .--> 1))) by FUNCT_4:98
.= DataPart ((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))) by FUNCT_4:71 ;
A31: DataPart (Comput (p,s,(LifeSpan ((p +* I),s)))) = DataPart ((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))) by A30, A2, A4, A18, Th24;
I c= p +* I by FUNCT_4:25;
then YY: p +* I halts_on s by A1, Def2;
then A33: DataPart (Comput (p,s,((LifeSpan ((p +* I),s)) + 1))) = DataPart ((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))) by A1, Th22, A31, A27;
A35: Comput (p,s,(((LifeSpan ((p +* I),s)) + 1) + (LifeSpan (((p +* I) +* J),((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))))))) = Comput (p,(Comput (p,s,((LifeSpan ((p +* I),s)) + 1))),(LifeSpan (((p +* I) +* J),((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1)))))) by EXTPRO_1:4;
A36: Reloc (J,(card I)) c= p by A9, A3, XBOOLE_1:1;
X: Initialize ((intloc 0) .--> 1) c= (Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1)) by FUNCT_4:25;
x: IC (Comput (p,s,((LifeSpan ((p +* I),s)) + 1))) = card I by YY, A27, Th21, A1;
A37: IncAddr ((CurInstr (((p +* I) +* J),(Comput (((p +* I) +* J),((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))),(LifeSpan (((p +* I) +* J),((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))))))))),(card I)) = CurInstr (p,(Comput (p,s,(((LifeSpan ((p +* I),s)) + 1) + (LifeSpan (((p +* I) +* J),((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))))))))) by A35, X, A7, A36, Th12, x, A33;
(p +* I) +* J halts_on (Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1)) by A7, Def2, X;
then CurInstr (p,(Comput (p,s,m))) = IncAddr ((halt SCM+FSA),(card I)) by A37, EXTPRO_1:def 15
.= halt SCM+FSA by COMPOS_1:11 ;
hence CurInstr (p,(Comput (p,s,m))) = halt SCM+FSA ; :: thesis: verum