let P be Instruction-Sequence of SCMPDS; :: thesis:
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-up (a,i,n,I);
set pFOR = stop (for-up (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
for Q being Instruction-Sequence 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-up (a,i,n,I) is_closed_on t,Q & for-up (a,i,n,I) is_halting_on t,Q );
assume A2: not DataLoc ((s . a),i) in X ; :: thesis:
assume A3: n > 0 ; :: thesis:
assume A5: a <> DataLoc ((s . a),i) ; :: thesis:
assume A6: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a holds
( (IExec (I,Q,(Initialize t))) . a = t . a & (IExec (I,Q,(Initialize t))) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for y being Int_position st y in X holds
(IExec (I,Q,(Initialize 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:
let t be State of SCMPDS; :: thesis:
let Q be Instruction-Sequence 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-up (a,i,n,I) is_closed_on t,Q & for-up (a,i,n,I) is_halting_on t,Q ) by A11, Th54; :: thesis:

end;

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

set t2 = Initialize t;
set t3 = Initialize t;
set Q2 = Q +* (stop I);
set Q3 = Q +* (stop (for-up (a,i,n,I)));
set t4 = Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1);
set Q4 = Q +* (stop (for-up (a,i,n,I)));
A15: stop I c= Q +* (stop I) by FUNCT_4:25;
A18: for-up (a,i,n,I) = ((a,i) >=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,n))) ';' (goto (- ((card I) + 2)))) by Th15;
A19: Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(0 + 1)) = Following ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),0))) by EXTPRO_1:3
.= Following ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t)) by EXTPRO_1:2
.= Exec (((a,i) >=0_goto ((card I) + 3)),(Initialize t)) by A18, SCMPDS_6:11 ;
for a being Int_position holds (Initialize t) . a = (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)) . a by A19, SCMPDS_2:57;
then A21: DataPart (Initialize t) = DataPart (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)) by SCMPDS_4:8;
A23: (IExec (I,Q,(Initialize 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_1:8;
then A24: (- n) - (t . (DataLoc ((s . a),i))) <= (- 1) - (t . (DataLoc ((s . a),i))) by XREAL_1:9;
(- (t . (DataLoc ((s . a),i)))) - 1 <= k by A9, XREAL_1:20;
then A25: (- n) - (t . (DataLoc ((s . a),i))) <= k by A24, XXREAL_0:2;
A26: I is_closed_on t,Q by A6, A10, A11;
then A27: I is_closed_on Initialize t,Q +* (stop I) by SCMPDS_6:24;
A28: not DataLoc ((s . a),i) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
set m2 = LifeSpan ((Q +* (stop I)),(Initialize t));
set t5 = Comput ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))));
set Q5 = Q +* (stop (for-up (a,i,n,I)));
set l1 = (card I) + 1;
A29: IC (Initialize t) = 0 by MEMSTR_0:def 8;
set m3 = (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1;
set t6 = Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1));
set Q6 = Q +* (stop (for-up (a,i,n,I)));
(card I) + 1 < (card I) + 3 by XREAL_1:6;
then A30: (card I) + 1 in dom (for-up (a,i,n,I)) by Th52;
set m5 = (((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1;
set t8 = Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1));
set Q8 = Q +* (stop (for-up (a,i,n,I)));
set t7 = Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1));
set Q7 = Q +* (stop (for-up (a,i,n,I)));
A31: (IExec (I,Q,(Initialize t))) . a = t . a by A6, A10, A11;
set l2 = (card I) + 2;
A32: 0 in dom (stop (for-up (a,i,n,I))) by COMPOS_1:36;
(card I) + 2 < (card I) + 3 by XREAL_1:6;
then A33: (card I) + 2 in dom (for-up (a,i,n,I)) by Th52;
A34: stop (for-up (a,i,n,I)) c= Q +* (stop (for-up (a,i,n,I))) by FUNCT_4:25;
for-up (a,i,n,I) c= stop (for-up (a,i,n,I)) by AFINSQ_1:74;
then A35: for-up (a,i,n,I) c= Q +* (stop (for-up (a,i,n,I))) by A34, XBOOLE_1:1;
Shift (I,1) c= for-up (a,i,n,I) by Lm3;
then A36: Shift (I,1) c= Q +* (stop (for-up (a,i,n,I))) by A35, XBOOLE_1:1;
I is_halting_on t,Q by A6, A10, A11;
then A37: Q +* (stop I) halts_on Initialize t by SCMPDS_6:def 3;
(Q +* (stop I)) +* (stop I) halts_on Initialize (Initialize t) by A37, FUNCT_4:93;
then A39: I is_halting_on Initialize t,Q +* (stop I) by SCMPDS_6:def 3;
not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then (Initialize t) . (DataLoc (((Initialize t) . a),i)) = (Initialize t) . (DataLoc ((s . a),i)) by A11, FUNCT_4:11
.= t . (DataLoc ((s . a),i)) by A28, FUNCT_4:11 ;
then A40: IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)) = succ (IC (Initialize t)) by A12, A19, SCMPDS_2:57
.= 0 + 1 by A29 ;
then A41: IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) = (card I) + 1 by A15, A39, A27, A21, A36, Th36;
A42: (Q +* (stop (for-up (a,i,n,I)))) /. (IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) = (Q +* (stop (for-up (a,i,n,I)))) . (IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by PBOOLE:143;
A43: Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)) = Comput ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t)))) by EXTPRO_1:4;
then A44: CurInstr ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) = (Q +* (stop (for-up (a,i,n,I)))) . ((card I) + 1) by A15, A39, A27, A40, A21, A36, Th36, A42
.= (for-up (a,i,n,I)) . ((card I) + 1) by A30, A35, GRFUNC_1:2
.= AddTo (a,i,n) by Th53 ;
A47: Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)) = Following ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by EXTPRO_1:3
.= Exec ((AddTo (a,i,n)),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by A44 ;
then A48: IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) = succ (IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by SCMPDS_2:48
.= ((card I) + 1) + 1 by A41, A43, NAT_1:38
.= (card I) + (1 + 1) ;
then A49: CurInstr ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)))) = (Q +* (stop (for-up (a,i,n,I)))) . ((card I) + 2) by PBOOLE:143
.= (for-up (a,i,n,I)) . ((card I) + 2) by A35, A33, GRFUNC_1:2
.= goto (- ((card I) + 2)) by Th53 ;
A51: Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1)) = Following ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 2))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)))) by A49 ;
then IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))),(0 - ((card I) + 2))) by SCMPDS_2:54
.= 0 by A48, Th1 ;
then A52: Initialize (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) = Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1)) by MEMSTR_0:46;
A54: DataPart (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) = DataPart (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) by A15, A39, A27, A40, A21, A36, Th36;
then A55: (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . a = (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . a by SCMPDS_4:8
.= s . a by A11, A31, A37, EXTPRO_1:23 ;
then DataLoc (((Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1))) . a),i) = DataLoc ((s . a),i) by EXTPRO_1:4;
then (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . a = (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1))) . a by A5, A47, SCMPDS_2:48
.= s . a by A55, EXTPRO_1:4 ;
then A56: (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) . a = s . a by A51, SCMPDS_2:54;

