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 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 pI = 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 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
( 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 SCMPDS 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 SCMPDS ; :: 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 = (Initialize t) +* (stop I);
set t3 = (Initialize t) +* (stop (for-down a,i,n,I));
set t4 = Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1;
I1: t +* (Initialize (stop I)) = (Initialize t) +* (stop I) by SCMPDS_4:5;
I2: t +* (Initialize (stop (for-down a,i,n,I))) = (Initialize t) +* (stop (for-down a,i,n,I)) by SCMPDS_4:5;
A13: Initialize (stop I) c= (Initialize t) +* (stop I) by I1, FUNCT_4:26;
A14: dom (ProgramPart t) = NAT by COMPOS_1:34;
A15: not a in dom (t | NAT ) by A14, SCMPDS_2:53;
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: Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(0 + 1) = Following (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),0 ) by AMI_1:14
.= Following (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))) by AMI_1:13
.= Exec (a,i <=0_goto ((card I) + 3)),((Initialize t) +* (stop (for-down a,i,n,I))) by A16, I2, SCMPDS_6:22 ;
A18: DataPart ((Initialize t) +* (stop I)) = DataPart ((Initialize t) +* (stop (for-down a,i,n,I))) by FUNCT_7:134, SCMPDS_4:24;

now

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

end;

then A19: DataPart ((Initialize t) +* (stop I)) = DataPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1) by SCMPDS_4:23;
A20: not DataLoc (s . a),i in dom (t | NAT ) by A14, SCMPDS_2:53;
A21: (IExec I,t) . (DataLoc (s . a),i) = t . (DataLoc (s . a),i) by A6, A10, A11;
- (- n) > 0 by A3;
then - n < 0 ;
then - n <= - 1 by INT_1:21;
then A22: (- n) + (t . (DataLoc (s . a),i)) <= (- 1) + (t . (DataLoc (s . a),i)) by XREAL_1:8;
(t . (DataLoc (s . a),i)) - 1 <= k by A9, XREAL_1:22;
then A23: (- n) + (t . (DataLoc (s . a),i)) <= k by A22, XXREAL_0:2;
A24: I is_closed_on t by A6, A10, A11;
then A25: I is_closed_on (Initialize t) +* (stop I) by SCMPDS_6:38;
A26: not DataLoc (s . a),i in dom (Initialize (stop (for-down a,i,n,I))) by SCMPDS_4:31;
set m2 = LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I));
set t5 = Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)));
set l1 = (card I) + 1;
A27: IC ((Initialize t) +* (stop (for-down a,i,n,I))) = 0 by SCMPDS_6:21;
set m3 = (LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1;
set t6 = Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1);
(card I) + 1 < (card I) + 3 by XREAL_1:8;
then A28: (card I) + 1 in dom (for-down a,i,n,I) by Th61;
set m5 = (((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1;
set t8 = Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1);
set t7 = Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1);
A29: (IExec I,t) . a = t . a by A6, A10, A11;
set l2 = (card I) + 2;
A30: 0 in dom (stop (for-down a,i,n,I)) by SCMPDS_4:75;
(card I) + 2 < (card I) + 3 by XREAL_1:8;
then A31: (card I) + 2 in dom (for-down a,i,n,I) by Th61;
A32: Initialize (stop (for-down a,i,n,I)) c= (Initialize t) +* (stop (for-down a,i,n,I)) by I2, FUNCT_4:26;
for-down a,i,n,I c= Initialize (stop (for-down a,i,n,I)) by SCMPDS_6:17;
then A33: for-down a,i,n,I c= (Initialize t) +* (stop (for-down a,i,n,I)) by A32, XBOOLE_1:1;
Shift I,1 c= for-down a,i,n,I by Lm5;
then Shift I,1 c= (Initialize t) +* (stop (for-down a,i,n,I)) by A33, XBOOLE_1:1;
then A34: Shift I,1 c= Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1 by AMI_1:81;
I is_halting_on t by A6, A10, A11;
then A35: ProgramPart ((Initialize t) +* (stop I)) halts_on (Initialize t) +* (stop I) by SCMPDS_6:def 3;
I3: ((Initialize t) +* (stop I)) +* (Initialize (stop I)) = (Initialize ((Initialize t) +* (stop I))) +* (stop I) by SCMPDS_4:5;
(Initialize t) +* (stop I) = (Initialize ((Initialize t) +* (stop I))) +* (stop I) by A13, I3, FUNCT_4:79;
then ProgramPart ((Initialize ((Initialize t) +* (stop I))) +* (stop I)) halts_on (Initialize ((Initialize t) +* (stop I))) +* (stop I) by A35;
then A36: I is_halting_on (Initialize t) +* (stop I) by SCMPDS_6:def 3;
not a in dom (Initialize (stop (for-down a,i,n,I))) by SCMPDS_4:31;
