let s be State of ; :: thesis:
let I be shiftable No-StopCode Program of ; :: 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 iWHL = Initialized (stop (while>0 a,i,I));
set pI = stop I;
set IsI = Initialized (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 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 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 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 ; :: 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:

A12: dom (t | NAT ) = NAT by SCMPDS_6:1;

A13: now

assume a in dom (t | NAT ) ; :: thesis:
then reconsider l = a as Instruction-Location of SCMPDS by A12, AMI_1:def 4;
l = a ;
hence contradiction by SCMPDS_2:53; :: thesis:

end;

now

let a be Int_position ; :: thesis:
thus (t +* (Initialized (stop I))) . a = (t +* (Initialized (stop (while>0 a,i,I)))) . a by A21, SCMPDS_4:23
.= (Computation (t +* (Initialized (stop (while>0 a,i,I)))),1) . a by A20, SCMPDS_2:68 ; :: thesis:

end;

A40: now

let x be Int_position ; :: thesis:
assume A41: x in Y ; :: thesis:
thus (Computation (t +* (Initialized (stop (while>0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . x = (Computation (Computation (t +* (Initialized (stop (while>0 a,i,I)))),1),(LifeSpan (t +* (Initialized (stop I))))) . x by A34, A36, SCMPDS_2:66
.= (IExec I,t) . x by A39, SCMPDS_3:4
.= t . x by A4, A8, A9, A10, A11, A41
.= s . x by A9, A41 ; :: thesis:

end;

InsCode (goto (- ((card I) + 1))) = 0 by SCMPDS_2:21;
then InsCode (goto (- ((card I) + 1))) in {0 ,4,5,6} by ENUMSET1:def 2;
then A42: Dstate (Computation (t +* (Initialized (stop (while>0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) = Dstate (Computation (t +* (Initialized (stop (while>0 a,i,I)))),((LifeSpan (t +* (Initialized (stop I)))) + 1)) by A36, Th3
.= Dstate (IExec I,t) by A39, A34, Th2 ;

A43: now

f . (Dstate (IExec I,t)) < f . (Dstate t) by A4, A8, A9, A10, A11;
then A44: f . (Dstate (Computation (t +* (Initialized (stop (while>0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1))) < k + 1 by A7, A42, XXREAL_0:2;
assume f . (Dstate (Computation (t +* (Initialized (stop (while>0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1))) > k ; :: thesis:
hence contradiction by A44, INT_1:20; :: thesis:

end;

A45: (Computation (t +* (Initialized (stop (while>0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . (DataLoc (s . a),i) = (Computation (Computation (t +* (Initialized (stop (while>0 a,i,I)))),1),(LifeSpan (t +* (Initialized (stop I))))) . (DataLoc (s . a),i) by A34, A36, SCMPDS_2:66
.= (IExec I,t) . (DataLoc (s . a),i) by A39, SCMPDS_3:4 ;

A46: now

end;

now

let k be Element of NAT ; :: thesis:

per cases ;

suppose k < ((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1 ; :: thesis:

then A51: k <= (LifeSpan (t +* (Initialized (stop I)))) + 1 by INT_1:20;

hereby :: thesis:

per cases by A51, NAT_1:8;

suppose A52: k <= LifeSpan (t +* (Initialized (stop I))) ; :: thesis:

hereby :: thesis:

per cases ;

suppose k = 0 ; :: thesis:

hence IC (Computation (t +* (Initialized (stop (while>0 a,i,I)))),k) in dom (stop (while>0 a,i,I)) by A15, A26, AMI_1:13; :: thesis:

end;

suppose k <> 0 ; :: thesis:

then consider kn being Nat such that
A53: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def 13;
reconsider lm = IC (Computation (t +* (Initialized (stop I))),kn) as Element of NAT by ORDINAL1:def 13;
kn < k by A53, XREAL_1:31;
then kn < LifeSpan (t +* (Initialized (stop I))) by A52, XXREAL_0:2;
then (IC (Computation (t +* (Initialized (stop I))),kn)) + 1 = IC (Computation (Computation (t +* (Initialized (stop (while>0 a,i,I)))),1),kn) by A1, A19, A31, A29, A32, A22, A24, SCMPDS_7:34;
then A54: IC (Computation (t +* (Initialized (stop (while>0 a,i,I)))),k) = inspos (lm + 1) by A53, AMI_1:51;
IC (Computation (t +* (Initialized (stop I))),kn) in dom (stop I) by A28, SCMPDS_6:def 2;
then lm < card (stop I) by SCMPDS_4:1;
then lm < (card I) + 1 by SCMPDS_5:7;
then A55: lm + 1 <= (card I) + 1 by INT_1:20;
(card I) + 1 < (card I) + 3 by XREAL_1:8;
then lm + 1 < (card I) + 3 by A55, XXREAL_0:2;
then lm + 1 < card (stop (while>0 a,i,I)) by Lm3;
hence IC (Computation (t +* (Initialized (stop (while>0 a,i,I)))),k) in dom (stop (while>0 a,i,I)) by A54, SCMPDS_4:1; :: thesis:

end;

suppose A56: k = (LifeSpan (t +* (Initialized (stop I)))) + 1 ; :: thesis:

inspos ((card I) + 1) in dom (stop (while>0 a,i,I)) by A27, SCMPDS_6:18;
hence IC (Computation (t +* (Initialized (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 (t +* (Initialized (stop I)))) + 1) + 1 ; :: thesis:

then consider nn being Nat such that
A57: k = (((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + nn by NAT_1:10;
A58: nn in NAT by ORDINAL1:def 13;
then Computation (t +* (Initialized (stop (while>0 a,i,I)))),k = Computation ((Computation (t +* (Initialized (stop (while>0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) +* (Initialized (stop (while>0 a,i,I)))),nn by A37, A57, AMI_1:51;
hence IC (Computation (t +* (Initialized (stop (while>0 a,i,I)))),k) in dom (stop (while>0 a,i,I)) by A50, A58, SCMPDS_6:def 2; :: 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 ; :: 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: