let s be State of SCMPDS; :: thesis:
let I be halt-free shiftable Program of SCMPDS; :: thesis:
let a be Int_position ; :: thesis:
let i be Integer; :: thesis:
let X be set ; :: thesis:
set b = DataLoc ((s . a),i);
set WHL = while<0 (a,i,I);
set pWHL = stop (while<0 (a,i,I));
set pI = stop I;
set i1 = (a,i) >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
assume A1: card I > 0 ; :: thesis:
defpred S₁[ Element of NAT ] means for t being State of SCMPDS st - (t . (DataLoc ((s . a),i))) <= $1 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a holds
( while<0 (a,i,I) is_closed_on t & while<0 (a,i,I) is_halting_on t );
assume A2: for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec (I,t)) . x = t . x ) ) ; :: thesis:
A3: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis:
assume A4: S₁[k] ; :: thesis:

now

let t be State of SCMPDS; :: thesis:
assume A5: - (t . (DataLoc ((s . a),i))) <= k + 1 ; :: thesis:
assume A6: for x being Int_position st x in X holds
t . x = s . x ; :: thesis:
assume A7: t . a = s . a ; :: thesis:

per cases ;

suppose t . (DataLoc ((s . a),i)) >= 0 ; :: thesis:

hence ( while<0 (a,i,I) is_closed_on t & while<0 (a,i,I) is_halting_on t ) by A7, Th9; :: thesis:

end;

suppose A8: t . (DataLoc ((s . a),i)) < 0 ; :: thesis:

now

let a be Int_position ; :: thesis:
thus ((Initialize t) +* (stop I)) . a = ((Initialize t) +* (stop (while<0 (a,i,I)))) . a by A19, SCMPDS_4:23
.= (Comput ((ProgramPart ((Initialize t) +* (stop (while<0 (a,i,I))))),((Initialize t) +* (stop (while<0 (a,i,I)))),1)) . a by A18, SCMPDS_2:69 ; :: thesis:

end;

A38: now

end;

A42: now

end;

now

let k be Element of NAT ; :: thesis:

per cases ;

suppose k < ((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1 ; :: thesis:

then A46: k <= (LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1 by INT_1:20;

hereby :: thesis:

per cases by A46, NAT_1:8;

suppose A47: k <= LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) ; :: thesis:

hereby :: thesis:

per cases ;

suppose k = 0 ; :: thesis:

hence IC (Comput ((ProgramPart ((Initialize t) +* (stop (while<0 (a,i,I))))),((Initialize t) +* (stop (while<0 (a,i,I)))),k)) in dom (stop (while<0 (a,i,I))) by A13, A23, EXTPRO_1:3; :: thesis:

end;

suppose k <> 0 ; :: thesis:

end;

suppose A51: k = (LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1 ; :: thesis:

(card I) + 1 in dom (stop (while<0 (a,i,I))) by A24, SCMPDS_6:18;
hence IC (Comput ((ProgramPart ((Initialize t) +* (stop (while<0 (a,i,I))))),((Initialize t) +* (stop (while<0 (a,i,I)))),k)) in dom (stop (while<0 (a,i,I))) by A1, A17, A29, A27, A30, A20, A22, A32, A51, SCMPDS_7:36; :: thesis:

end;

suppose k >= ((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1 ; :: thesis:

end;

end;

hence S₁[k + 1] ; :: thesis:

end;

A53: S₁[ 0 ]

proof

let t be State of SCMPDS; :: thesis:
assume - (t . (DataLoc ((s . a),i))) <= 0 ; :: thesis:
then - (t . (DataLoc ((s . a),i))) <= - 0 ;
then A54: t . (DataLoc ((s . a),i)) >= 0 by XREAL_1:26;
assume for x being Int_position st x in X holds
t . x = s . x ; :: thesis:
assume t . a = s . a ; :: thesis:
hence ( while<0 (a,i,I) is_closed_on t & while<0 (a,i,I) is_halting_on t ) by A54, Th9; :: thesis:

end;

A55: for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A53, A3);

per cases ;

suppose s . (DataLoc ((s . a),i)) >= 0 ; :: thesis:

hence ( while<0 (a,i,I) is_closed_on s & while<0 (a,i,I) is_halting_on s ) by Th9; :: thesis:

end;

suppose s . (DataLoc ((s . a),i)) < 0 ; :: thesis:

then reconsider n = - (s . (DataLoc ((s . a),i))) as Element of NAT by INT_1:16;
( S₁[n] & ( for x being Int_position st x in X holds
s . x = s . x ) ) by A55;
hence ( while<0 (a,i,I) is_closed_on s & while<0 (a,i,I) is_halting_on s ) ; :: thesis:

end;