let s be State of SCMPDS ; :: thesis: for I being Program of SCMPDS
for a being Int_position
for i being Integer
for n being Element of NAT st s . (DataLoc (s . a),i) >= 0 holds
( for-up a,i,n,I is_closed_on s & for-up a,i,n,I is_halting_on s )
let I be Program of SCMPDS ; :: thesis: for a being Int_position
for i being Integer
for n being Element of NAT st s . (DataLoc (s . a),i) >= 0 holds
( for-up a,i,n,I is_closed_on s & for-up a,i,n,I is_halting_on s )
let a be Int_position ; :: thesis: for i being Integer
for n being Element of NAT st s . (DataLoc (s . a),i) >= 0 holds
( for-up a,i,n,I is_closed_on s & for-up a,i,n,I is_halting_on s )
let i be Integer; :: thesis: for n being Element of NAT st s . (DataLoc (s . a),i) >= 0 holds
( for-up a,i,n,I is_closed_on s & for-up a,i,n,I is_halting_on s )
let n be Element of NAT ; :: thesis: ( s . (DataLoc (s . a),i) >= 0 implies ( for-up a,i,n,I is_closed_on s & for-up a,i,n,I is_halting_on s ) )
set d1 = DataLoc (s . a),i;
assume A1: s . (DataLoc (s . a),i) >= 0 ; :: thesis: ( for-up a,i,n,I is_closed_on s & for-up a,i,n,I is_halting_on s )
set i1 = a,i >=0_goto ((card I) + 3);
set i2 = AddTo a,i,n;
set i3 = goto (- ((card I) + 2));
set FOR = for-up a,i,n,I;
set pFOR = stop (for-up a,i,n,I);
set s3 = (Initialize s) +* (stop (for-up a,i,n,I));
set s4 = Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1;
I1: s +* (Initialize (stop (for-up a,i,n,I))) = (Initialize s) +* (stop (for-up a,i,n,I)) by SCMPDS_4:5;
A2: IC ((Initialize s) +* (stop (for-up a,i,n,I))) = 0 by SCMPDS_6:21;
A3: not DataLoc (s . a),i in dom (Initialize (stop (for-up a,i,n,I))) by SCMPDS_4:31;
not a in dom (Initialize (stop (for-up a,i,n,I))) by SCMPDS_4:31;
then A4: ((Initialize s) +* (stop (for-up a,i,n,I))) . (DataLoc (((Initialize s) +* (stop (for-up a,i,n,I))) . a),i) = ((Initialize s) +* (stop (for-up a,i,n,I))) . (DataLoc (s . a),i) by I1, FUNCT_4:12
.= s . (DataLoc (s . a),i) by A3, I1, FUNCT_4:12 ;
A5: for-up a,i,n,I = (a,i >=0_goto ((card I) + 3)) ';' ((I ';' (AddTo a,i,n)) ';' (goto (- ((card I) + 2)))) by Th15;
Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),(0 + 1) = Following (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),0 ) by AMI_1:14
.= Following (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))) by AMI_1:13
.= Exec (a,i >=0_goto ((card I) + 3)),((Initialize s) +* (stop (for-up a,i,n,I))) by A5, I1, SCMPDS_6:22 ;
then A6: IC (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1) = ICplusConst ((Initialize s) +* (stop (for-up a,i,n,I))),((card I) + 3) by A1, A4, SCMPDS_2:69
.= 0 + ((card I) + 3) by A2, SCMPDS_6:23 ;
A7: card (for-up a,i,n,I) = (card I) + 3 by Th51;
then A8: (card I) + 3 in dom (stop (for-up a,i,n,I)) by SCMPDS_6:25;
Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1)) = (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1) . (IC (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1)) by COMPOS_1:38;
Initialize (stop (for-up a,i,n,I)) c= (Initialize s) +* (stop (for-up a,i,n,I)) by I1, FUNCT_4:26;
then stop (for-up a,i,n,I) c= Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1 by AMI_1:81, SCMPDS_4:57;
then (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1) . ((card I) + 3) = (stop (for-up a,i,n,I)) . ((card I) + 3) by A8, GRFUNC_1:8
.= halt SCMPDS by A7, SCMPDS_6:25 ;
then A9: CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1)),(Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1) = halt SCMPDS by A6, Y;
TX: ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),1) by AMI_1:123;

now

let k be Element of NAT ; :: thesis: IC (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),b₁) in dom (stop (for-up a,i,n,I))

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

suppose 0 < k ; :: thesis: IC (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),b₁) in dom (stop (for-up a,i,n,I))

then 1 + 0 <= k by INT_1:20;
hence IC (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),k) in dom (stop (for-up a,i,n,I)) by A8, A6, A9, TX, AMI_1:52; :: thesis: verum

end;

suppose k = 0 ; :: thesis: IC (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),b₁) in dom (stop (for-up a,i,n,I))

then Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),k = (Initialize s) +* (stop (for-up a,i,n,I)) by AMI_1:13;
hence IC (Comput (ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I)))),((Initialize s) +* (stop (for-up a,i,n,I))),k) in dom (stop (for-up a,i,n,I)) by A2, SCMPDS_4:75; :: thesis: verum

end;

end;

end;

hence for-up a,i,n,I is_closed_on s by SCMPDS_6:def 2; :: thesis: for-up a,i,n,I is_halting_on s
ProgramPart ((Initialize s) +* (stop (for-up a,i,n,I))) halts_on (Initialize s) +* (stop (for-up a,i,n,I)) by A9, TX, AMI_1:146;
hence for-up a,i,n,I is_halting_on s by SCMPDS_6:def 3; :: thesis: verum