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, c be Integer; :: thesis:
let X, Y be set ; :: thesis:
let f be Function of (product the Object-Kind of SCMPDS),NAT; :: 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 f . (Dstate t) <= $1 & ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ; :: thesis:
assume A3: for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ; :: thesis:
assume A4: for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,t)) . a = t . a & I is_closed_on t & I is_halting_on t & f . (Dstate (IExec (I,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,t)) . x >= c + ((IExec (I,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,t)) . x = t . x ) ) ; :: thesis:
A5: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis:
assume A6: S₁[k] ; :: thesis:

now

let t be State of SCMPDS; :: thesis:
assume A7: f . (Dstate t) <= k + 1 ; :: thesis:
assume A8: for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ; :: thesis:
assume A9: for x being Int_position st x in Y holds
t . x = s . x ; :: thesis:
assume A10: 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 A10, Th20; :: thesis:

end;

suppose A11: 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 A21, SCMPDS_4:23
.= (Comput ((ProgramPart ((Initialize t) +* (stop (while>0 (a,i,I))))),((Initialize t) +* (stop (while>0 (a,i,I)))),1)) . a by A20, SCMPDS_2:68 ; :: thesis:

end;

A40: now

end;

A43: now

end;

A46: 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 A51: k <= (LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1 by INT_1:20;

hereby :: thesis:

per cases by A51, NAT_1:8;

suppose A52: 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 A15, A26, EXTPRO_1:3; :: thesis:

end;

suppose k <> 0 ; :: thesis:

end;

suppose A56: 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 A27, 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, A19, A31, A29, A32, A22, A24, A34, A56, 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;

set n = f . (Dstate s);
A59: for x being Int_position st x in Y holds
s . x = s . x ;
A60: S₁[ 0 ]

proof

let t be State of SCMPDS; :: thesis:
assume f . (Dstate t) <= 0 ; :: thesis:
then f . (Dstate t) = 0 ;
then A61: t . (DataLoc ((s . a),i)) <= 0 by A2;
assume for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ; :: thesis:
assume for x being Int_position st x in Y 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 A61, Th20; :: thesis:

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A60, A5);
then S₁[f . (Dstate s)] ;
hence ( while>0 (a,i,I) is_closed_on s & while>0 (a,i,I) is_halting_on s ) by A3, A59; :: thesis: