let s be State of SCMPDS ; :: thesis: for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc (s . a),i) = 0 holds
( while<>0 a,i,I is_closed_on s & while<>0 a,i,I is_halting_on s )
let I be Program of SCMPDS ; :: thesis: for a being Int_position
for i being Integer st s . (DataLoc (s . a),i) = 0 holds
( while<>0 a,i,I is_closed_on s & while<>0 a,i,I is_halting_on s )
let a be Int_position ; :: thesis: for i being Integer st s . (DataLoc (s . a),i) = 0 holds
( while<>0 a,i,I is_closed_on s & while<>0 a,i,I is_halting_on s )
let i be Integer; :: thesis: ( s . (DataLoc (s . a),i) = 0 implies ( while<>0 a,i,I is_closed_on s & while<>0 a,i,I is_halting_on s ) )
set b = DataLoc (s . a),i;
assume A1: s . (DataLoc (s . a),i) = 0 ; :: thesis: ( while<>0 a,i,I is_closed_on s & while<>0 a,i,I is_halting_on s )
set WHL = while<>0 a,i,I;
set pWH = stop (while<>0 a,i,I);
set iWH = Initialize (stop (while<>0 a,i,I));
set s3 = (Initialize s) +* (stop (while<>0 a,i,I));
set s4 = Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1;
set s5 = Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2;
I1: (Initialize s) +* (stop (while<>0 a,i,I)) = s +* (Initialize (stop (while<>0 a,i,I))) by SCMPDS_4:5;
A2: not DataLoc (s . a),i in dom (Initialize (stop (while<>0 a,i,I))) by SCMPDS_4:31;
not a in dom (Initialize (stop (while<>0 a,i,I))) by SCMPDS_4:31;
then A3: ((Initialize s) +* (stop (while<>0 a,i,I))) . (DataLoc (((Initialize s) +* (stop (while<>0 a,i,I))) . a),i) = ((Initialize s) +* (stop (while<>0 a,i,I))) . (DataLoc (s . a),i) by I1, FUNCT_4:12
.= 0 by A1, A2, I1, FUNCT_4:12 ;
set i1 = a,i <>0_goto 2;
set i2 = goto ((card I) + 2);
set i3 = goto (- ((card I) + 2));
A4: IC ((Initialize s) +* (stop (while<>0 a,i,I))) = 0 by SCMPDS_6:21;
A5: while<>0 a,i,I = (a,i <>0_goto 2) ';' (((goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 2)))) by Lm10;
Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),(0 + 1) = Following (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),0 ) by AMI_1:14
.= Following (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))) by AMI_1:13
.= Exec (a,i <>0_goto 2),((Initialize s) +* (stop (while<>0 a,i,I))) by A5, I1, SCMPDS_6:22 ;
then A6: IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) = succ (IC ((Initialize s) +* (stop (while<>0 a,i,I)))) by A3, SCMPDS_2:67
.= 0 + 1 by A4 ;
A7: Initialize (stop (while<>0 a,i,I)) c= (Initialize s) +* (stop (while<>0 a,i,I)) by I1, FUNCT_4:26;
then A8: stop (while<>0 a,i,I) c= Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1 by AMI_1:81, SCMPDS_4:57;
A9: 1 in dom (while<>0 a,i,I) by Th14;
then 1 in dom (stop (while<>0 a,i,I)) by SCMPDS_6:18;
then A10: (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) . 1 = (stop (while<>0 a,i,I)) . 1 by A8, GRFUNC_1:8
.= (while<>0 a,i,I) . 1 by A9, SCMPDS_6:19
.= goto ((card I) + 2) by Th15 ;
A11: card (while<>0 a,i,I) = (card I) + 3 by Th12;
then A12: (card I) + 3 in dom (stop (while<>0 a,i,I)) by SCMPDS_6:25;
Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1)) = (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) . (IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1)) by COMPOS_1:38;
T: ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) by AMI_1:123;
Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),(1 + 1) = Following (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) by AMI_1:14
.= Exec (goto ((card I) + 2)),(Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) by A6, A10, Y, T ;
then A13: IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) = ICplusConst (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1),((card I) + 2) by SCMPDS_2:66
.= ((card I) + 2) + 1 by A6, SCMPDS_6:23
.= (card I) + (2 + 1) ;
Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2)) = (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) . (IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2)) by COMPOS_1:38;
stop (while<>0 a,i,I) c= Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2 by A7, AMI_1:81, SCMPDS_4:57;
then (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) . ((card I) + 3) = (stop (while<>0 a,i,I)) . ((card I) + 3) by A12, GRFUNC_1:8
.= halt SCMPDS by A11, SCMPDS_6:25 ;
then A14: CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2)),(Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) = halt SCMPDS by A13, Y;
TX: ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) by AMI_1:123;

now

let k be Element of NAT ; :: thesis: IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),b₁) in dom (stop (while<>0 a,i,I))
( k = 0 or 0 < k ) by NAT_1:3;
then A15: ( k = 0 or 0 + 1 <= k ) by INT_1:20;

per cases ( k = 0 or k = 1 or 1 < k ) by A15, XXREAL_0:1;

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

then Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),k = (Initialize s) +* (stop (while<>0 a,i,I)) by AMI_1:13;
hence IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),k) in dom (stop (while<>0 a,i,I)) by A4, SCMPDS_4:75; :: thesis: verum

end;

suppose k = 1 ; :: thesis: IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),b₁) in dom (stop (while<>0 a,i,I))

hence IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),k) in dom (stop (while<>0 a,i,I)) by A9, A6, SCMPDS_6:18; :: thesis: verum

end;

suppose 1 < k ; :: thesis: IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),b₁) in dom (stop (while<>0 a,i,I))

then 1 + 1 <= k by INT_1:20;
hence IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),k) in dom (stop (while<>0 a,i,I)) by A12, A13, A14, TX, AMI_1:52; :: thesis: verum

end;

end;

end;

hence while<>0 a,i,I is_closed_on s by SCMPDS_6:def 2; :: thesis: while<>0 a,i,I is_halting_on s
ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I))) halts_on (Initialize s) +* (stop (while<>0 a,i,I)) by A14, TX, AMI_1:146;
hence while<>0 a,i,I is_halting_on s by SCMPDS_6:def 3; :: thesis: verum