let s be State of ; :: thesis:
let I be shiftable No-StopCode Program of ; :: 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 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 - (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 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 ; :: 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:

A9: dom (t | NAT ) = NAT by SCMPDS_6:1;

A10: now

assume a in dom (t | NAT ) ; :: thesis:
then reconsider l = a as Instruction-Location of SCMPDS by A9, AMI_1:def 4;
l = a ;
hence contradiction by SCMPDS_2:53; :: thesis:

end;

A11: now

assume DataLoc (s . a),i in dom (t | NAT ) ; :: thesis:
then reconsider l = DataLoc (s . a),i as Instruction-Location of SCMPDS by A9, AMI_1:def 4;
l = DataLoc (s . a),i ;
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 A19, SCMPDS_4:23
.= (Computation (t +* (Initialized (stop (while<0 a,i,I)))),1) . a by A18, SCMPDS_2:69 ; :: thesis:

end;

A38: now

let x be Int_position ; :: thesis:
assume A39: x in X ; :: thesis:

A40: now

assume x in dom (t | NAT ) ; :: thesis:
then reconsider l = x as Instruction-Location of SCMPDS by A9, AMI_1:def 4;
l = x ;
hence contradiction by SCMPDS_2:53; :: thesis:

end;

(Computation (Computation (t +* (Initialized (stop (while<0 a,i,I)))),1),(LifeSpan (t +* (Initialized (stop I))))) . x = (Computation (t +* (Initialized (stop I))),(LifeSpan (t +* (Initialized (stop I))))) . x by A34, SCMPDS_4:23
.= (Result (t +* (Initialized (stop I)))) . x by A28, AMI_1:122
.= (IExec I,t) . x by A25, A40, FUNCT_4:12
.= t . x by A2, A6, A7, A8, A39
.= s . x by A6, A39 ;
hence (Computation (t +* (Initialized (stop (while<0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . x = s . x by A32, A36, SCMPDS_2:66; :: thesis:

end;

(Computation (Computation (t +* (Initialized (stop (while<0 a,i,I)))),1),(LifeSpan (t +* (Initialized (stop I))))) . (DataLoc (s . a),i) = (Computation (t +* (Initialized (stop I))),(LifeSpan (t +* (Initialized (stop I))))) . (DataLoc (s . a),i) by A34, SCMPDS_4:23
.= (Result (t +* (Initialized (stop I)))) . (DataLoc (s . a),i) by A28, AMI_1:122
.= (IExec I,t) . (DataLoc (s . a),i) by A25, A11, FUNCT_4:12 ;
then A41: (Computation (t +* (Initialized (stop (while<0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . (DataLoc (s . a),i) = (IExec I,t) . (DataLoc (s . a),i) by A32, A36, SCMPDS_2:66;

A42: now

- ((Computation (t +* (Initialized (stop (while<0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . (DataLoc (s . a),i)) < - (t . (DataLoc (s . a),i)) by A12, A41, XREAL_1:26;
then A43: - ((Computation (t +* (Initialized (stop (while<0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . (DataLoc (s . a),i)) < k + 1 by A5, XXREAL_0:2;
assume - ((Computation (t +* (Initialized (stop (while<0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . (DataLoc (s . a),i)) > k ; :: thesis:
hence contradiction by A43, INT_1:20; :: thesis:

end;

A44: (Computation (t +* (Initialized (stop (while<0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . a = (Computation (t +* (Initialized (stop (while<0 a,i,I)))),((LifeSpan (t +* (Initialized (stop I)))) + 1)) . a by A36, SCMPDS_2:66
.= s . a by A35, AMI_1:51 ;
then A45: while<0 a,i,I is_closed_on Computation (t +* (Initialized (stop (while<0 a,i,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) by A4, A38, A42;

now

let k be Element of NAT ; :: thesis:

per cases ;

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

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

hereby :: thesis:

per cases by A46, NAT_1:8;

suppose A47: 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 A13, A23, AMI_1:13; :: thesis:

end;

suppose k <> 0 ; :: thesis:

then consider kn being Nat such that
A48: 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 A48, XREAL_1:31;
then kn < LifeSpan (t +* (Initialized (stop I))) by A47, 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, A17, A29, A27, A30, A20, A22, SCMPDS_7:34;
then A49: IC (Computation (t +* (Initialized (stop (while<0 a,i,I)))),k) = inspos (lm + 1) by A48, AMI_1:51;
IC (Computation (t +* (Initialized (stop I))),kn) in dom (stop I) by A26, SCMPDS_6:def 2;
then lm < card (stop I) by SCMPDS_4:1;
then lm < (card I) + 1 by SCMPDS_5:7;
then A50: 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 A50, XXREAL_0:2;
then lm + 1 < card (stop (while<0 a,i,I)) by Lm1;
hence IC (Computation (t +* (Initialized (stop (while<0 a,i,I)))),k) in dom (stop (while<0 a,i,I)) by A49, SCMPDS_4:1; :: thesis:

end;

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

inspos ((card I) + 1) in dom (stop (while<0 a,i,I)) by A24, 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, A17, A29, A27, A30, A20, A22, A32, A51, SCMPDS_7:36; :: thesis:

end;

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

then consider nn being Nat such that
A52: k = (((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 13;
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, A52, AMI_1:51;
hence IC (Computation (t +* (Initialized (stop (while<0 a,i,I)))),k) in dom (stop (while<0 a,i,I)) by A45, SCMPDS_6:def 2; :: thesis:

end;

end;

hence S₁[k + 1] ; :: thesis:

end;

A53: S₁[ 0 ]

proof

let t be State of ; :: 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;