then ((Initialize t) +* (stop (for-down a,i,n,I))) . (DataLoc (((Initialize t) +* (stop (for-down a,i,n,I))) . a),i) = ((Initialize t) +* (stop (for-down a,i,n,I))) . (DataLoc (s . a),i) by A11, I2, FUNCT_4:12
.= t . (DataLoc (s . a),i) by A26, I2, FUNCT_4:12 ;
then A37: IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1) = succ (IC ((Initialize t) +* (stop (for-down a,i,n,I)))) by A12, A17, SCMPDS_2:68
.= 0 + 1 by A27 ;
then A38: IC (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) = (card I) + 1 by A4, A13, A36, A25, A19, A34, Th36;
Y: (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1))) /. (IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1))) = (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) . (IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1))) by COMPOS_1:38;
ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I))) = ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1) by AMI_1:123;
then A39: Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) = Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) by AMI_1:51;
then A40: CurInstr (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1))),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) = (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . ((card I) + 1) by A4, A13, A36, A25, A37, A19, A34, Th36, Y
.= (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1) . ((card I) + 1) by AMI_1:54
.= ((Initialize t) +* (stop (for-down a,i,n,I))) . ((card I) + 1) by AMI_1:54
.= (for-down a,i,n,I) . ((card I) + 1) by A28, A33, GRFUNC_1:8
.= AddTo a,i,(- n) by Th62 ;
Y: (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1))) /. (IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1))) = (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) . (IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1))) by COMPOS_1:38;
T: ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I))) = ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) by AMI_1:123;
A41: Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) = Following (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) by AMI_1:14
.= Exec (AddTo a,i,(- n)),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) by A40, T ;
then A42: IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) = succ (IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1))) by SCMPDS_2:60
.= ((card I) + 1) + 1 by A38, A39, NAT_1:39
.= (card I) + (1 + 1) ;
then A43: CurInstr (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1))),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) = ((Initialize t) +* (stop (for-down a,i,n,I))) . ((card I) + 2) by Y, AMI_1:54
.= (for-down a,i,n,I) . ((card I) + 2) by A33, A31, GRFUNC_1:8
.= goto (- ((card I) + 2)) by Th62 ;
T: ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I))) = ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) by AMI_1:123;
A44: Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1) = Following (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) by AMI_1:14
.= Exec (goto (- ((card I) + 2))),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) by A43, T ;
then IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1)) = ICplusConst (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)),(0 - ((card I) + 2)) by SCMPDS_2:66
.= 0 by A42, Th1 ;
then A45: (Initialize (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1))) +* (stop (for-down a,i,n,I)) = Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1) by Th37;
t: ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I))) = ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1) by AMI_1:123;
A46: DataPart (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) = DataPart (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) by A4, A13, A36, A25, A37, A19, A34, Th36;
then A47: (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . a = (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . a by SCMPDS_4:23
.= (Result (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) . a by A35, AMI_1:122
.= s . a by A11, A29, A15, FUNCT_4:12 ;
then DataLoc ((Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) . a),i = DataLoc (s . a),i by t, AMI_1:51;
then (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) . a = (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) . a by A5, A41, SCMPDS_2:60
.= s . a by A47, t, AMI_1:51 ;
then A48: (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1)) . a = s . a by A44, SCMPDS_2:66;