A57: now

let x be Int_position ; :: thesis:
assume A58: x in X ; :: thesis:
(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . x = (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . x by A54, SCMPDS_4:8
.= (IExec (I,Q,(Initialize t))) . x by A37, EXTPRO_1:23
.= t . x by A6, A10, A11, A58
.= s . x by A10, A58 ;
then (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . x = s . x by A2, A55, A43, A47, A58, SCMPDS_2:48;
hence (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) . x = s . x by A51, SCMPDS_2:54; :: thesis:

end;

A60: (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . (DataLoc ((s . a),i)) = (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . (DataLoc ((s . a),i)) by A54, SCMPDS_4:8
.= t . (DataLoc ((s . a),i)) by A23, A37, EXTPRO_1:23 ;
(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) . (DataLoc ((s . a),i)) = (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . (DataLoc ((s . a),i)) by A51, SCMPDS_2:54
.= (t . (DataLoc ((s . a),i))) + n by A55, A60, A43, A47, SCMPDS_2:48 ;
then A61: - ((Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) . (DataLoc ((s . a),i))) = (- n) - (t . (DataLoc ((s . a),i))) ;
then A62: for-up (a,i,n,I) is_closed_on Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1)),Q +* (stop (for-up (a,i,n,I))) by A8, A56, A57, A25;

now

let k be Element of NAT ; :: thesis:

per cases ;

