set D = SCM-Data-Loc ;
let I be halt-free parahalting Program of SCMPDS ; :: thesis: for J being parahalting shiftable Program of SCMPDS
for s, s1 being State of SCMPDS st Initialize (stop (I ';' J)) c= s & s1 = (Initialize s) +* (stop I) holds
( IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = card I & DataPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = DataPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) & Shift (stop J),(card I) c= Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1) & LifeSpan (ProgramPart s),s = (LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Result (ProgramPart s1),s1)) +* (stop J))),((Initialize (Result (ProgramPart s1),s1)) +* (stop J))) )
let J be parahalting shiftable Program of SCMPDS ; :: thesis: for s, s1 being State of SCMPDS st Initialize (stop (I ';' J)) c= s & s1 = (Initialize s) +* (stop I) holds
( IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = card I & DataPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = DataPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) & Shift (stop J),(card I) c= Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1) & LifeSpan (ProgramPart s),s = (LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Result (ProgramPart s1),s1)) +* (stop J))),((Initialize (Result (ProgramPart s1),s1)) +* (stop J))) )
let s, s1 be State of SCMPDS ; :: thesis: ( Initialize (stop (I ';' J)) c= s & s1 = (Initialize s) +* (stop I) implies ( IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = card I & DataPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = DataPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) & Shift (stop J),(card I) c= Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1) & LifeSpan (ProgramPart s),s = (LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Result (ProgramPart s1),s1)) +* (stop J))),((Initialize (Result (ProgramPart s1),s1)) +* (stop J))) ) )
set spJ = stop J;
set sIJ = stop (I ';' J);
set m1 = LifeSpan (ProgramPart s1),s1;
set s3 = (Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J);
I1: (Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J) = (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1)) +* (Initialize (stop J)) by SCMPDS_4:5;
set m3 = LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J));
assume that
A1: Initialize (stop (I ';' J)) c= s and
A2: s1 = (Initialize s) +* (stop I) ; :: thesis: ( IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = card I & DataPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = DataPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) & Shift (stop J),(card I) c= Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1) & LifeSpan (ProgramPart s),s = (LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Result (ProgramPart s1),s1)) +* (stop J))),((Initialize (Result (ProgramPart s1),s1)) +* (stop J))) )
I2: (Initialize s) +* (stop I) = s +* (Initialize (stop I)) by SCMPDS_4:5;
A3: DataPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) = (DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (DataPart (Initialize (stop J))) by I1, FUNCT_4:75;
set s4 = Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1);
A4: Initialize I c= Initialize (stop (I ';' J)) by Th15;
hence A5: IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = card I by A1, A2, Th30, XBOOLE_1:1; :: thesis: ( DataPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = DataPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) & Shift (stop J),(card I) c= Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1) & LifeSpan (ProgramPart s),s = (LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Result (ProgramPart s1),s1)) +* (stop J))),((Initialize (Result (ProgramPart s1),s1)) +* (stop J))) )

A6: now

let x be set ; :: thesis: ( x in dom (DataPart (Initialize (stop J))) implies (DataPart (Initialize (stop J))) . b₁ = (DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) . b₁ )
assume x in dom (DataPart (Initialize (stop J))) ; :: thesis: (DataPart (Initialize (stop J))) . b₁ = (DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) . b₁
then A7: x in (dom (Initialize (stop J))) /\ SCM-Data-Loc by RELAT_1:90, SCMPDS_2:100;
then A8: x in dom (Initialize (stop J)) by XBOOLE_0:def 4;
A9: x in SCM-Data-Loc by A7, XBOOLE_0:def 4;

per cases ( x in dom (stop J) or x = IC SCMPDS ) by A8, COMPOS_1:77;

suppose A10: x in dom (stop J) ; :: thesis: (DataPart (Initialize (stop J))) . b₁ = (DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) . b₁

dom (stop J) c= NAT by RELAT_1:def 18;
then reconsider l = x as Element of NAT by A10;
(DataPart (Initialize (stop J))) . l = (DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) . l by A9, SCMPDS_4:22;
hence (DataPart (Initialize (stop J))) . x = (DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) . x ; :: thesis: verum

end;

suppose x = IC SCMPDS ; :: thesis: (DataPart (Initialize (stop J))) . b₁ = (DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) . b₁

hence (DataPart (Initialize (stop J))) . x = (DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) . x by A7, SCMPDS_3:6, XBOOLE_0:def 4; :: thesis: verum

end;

end;

end;

