let s be State of SCMPDS; :: thesis: for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s holds
( if<=0 (a,k1,I) is_closed_on s & if<=0 (a,k1,I) is_halting_on s )
let I be shiftable Program of SCMPDS; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s holds
( if<=0 (a,k1,I) is_closed_on s & if<=0 (a,k1,I) is_halting_on s )
let a be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s holds
( if<=0 (a,k1,I) is_closed_on s & if<=0 (a,k1,I) is_halting_on s )
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s implies ( if<=0 (a,k1,I) is_closed_on s & if<=0 (a,k1,I) is_halting_on s ) )
set b = DataLoc ((s . a),k1);
set IF = if<=0 (a,k1,I);
set pIF = stop (if<=0 (a,k1,I));
set pI = stop I;
set s2 = (Initialize s) +* (stop I);
set s3 = (Initialize s) +* (stop (if<=0 (a,k1,I)));
set s4 = Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1);
set i = (a,k1) <=0_goto 2;
set j = goto ((card I) + 1);
A1: s +* (Initialize (stop I)) = (Initialize s) +* (stop I) by COMPOS_1:125;
A2: s +* (Initialize (stop (if<=0 (a,k1,I)))) = (Initialize s) +* (stop (if<=0 (a,k1,I))) by COMPOS_1:125;
A3: if<=0 (a,k1,I) = ((a,k1) <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:52;
A4: Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),(0 + 1)) = Following ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),0))) by EXTPRO_1:4
.= Following ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I))))) by EXTPRO_1:3
.= Exec (((a,k1) <=0_goto 2),((Initialize s) +* (stop (if<=0 (a,k1,I))))) by A3, Th22, A2 ;
A5: DataPart ((Initialize s) +* (stop I)) = DataPart ((Initialize s) +* (stop (if<=0 (a,k1,I)))) by COMPOS_1:138, FUNCT_7:134;

now

let a be Int_position ; :: thesis: ((Initialize s) +* (stop I)) . a = (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) . a
thus ((Initialize s) +* (stop I)) . a = ((Initialize s) +* (stop (if<=0 (a,k1,I)))) . a by A5, SCMPDS_4:23
.= (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) . a by A4, SCMPDS_2:68 ; :: thesis: verum

end;

then A6: DataPart ((Initialize s) +* (stop I)) = DataPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) by SCMPDS_4:23;
A7: not DataLoc ((s . a),k1) in dom (Initialize (stop (if<=0 (a,k1,I)))) by SCMPDS_4:31;
not a in dom (Initialize (stop (if<=0 (a,k1,I)))) by SCMPDS_4:31;
then A8: ((Initialize s) +* (stop (if<=0 (a,k1,I)))) . (DataLoc ((((Initialize s) +* (stop (if<=0 (a,k1,I)))) . a),k1)) = ((Initialize s) +* (stop (if<=0 (a,k1,I)))) . (DataLoc ((s . a),k1)) by A2, FUNCT_4:12
.= s . (DataLoc ((s . a),k1)) by A7, A2, FUNCT_4:12 ;
A9: IC ((Initialize s) +* (stop (if<=0 (a,k1,I)))) = 0 by A2, FUNCT_4:26, SCMPDS_5:18;
assume s . (DataLoc ((s . a),k1)) <= 0 ; :: thesis: ( not I is_closed_on s or not I is_halting_on s or ( if<=0 (a,k1,I) is_closed_on s & if<=0 (a,k1,I) is_halting_on s ) )
then A10: IC (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) = ICplusConst (((Initialize s) +* (stop (if<=0 (a,k1,I)))),2) by A4, A8, SCMPDS_2:68
.= 0 + 2 by A9, Th23 ;
assume A11: I is_closed_on s ; :: thesis: ( not I is_halting_on s or ( if<=0 (a,k1,I) is_closed_on s & if<=0 (a,k1,I) is_halting_on s ) )
then A12: I is_closed_on (Initialize s) +* (stop I) by Th38;
assume I is_halting_on s ; :: thesis: ( if<=0 (a,k1,I) is_closed_on s & if<=0 (a,k1,I) is_halting_on s )
then A13: ProgramPart ((Initialize s) +* (stop I)) halts_on (Initialize s) +* (stop I) by Def3;
A14: ( Initialize (stop I) c= (Initialize s) +* (stop I) & Shift ((stop I),2) c= Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1) ) by Lm7, A1, FUNCT_4:26;
A15: 0 in dom (stop (if<=0 (a,k1,I))) by COMPOS_1:135;

now

let k be Element of NAT ; :: thesis: IC (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),b₁)) in dom (stop (if<=0 (a,k1,I)))

per cases ( 0 < k or k = 0 ) ;

suppose 0 < k ; :: thesis: IC (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),b₁)) in dom (stop (if<=0 (a,k1,I)))

then consider k1 being Nat such that
A16: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def 13;
reconsider m = IC (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k1)) as Element of NAT ;
A17: card (stop (if<=0 (a,k1,I))) = 1 + (card (if<=0 (a,k1,I))) by SCMPDS_5:7
.= 1 + ((card I) + 2) by Lm8
.= (1 + (card I)) + 2
.= (card (stop I)) + 2 by SCMPDS_5:7 ;
m in dom (stop I) by A11, Def2;
then m < card (stop I) by AFINSQ_1:70;
then A18: m + 2 < card (stop (if<=0 (a,k1,I))) by A17, XREAL_1:8;
A19: ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I)))) = ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) by AMI_1:123;
IC (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),k)) = IC (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)),k1)) by A16, EXTPRO_1:5
.= m + 2 by A12, A14, A10, A6, Th45, A19 ;
hence IC (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),k)) in dom (stop (if<=0 (a,k1,I))) by A18, AFINSQ_1:70; :: thesis: verum

end;

suppose k = 0 ; :: thesis: IC (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),b₁)) in dom (stop (if<=0 (a,k1,I)))

hence IC (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),k)) in dom (stop (if<=0 (a,k1,I))) by A15, A9, EXTPRO_1:3; :: thesis: verum

end;

end;

end;

hence if<=0 (a,k1,I) is_closed_on s by Def2; :: thesis: if<=0 (a,k1,I) is_halting_on s
A20: ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I)))) = ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) by AMI_1:123;
A21: Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),((LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1)) = Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))) by EXTPRO_1:5;
A22: ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) = ProgramPart (Comput ((ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1))),(Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by AMI_1:123;
A23: ProgramPart ((Initialize s) +* (stop I)) = ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by AMI_1:123;
CurInstr ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),((LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1)))) = CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1))),(Comput ((ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1))),(Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))))) by A21, A20
.= CurInstr ((ProgramPart ((Initialize s) +* (stop I))),(Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))))) by A12, A14, A10, A6, Th45, A23, A22
.= halt SCMPDS by A13, EXTPRO_1:def 14 ;
then ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I)))) halts_on (Initialize s) +* (stop (if<=0 (a,k1,I))) by EXTPRO_1:30;
hence if<=0 (a,k1,I) is_halting_on s by Def3; :: thesis: verum