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);
I1: s +* (Initialize (stop I)) = (Initialize s) +* (stop I) by SCMPDS_4:5;
I2: s +* (Initialize (stop (if<=0 a,k1,I))) = (Initialize s) +* (stop (if<=0 a,k1,I)) by SCMPDS_4:5;
A1: if<=0 a,k1,I = (a,k1 <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:52;
A2: 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 AMI_1:14
.= Following (ProgramPart ((Initialize s) +* (stop (if<=0 a,k1,I)))),((Initialize s) +* (stop (if<=0 a,k1,I))) by AMI_1:13
.= Exec (a,k1 <=0_goto 2),((Initialize s) +* (stop (if<=0 a,k1,I))) by A1, Th22, I2 ;
A3: DataPart ((Initialize s) +* (stop I)) = DataPart ((Initialize s) +* (stop (if<=0 a,k1,I))) by FUNCT_7:134, SCMPDS_4:24;

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 A3, SCMPDS_4:23
.= (Comput (ProgramPart ((Initialize s) +* (stop (if<=0 a,k1,I)))),((Initialize s) +* (stop (if<=0 a,k1,I))),1) . a by A2, SCMPDS_2:68 ; :: thesis: verum

end;

then A4: 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;
A5: 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 A6: ((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 I2, FUNCT_4:12
.= s . (DataLoc (s . a),k1) by A5, I2, FUNCT_4:12 ;
A7: IC ((Initialize s) +* (stop (if<=0 a,k1,I))) = 0 by I2, 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 A8: 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 A2, A6, SCMPDS_2:68
.= 0 + 2 by A7, Th23 ;
assume A9: 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 A10: 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 A11: ProgramPart ((Initialize s) +* (stop I)) halts_on (Initialize s) +* (stop I) by Def3;
A12: ( 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 Lm6, I1, FUNCT_4:26;
A13: 0 in dom (stop (if<=0 a,k1,I)) by SCMPDS_4:75;

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
A14: 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 ;
A15: 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 A9, Def2;
then m < card (stop I) by AFINSQ_1:70;
then A16: m + 2 < card (stop (if<=0 a,k1,I)) by A15, XREAL_1:8;
T: 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 A14, AMI_1:51
.= m + 2 by A10, A12, A8, A4, Th45, T ;
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 A16, 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 A13, A7, AMI_1:13; :: 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
T: 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;
x: 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 AMI_1:51;
TX4: 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;
TX2: 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 x, T
.= 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 A10, A12, A8, A4, Th45, TX2, TX4
.= halt SCMPDS by A11, AMI_1:def 46 ;
then ProgramPart ((Initialize s) +* (stop (if<=0 a,k1,I))) halts_on (Initialize s) +* (stop (if<=0 a,k1,I)) by AMI_1:146;
hence if<=0 a,k1,I is_halting_on s by Def3; :: thesis: verum