A11: dom (DataPart (Initialize (stop J))) = (dom (Initialize (stop J))) /\ SCM-Data-Loc by RELAT_1:90, SCMPDS_2:100;
reconsider m = (LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))) as Element of NAT ;
stop (I ';' J) = I ';' (J ';' (Stop SCMPDS )) by SCMPDS_4:46
.= I +* (Shift (stop J),(card I)) ;
then A12: Shift (stop J),(card I) c= stop (I ';' J) by FUNCT_4:26;
A13: Initialize (stop J) c= (Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J) by I1, FUNCT_4:26;
then A14: ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) halts_on (Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J) by AMI_1:def 26;
I3: (Initialize s) +* (stop (I ';' J)) = s +* (Initialize (stop (I ';' J))) by SCMPDS_4:5;
dom (Initialize (stop J)) c= dom ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) by A13, GRFUNC_1:8;
then dom (Initialize (stop J)) c= the carrier of SCMPDS by PARTFUN1:def 4;
then dom (DataPart (Initialize (stop J))) c= the carrier of SCMPDS /\ SCM-Data-Loc by A11, XBOOLE_1:26;
then dom (DataPart (Initialize (stop J))) c= (dom (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) /\ SCM-Data-Loc by PARTFUN1:def 4;
then dom (DataPart (Initialize (stop J))) c= dom (DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) by RELAT_1:90, SCMPDS_2:100;
then DataPart (Initialize (stop J)) c= DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1)) by A6, GRFUNC_1:8;
then A15: DataPart (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1)) = DataPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) by A3, LATTICE2:8;
s = (Initialize s) +* (stop (I ';' J)) by A1, I3, FUNCT_4:79
.= s +* ((I ';' (J ';' (Stop SCMPDS ))) +* (Start-At 0 ,SCMPDS )) by I3, SCMPDS_4:46 ;
hence A16: DataPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = DataPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) by A2, A15, Th33, SCMPDS_4:24; :: thesis: ( Shift (stop J),(card I) c= Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1) & LifeSpan (ProgramPart s),s = (LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Result (ProgramPart s1),s1)) +* (stop J))),((Initialize (Result (ProgramPart s1),s1)) +* (stop J))) )
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) by AMI_1:123;
x: Comput (ProgramPart s),s,((LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)))) = Comput (ProgramPart s),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)),(LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))) by AMI_1:51;
TX3: ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) = ProgramPart (Comput (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)),(LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)))) by AMI_1:123;
TX4: ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = ProgramPart (Comput (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1))),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)),(LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)))) by AMI_1:123;
stop (I ';' J) c= s by A1, SCMPDS_4:57;
then Shift (stop J),(card I) c= s by A12, XBOOLE_1:1;
hence A17: Shift (stop J),(card I) c= Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1) by AMI_1:81; :: thesis: LifeSpan (ProgramPart s),s = (LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Result (ProgramPart s1),s1)) +* (stop J))),((Initialize (Result (ProgramPart s1),s1)) +* (stop J)))
then CurInstr (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),(Comput (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)),(LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)))) = CurInstr (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1))),(Comput (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1))),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)),(LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)))) by A13, A5, A16, TX4, TX3, SCMPDS_4:84;
then CurInstr (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),(Comput (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)),(LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)))) = CurInstr (ProgramPart s),(Comput (ProgramPart s),s,((LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))))) by x, T;
then A18: CurInstr (ProgramPart s),(Comput (ProgramPart s),s,m) = halt SCMPDS by A14, AMI_1:def 46;