A49: now

let x be Int_position ; :: thesis:
assume A50: x in X ; :: thesis:
A51: not x in dom (t | NAT ) by A14, SCMPDS_2:53;
(Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . x = (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . x by A46, SCMPDS_4:23
.= (Result (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) . x by A35, AMI_1:122
.= (IExec I,t) . x by A51, FUNCT_4:12
.= t . x by A6, A10, A11, A50
.= s . x by A10, A50 ;
then (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) . x = s . x by A2, A47, A39, A41, A50, SCMPDS_2:60;
hence (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1)) . x = s . x by A44, SCMPDS_2:66; :: thesis:

end;

A52: (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . (DataLoc (s . a),i) = (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . (DataLoc (s . a),i) by A46, SCMPDS_4:23
.= (Result (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) . (DataLoc (s . a),i) by A35, AMI_1:122
.= t . (DataLoc (s . a),i) by A21, A20, FUNCT_4:12 ;
A53: (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1)) . (DataLoc (s . a),i) = (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) . (DataLoc (s . a),i) by A44, SCMPDS_2:66
.= (t . (DataLoc (s . a),i)) + (- n) by A47, A52, A39, A41, SCMPDS_2:60 ;
then A54: for-down a,i,n,I is_closed_on Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1) by A8, A48, A49, A23;

now

let k be Element of NAT ; :: thesis:

per cases ;

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

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

hereby :: thesis:

per cases by A55, NAT_1:8;

suppose A56: 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 (for-down a,i,n,I)))),((Initialize t) +* (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 (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),kn) as Element of NAT ;
kn < k by A57, XREAL_1:31;
then kn < LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)) by A56, XXREAL_0:2;
then (IC (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),kn)) + 1 = IC (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),1),kn) by A4, A13, A36, A25, A37, A19, A34, Th34;
then A58: IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),k) = lm + 1 by A57, t, AMI_1:51
.= lm + 1 ;
IC (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),kn) in dom (stop I) by A24, SCMPDS_6:def 2;
then lm < card (stop I) by AFINSQ_1:70;
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 (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),k) in dom (stop (for-down a,i,n,I)) by A58, AFINSQ_1:70; :: thesis:

end;

end;

end;

end;

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

(card I) + 1 in dom (stop (for-down a,i,n,I)) by A28, SCMPDS_6:18;
hence IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (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 (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1 ; :: thesis:

hence IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (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;

end;

end;

end;

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

then consider nn being Nat such that
A61: k = ((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 13;
ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I))) = ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1)) by AMI_1:123;
then Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),k = Comput (ProgramPart ((Initialize (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1))) +* (stop (for-down a,i,n,I)))),((Initialize (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1))) +* (stop (for-down a,i,n,I))),nn by A45, A61, AMI_1:51;
hence IC (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (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;

end;

hence for-down a,i,n,I is_closed_on t by SCMPDS_6:def 2; :: thesis:
for-down a,i,n,I is_halting_on Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1) by A8, A48, A49, A53, A23;
then ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1)) halts_on Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1) by A45, SCMPDS_6:def 3;
then ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I))) halts_on Comput (ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I)))),((Initialize t) +* (stop (for-down a,i,n,I))),((((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + 1) by AMI_1:123;
then ProgramPart ((Initialize t) +* (stop (for-down a,i,n,I))) halts_on (Initialize t) +* (stop (for-down a,i,n,I)) by AMI_1:93;
hence for-down a,i,n,I is_halting_on t by SCMPDS_6:def 3; :: thesis:

end;

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: