let I, J be Program of SCMPDS ; :: thesis:
let s be State of SCMPDS ; :: thesis:
assume A1: I is_closed_on s ; :: thesis:
set pI = stop I;
set IsI = Initialized (stop I);
set pIJ = stop (I ';' J);
set IsJ = Initialized (stop (I ';' J));
set s1 = s +* (Initialized (stop I));
set IL = NAT ;
A2: (s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))) = (s +* (Initialized (stop I))) +* (stop (I ';' J)) by FUNCT_4:26, SCMPDS_4:34;
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan (s +* (Initialized (stop I))) implies Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),$1, Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),$1 equal_outside NAT );
assume I is_halting_on s ; :: thesis:
then A3: ProgramPart (s +* (Initialized (stop I))) halts_on s +* (Initialized (stop I)) by Def3;
A4: ( Initialized (stop I) = (stop I) +* (Start-At 0 ,SCMPDS ) & Initialized (stop I) c= s +* (Initialized (stop I)) ) by FUNCT_4:26, SCMPDS_4:def 2;
A5: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

set JS = J ';' (Stop SCMPDS );
set S1 = s +* (Initialized (stop I));
set S2 = (s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)));
let m be Element of NAT ; :: thesis:
assume A6: ( m <= LifeSpan (s +* (Initialized (stop I))) implies Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m, Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m equal_outside NAT ) ; :: thesis:
A7: stop (I ';' J) c= Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m by A2, AMI_1:81, FUNCT_4:26;
T: ProgramPart (s +* (Initialized (stop I))) = ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) by AMI_1:144;
A8: Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(m + 1) = Following (ProgramPart (s +* (Initialized (stop I)))),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) by T ;
A9: stop (I ';' J) = (I ';' J) ';' (Stop SCMPDS ) by SCMPDS_4:def 7
.= I ';' (J ';' (Stop SCMPDS )) by SCMPDS_4:46 ;
dom (I ';' (J ';' (Stop SCMPDS ))) = dom (I +* (Shift (J ';' (Stop SCMPDS )),(card I))) by SCMPDS_4:def 3
.= (dom I) \/ (dom (Shift (J ';' (Stop SCMPDS )),(card I))) by FUNCT_4:def 1 ;
then A10: dom I c= dom (I ';' (J ';' (Stop SCMPDS ))) by XBOOLE_1:7;
T: ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))) = ProgramPart (Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m) by AMI_1:144;
A11: Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),(m + 1) = Following (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),(Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m)),(Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m)),(Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m) by T ;
A12: IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) in dom (stop I) by A1, Def2;
Y: (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) /. (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) = (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by AMI_1:150;
dom (stop I) misses dom (Start-At 0 ,SCMPDS ) by SCMPDS_4:54;
then stop I c= (stop I) +* (Start-At 0 ,SCMPDS ) by FUNCT_4:33;
then stop I c= s +* (Initialized (stop I)) by A4, XBOOLE_1:1;
then stop I c= Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m by AMI_1:81;
then A13: CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) = (stop I) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by A12, GRFUNC_1:8, Y;
assume A14: m + 1 <= LifeSpan (s +* (Initialized (stop I))) ; :: thesis:
then m < LifeSpan (s +* (Initialized (stop I))) by NAT_1:13;
then (stop I) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) <> halt SCMPDS by A3, A13, AMI_1:def 46;
then A15: IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) in dom I by A12, SCMPDS_5:3;
Y: (ProgramPart (Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m)) /. (IC (Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m)) = (Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m) . (IC (Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m)) by AMI_1:150;
CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) = (I ';' (Stop SCMPDS )) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by A13, SCMPDS_4:def 7
.= I . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by A15, SCMPDS_4:37
.= (stop (I ';' J)) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by A15, A9, SCMPDS_4:37
.= (Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by A7, A15, A9, A10, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m)),(Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),m) by A6, A14, AMI_1:121, NAT_1:13, Y ;
hence Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(m + 1), Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),(m + 1) equal_outside NAT by A6, A14, A8, A11, NAT_1:13, SCMPDS_4:15; :: thesis:

end;

( Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),0 = s +* (Initialized (stop I)) & Comput (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))))),((s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J)))),0 = (s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))) ) by AMI_1:13;
then A16: S₁[ 0 ] by A2, AMI_1:120;
A17: for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A16, A5);
A18: (s +* (Initialized (stop I))) +* (Initialized (stop (I ';' J))) = s +* ((Initialized (stop I)) +* (Initialized (stop (I ';' J)))) by FUNCT_4:15
.= s +* (Initialized (stop (I ';' J))) by SCMPDS_5:17 ;
hence for k being Element of NAT st k <= LifeSpan (s +* (Initialized (stop I))) holds
IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) = IC (Comput (ProgramPart (s +* (Initialized (stop (I ';' J))))),(s +* (Initialized (stop (I ';' J)))),k) by A17, AMI_1:121; :: thesis:
thus DataPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))))) = DataPart (Comput (ProgramPart (s +* (Initialized (stop (I ';' J))))),(s +* (Initialized (stop (I ';' J)))),(LifeSpan (s +* (Initialized (stop I))))) by A18, A17, SCMPDS_4:24; :: thesis:
set m = LifeSpan (s +* (Initialized (stop I)));