A19: now

let k be Element of NAT ; :: thesis: ( (LifeSpan (ProgramPart s1),s1) + k < m implies not CurInstr (ProgramPart s),(Comput (ProgramPart s),s,((LifeSpan (ProgramPart s1),s1) + k)) = halt SCMPDS )
assume (LifeSpan (ProgramPart s1),s1) + k < m ; :: thesis: not CurInstr (ProgramPart s),(Comput (ProgramPart s),s,((LifeSpan (ProgramPart s1),s1) + k)) = halt SCMPDS
then A20: k < LifeSpan (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) by XREAL_1:8;
assume A21: CurInstr (ProgramPart s),(Comput (ProgramPart s),s,((LifeSpan (ProgramPart s1),s1) + k)) = halt SCMPDS ; :: thesis: contradiction
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) by AMI_1:123;
TX3: ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)) = ProgramPart (Comput (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)),k) by AMI_1:123;
TX4: ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)) = ProgramPart (Comput (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1))),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)),k) by AMI_1:123;
x: Comput (ProgramPart s),s,((LifeSpan (ProgramPart s1),s1) + k) = Comput (ProgramPart s),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)),k by AMI_1:51;
CurInstr (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),(Comput (ProgramPart ((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J))),((Initialize (Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1))) +* (stop J)),k) = CurInstr (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1))),(Comput (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1))),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart s1),s1)),k) by A13, A5, A16, A17, TX4, TX3, SCMPDS_4:84
.= halt SCMPDS by A21, x, T ;
hence contradiction by A14, A20, AMI_1:def 46; :: thesis: verum

end;

now

let k be Element of NAT ; :: thesis: ( k < m implies CurInstr (ProgramPart s),(Comput (ProgramPart s),s,b₁) <> halt SCMPDS )
assume A22: k < m ; :: thesis: CurInstr (ProgramPart s),(Comput (ProgramPart s),s,b₁) <> halt SCMPDS
TX: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,k) by AMI_1:123;

per cases ( k < LifeSpan (ProgramPart s1),s1 or LifeSpan (ProgramPart s1),s1 <= k ) ;

suppose k < LifeSpan (ProgramPart s1),s1 ; :: thesis: CurInstr (ProgramPart s),(Comput (ProgramPart s),s,b₁) <> halt SCMPDS

hence CurInstr (ProgramPart s),(Comput (ProgramPart s),s,k) <> halt SCMPDS by A1, A2, A4, Th32, TX, XBOOLE_1:1; :: thesis: verum

end;

suppose LifeSpan (ProgramPart s1),s1 <= k ; :: thesis: CurInstr (ProgramPart s),(Comput (ProgramPart s),s,b₁) <> halt SCMPDS

then consider kk being Nat such that
A23: (LifeSpan (ProgramPart s1),s1) + kk = k by NAT_1:10;
kk in NAT by ORDINAL1:def 13;
hence CurInstr (ProgramPart s),(Comput (ProgramPart s),s,k) <> halt SCMPDS by A19, A22, A23; :: thesis: verum

end;

end;

end;

then A24: for k being Element of NAT st CurInstr (ProgramPart s),(Comput (ProgramPart s),s,k) = halt SCMPDS holds
m <= k ;
Initialize (stop I) c= s1 by A2, I2, FUNCT_4:26;
then ProgramPart s1 halts_on s1 by SCMPDS_4:63;
then X: Result (ProgramPart s1),s1 = Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1) by AMI_1:122;
ProgramPart s halts_on s by A1, SCMPDS_4:63;
then LifeSpan (ProgramPart s),s = m by A18, A24, AMI_1:def 46;
hence LifeSpan (ProgramPart s),s = (LifeSpan (ProgramPart s1),s1) + (LifeSpan (ProgramPart ((Initialize (Result (ProgramPart s1),s1)) +* (stop J))),((Initialize (Result (ProgramPart s1),s1)) +* (stop J))) by X; :: thesis: verum