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 d1 = 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 i1 = (a,i) >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set WHL = while<0 (a,i,I);
set pWHL = 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);
I1: s +* (Initialize (stop (while<0 (a,i,I)))) = (Initialize s) +* (stop (while<0 (a,i,I))) by COMPOS_1:125;
A2: IC ((Initialize s) +* (stop (while<0 (a,i,I)))) = 0 by SCMPDS_6:21;
A3: 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 A4: ((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
.= s . (DataLoc ((s . a),i)) by A3, I1, FUNCT_4:12 ;
A5: while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:51;
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 EXTPRO_1:4
.= Following ((ProgramPart ((Initialize s) +* (stop (while<0 (a,i,I))))),((Initialize s) +* (stop (while<0 (a,i,I))))) by EXTPRO_1:3
.= Exec (((a,i) >=0_goto ((card I) + 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)) = ICplusConst (((Initialize s) +* (stop (while<0 (a,i,I)))),((card I) + 2)) by A1, A4, SCMPDS_2:69
.= 0 + ((card I) + 2) by A2, SCMPDS_6:23 ;
A7: card (while<0 (a,i,I)) = (card I) + 2 by Th6;
then A8: (card I) + 2 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;
Initialize (stop (while<0 (a,i,I))) c= (Initialize s) +* (stop (while<0 (a,i,I))) by I1, FUNCT_4:26;
then 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, COMPOS_1:132;
then (Comput ((ProgramPart ((Initialize s) +* (stop (while<0 (a,i,I))))),((Initialize s) +* (stop (while<0 (a,i,I)))),1)) . ((card I) + 2) = (stop (while<0 (a,i,I))) . ((card I) + 2) by A8, GRFUNC_1:8
.= halt SCMPDS by A7, SCMPDS_6:25 ;
then A9: CurInstr ((ProgramPart (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))) = halt SCMPDS by A6, Y;
TX: ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (while<0 (a,i,I))))),((Initialize s) +* (stop (while<0 (a,i,I)))),1)) = ProgramPart ((Initialize s) +* (stop (while<0 (a,i,I)))) 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)))

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

suppose 0 < 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 + 0 <= 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 A8, A6, A9, TX, EXTPRO_1:6; :: thesis: verum

end;

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 EXTPRO_1:3;
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 A2, COMPOS_1:135; :: 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 A9, TX, EXTPRO_1:30;
hence while<0 (a,i,I) is_halting_on s by SCMPDS_6:def 3; :: thesis: verum