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

then k <= ((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1 by INT_1:7;
then A63: ( k <= (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1 or k = ((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1 ) by NAT_1:8;

hereby :: thesis:

per cases by A63, NAT_1:8;

suppose A64: k <= LifeSpan ((Q +* (stop I)),(Initialize t)) ; :: thesis:

hereby :: thesis:

per cases ;

suppose k = 0 ; :: thesis:

hence IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),k)) in dom (stop (for-up (a,i,n,I))) by A32, A29, EXTPRO_1:2; :: thesis:

end;

suppose k <> 0 ; :: thesis:

then consider kn being Nat such that
A65: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def 12;
reconsider lm = IC (Comput ((Q +* (stop I)),(Initialize t),kn)) as Element of NAT ;
kn < k by A65, XREAL_1:29;
then kn < LifeSpan ((Q +* (stop I)),(Initialize t)) by A64, XXREAL_0:2;
then (IC (Comput ((Q +* (stop I)),(Initialize t),kn))) + 1 = IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),1)),kn)) by A15, A39, A27, A40, A21, A36, Th34;
then A66: IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),k)) = lm + 1 by A65, EXTPRO_1:4;
IC (Comput ((Q +* (stop I)),(Initialize t),kn)) in dom (stop I) by A26, SCMPDS_6:def 2;
then lm < card (stop I) by AFINSQ_1:66;
then lm < (card I) + 1 by COMPOS_1:55;
then A67: lm + 1 <= (card I) + 1 by INT_1:7;
(card I) + 1 < (card I) + 4 by XREAL_1:6;
then lm + 1 < (card I) + 4 by A67, XXREAL_0:2;
then lm + 1 < card (stop (for-up (a,i,n,I))) by Lm2;
hence IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),k)) in dom (stop (for-up (a,i,n,I))) by A66, AFINSQ_1:66; :: thesis:

end;

suppose A68: k = (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1 ; :: thesis:

(card I) + 1 in dom (stop (for-up (a,i,n,I))) by A30, COMPOS_1:62;
hence IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),k)) in dom (stop (for-up (a,i,n,I))) by A15, A39, A27, A40, A21, A36, A43, A68, Th36; :: thesis:

end;

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

hence IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),k)) in dom (stop (for-up (a,i,n,I))) by A48, A33, COMPOS_1:62; :: thesis:

end;

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

then consider nn being Nat such that
A69: k = ((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 12;
(Q +* (stop (for-up (a,i,n,I)))) +* (stop (for-up (a,i,n,I))) = Q +* (stop (for-up (a,i,n,I))) by FUNCT_4:93;
then Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),k) = Comput (((Q +* (stop (for-up (a,i,n,I)))) +* (stop (for-up (a,i,n,I)))),(Initialize (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1)))),nn) by A52, A69, EXTPRO_1:4;
hence IC (Comput ((Q +* (stop (for-up (a,i,n,I)))),(Initialize t),k)) in dom (stop (for-up (a,i,n,I))) by A62, SCMPDS_6:def 2; :: thesis:

end;

end;

reconsider nn = - (s . (DataLoc ((s . a),i))) as Element of NAT by A1, INT_1:3;
A70: S₁[ 0 ]

proof

let t be State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
assume - (t . (DataLoc ((s . a),i))) <= 0 ; :: thesis:
then - (t . (DataLoc ((s . a),i))) <= - 0 ;
then A71: t . (DataLoc ((s . a),i)) >= 0 by XREAL_1:24;
assume for x being Int_position st x in X holds
t . x = s . x ; :: thesis:
assume t . a = s . a ; :: thesis:
hence ( for-up (a,i,n,I) is_closed_on t,Q & for-up (a,i,n,I) is_halting_on t,Q ) by A71, Th54; :: thesis:

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A70, A7);
then A72: S₁[nn] ;
for x being Int_position st x in X holds
s . x = s . x ;
hence ( for-up (a,i,n,I) is_closed_on s,P & for-up (a,i,n,I) is_halting_on s,P ) by A72; :: thesis: