let s be State of SCMPDS ; :: thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc (s . a),k1) >= 0 holds
( if<0 a,k1,I is_closed_on s & if<0 a,k1,I is_halting_on s )
let I be Program of SCMPDS ; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc (s . a),k1) >= 0 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 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 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;
assume A1: s . (DataLoc (s . a),k1) >= 0 ; :: thesis: ( if<0 a,k1,I is_closed_on s & if<0 a,k1,I is_halting_on s )
set i = a,k1 >=0_goto ((card I) + 1);
set IF = if<0 a,k1,I;
set pIF = stop (if<0 a,k1,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;
I2: s +* (Initialize (stop (if<0 a,k1,I))) = (Initialize s) +* (stop (if<0 a,k1,I)) by SCMPDS_4:5;
A2: IC ((Initialize s) +* (stop (if<0 a,k1,I))) = 0 by I2, FUNCT_4:26, SCMPDS_5:18;
A3: 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 A4: ((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 A3, I2, FUNCT_4:12 ;
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 ((card I) + 1)),((Initialize s) +* (stop (if<0 a,k1,I))) by Th22, I2 ;
then A5: 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))),((card I) + 1) by A1, A4, SCMPDS_2:69
.= 0 + ((card I) + 1) by A2, Th23 ;
A6: card (if<0 a,k1,I) = (card I) + 1 by Th15;
then A7: (card I) + 1 in dom (stop (if<0 a,k1,I)) by Th25;
Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I)))),((Initialize s) +* (stop (if<0 a,k1,I))),1)) /. (IC (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) . (IC (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I)))),((Initialize s) +* (stop (if<0 a,k1,I))),1)) by COMPOS_1:38;
TX: 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;
Initialize (stop (if<0 a,k1,I)) c= (Initialize s) +* (stop (if<0 a,k1,I)) by I2, FUNCT_4:26;
then stop (if<0 a,k1,I) c= Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I)))),((Initialize s) +* (stop (if<0 a,k1,I))),1 by AMI_1:81, SCMPDS_4:57;
then (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I)))),((Initialize s) +* (stop (if<0 a,k1,I))),1) . ((card I) + 1) = (stop (if<0 a,k1,I)) . ((card I) + 1) by A7, GRFUNC_1:8
.= halt SCMPDS by A6, Th25 ;
then A8: 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))),1) = halt SCMPDS by A5, Y, TX;

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 1 + 0 <= k by INT_1:20;
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 A7, A5, A8, AMI_1:52; :: 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))

then Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I)))),((Initialize s) +* (stop (if<0 a,k1,I))),k = (Initialize s) +* (stop (if<0 a,k1,I)) by AMI_1:13;
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 A2, SCMPDS_4:75; :: 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
ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I))) halts_on (Initialize s) +* (stop (if<0 a,k1,I)) by A8, AMI_1:146;
hence if<0 a,k1,I is_halting_on s by Def3; :: thesis: verum