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 n be Element of NAT ; :: thesis:
let X be set ; :: thesis:
set b = DataLoc (s . a),i;
set FOR = for-down a,i,n,I;
set pFOR = stop (for-down a,i,n,I);
set iFOR = Initialized (stop (for-down a,i,n,I));
set pI = stop I;
set IsI = Initialized (stop I);
set i1 = a,i <=0_goto ((card I) + 3);
set i2 = AddTo a,i,(- n);
set i3 = goto (- ((card I) + 2));
assume A1: s . (DataLoc (s . a),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
( for-down a,i,n,I is_closed_on t & for-down a,i,n,I is_halting_on t );
assume A2: not DataLoc (s . a),i in X ; :: thesis:
assume A3: n > 0 ; :: thesis:
assume A4: card I > 0 ; :: thesis:
assume A5: a <> DataLoc (s . a),i ; :: thesis:
assume A6: for t being State of st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a 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 y being Int_position st y in X holds
(IExec I,t) . y = t . y ) ) ; :: thesis:
A7: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis:
assume A8: S₁[k] ; :: thesis:

now

let t be State of ; :: thesis:
assume A9: t . (DataLoc (s . a),i) <= k + 1 ; :: thesis:
assume A10: for x being Int_position st x in X holds
t . x = s . x ; :: thesis:
assume A11: t . a = s . a ; :: thesis:

per cases ;

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

hence ( for-down a,i,n,I is_closed_on t & for-down a,i,n,I is_halting_on t ) by A11, Th63; :: thesis:

end;

suppose A12: t . (DataLoc (s . a),i) > 0 ; :: thesis:

set t2 = t +* (Initialized (stop I));
set t3 = t +* (Initialized (stop (for-down a,i,n,I)));
set t4 = Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1;
A13: Initialized (stop I) c= t +* (Initialized (stop I)) by FUNCT_4:26;
A14: dom (t | NAT ) = NAT by SCMPDS_6:1;

A15: now

assume a in dom (t | NAT ) ; :: thesis:
then reconsider l = a as Instruction-Location of SCMPDS by A14, AMI_1:def 4;
l = a ;
hence contradiction by SCMPDS_2:53; :: thesis:

end;

A16: for-down a,i,n,I = (a,i <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo a,i,(- n))) ';' (goto (- ((card I) + 2)))) by Th15;
A17: Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(0 + 1) = Following (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),0 ) by AMI_1:14
.= Following (t +* (Initialized (stop (for-down a,i,n,I)))) by AMI_1:13
.= Exec (a,i <=0_goto ((card I) + 3)),(t +* (Initialized (stop (for-down a,i,n,I)))) by A16, SCMPDS_6:22 ;
A18: DataPart (t +* (Initialized (stop I))) = DataPart (t +* (Initialized (stop (for-down a,i,n,I)))) by SCMPDS_4:24, SCMPDS_4:36;

now

let a be Int_position ; :: thesis:
thus (t +* (Initialized (stop I))) . a = (t +* (Initialized (stop (for-down a,i,n,I)))) . a by A18, SCMPDS_4:23
.= (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1) . a by A17, SCMPDS_2:68 ; :: thesis:

end;

then A19: DataPart (t +* (Initialized (stop I))) = DataPart (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1) by SCMPDS_4:23;

A20: now

assume DataLoc (s . a),i in dom (t | NAT ) ; :: thesis:
then reconsider l = DataLoc (s . a),i as Instruction-Location of SCMPDS by A14, AMI_1:def 4;
l = DataLoc (s . a),i ;
hence contradiction by SCMPDS_2:53; :: thesis:

end;

A49: now

let x be Int_position ; :: thesis:
assume A50: x in X ; :: thesis:

A51: now

assume x in dom (t | NAT ) ; :: thesis:
then reconsider l = x as Instruction-Location of SCMPDS by A14, AMI_1:def 4;
l = x ;
hence contradiction by SCMPDS_2:53; :: thesis:

end;

end;

now

let k be Element of NAT ; :: thesis:

per cases ;

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

then k <= ((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1 by INT_1:20;
then A55: ( k <= (LifeSpan (t +* (Initialized (stop I)))) + 1 or k = ((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1 ) by NAT_1:8;

hereby :: thesis:

per cases by A55, NAT_1:8;

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

hereby :: thesis:

per cases ;

suppose k = 0 ; :: thesis:

hence IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I)) by A30, A27, AMI_1:13; :: thesis:

end;

suppose k <> 0 ; :: thesis:

then consider kn being Nat such that
A57: 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 A57, XREAL_1:31;
then kn < LifeSpan (t +* (Initialized (stop I))) by A56, XXREAL_0:2;
then (IC (Computation (t +* (Initialized (stop I))),kn)) + 1 = IC (Computation (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1),kn) by A4, A13, A36, A25, A37, A19, A34, Th34;
then A58: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) = (inspos lm) + 1 by A57, AMI_1:51
.= inspos (lm + 1) ;
IC (Computation (t +* (Initialized (stop I))),kn) in dom (stop I) by A24, SCMPDS_6:def 2;
then lm < card (stop I) by SCMPDS_4:1;
then lm < (card I) + 1 by SCMPDS_5:7;
then A59: lm + 1 <= (card I) + 1 by INT_1:20;
(card I) + 1 < (card I) + 4 by XREAL_1:8;
then lm + 1 < (card I) + 4 by A59, XXREAL_0:2;
then lm + 1 < card (stop (for-down a,i,n,I)) by Lm4;
hence IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I)) by A58, SCMPDS_4:1; :: thesis:

end;

suppose A60: k = (LifeSpan (t +* (Initialized (stop I)))) + 1 ; :: thesis:

inspos ((card I) + 1) in dom (stop (for-down a,i,n,I)) by A28, SCMPDS_6:18;
hence IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I)) by A4, A13, A36, A25, A37, A19, A34, A39, A60, Th36; :: thesis:

end;

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

hence IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I)) by A42, A31, SCMPDS_6:18; :: thesis:

end;

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

then consider nn being Nat such that
A61: k = ((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 13;
Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k = Computation ((Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1)) +* (Initialized (stop (for-down a,i,n,I)))),nn by A45, A61, AMI_1:51;
hence IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I)) by A54, SCMPDS_6:def 2; :: thesis:

end;

end;

hence S₁[k + 1] ; :: thesis:

end;

reconsider n = s . (DataLoc (s . a),i) as Element of NAT by A1, INT_1:16;
A62: S₁[ 0 ] by Th63;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A62, A7);
then A63: S₁[n] ;
for x being Int_position st x in X holds
s . x = s . x ;
hence ( for-down a,i,n,I is_closed_on s & for-down a,i,n,I is_halting_on s ) by A63; :: thesis: