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 COMPOS_1:125;
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 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 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, COMPOS_1:132;
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 EXTPRO_1:4
.= 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, COMPOS_1:132;
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 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 A4, COMPOS_1:135; :: 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, EXTPRO_1:6; :: 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, EXTPRO_1:30;
hence while<>0 (a,i,I) is_halting_on s by SCMPDS_6:def 3; :: thesis: verum