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-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
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-down (a,i,n,I) is_closed_on t,Q & for-down (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 0 -started 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,t)) . a = t . a & (IExec (I,Q,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,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:
A11: for x being Int_position st x in X holds
(Initialize t) . x = s . x

proof

let x be Int_position ; :: thesis:
assume x in X ; :: thesis:
then t . x = s . x by A10;
hence (Initialize t) . x = s . x by SCMPDS_5:15; :: thesis:

end;

assume A12: t . a = s . a ; :: thesis:
then A13: (Initialize t) . a = s . a by SCMPDS_5:15;

per cases ;

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

hence ( for-down (a,i,n,I) is_closed_on t,Q & for-down (a,i,n,I) is_halting_on t,Q ) by A12, Th63; :: thesis:

end;

suppose A14: 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-down (a,i,n,I)));
set t4 = Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),1);
set Q4 = Q +* (stop (for-down (a,i,n,I)));
A17: stop I c= Q +* (stop I) by FUNCT_4:25;
A20: for-down (a,i,n,I) = ((a,i) <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2)))) by Th15;
A21: Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),(0 + 1)) = Following ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),0))) by EXTPRO_1:3
.= Following ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t)) by EXTPRO_1:2
.= Exec (((a,i) <=0_goto ((card I) + 3)),(Initialize t)) by A20, SCMPDS_6:11 ;
for a being Int_position holds (Initialize t) . a = (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),1)) . a by A21, SCMPDS_2:56;
then A23: DataPart (Initialize t) = DataPart (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),1)) by SCMPDS_4:8;
A26: (IExec (I,Q,(Initialize t))) . (DataLoc ((s . a),i)) = (Initialize t) . (DataLoc ((s . a),i)) by A6, A11, A13
.= t . (DataLoc ((s . a),i)) by SCMPDS_5:15 ;
- (- n) > 0 by A3;
then - n < 0 ;
then - n <= - 1 by INT_1:8;
then A27: (- n) + (t . (DataLoc ((s . a),i))) <= (- 1) + (t . (DataLoc ((s . a),i))) by XREAL_1:6;
(t . (DataLoc ((s . a),i))) - 1 <= k by A9, XREAL_1:20;
then A28: (- n) + (t . (DataLoc ((s . a),i))) <= k by A27, XXREAL_0:2;
I is_closed_on Initialize t,Q by A6, A11, A13;
then A29: I is_closed_on t,Q by SCMPDS_6:125;
then A30: I is_closed_on Initialize t,Q +* (stop I) by SCMPDS_6:24;
A31: 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-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))));
set Q5 = Q +* (stop (for-down (a,i,n,I)));
set l1 = (card I) + 1;
A32: IC (Initialize t) = 0 by MEMSTR_0:def 8;
set m3 = (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1;
set t6 = Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1));
set Q6 = Q +* (stop (for-down (a,i,n,I)));
(card I) + 1 < (card I) + 3 by XREAL_1:6;
then A33: (card I) + 1 in dom (for-down (a,i,n,I)) by Th61;
set m5 = (((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1;
set t8 = Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1));
set Q8 = Q +* (stop (for-down (a,i,n,I)));
set t7 = Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1));
set Q7 = Q +* (stop (for-down (a,i,n,I)));
A34: (IExec (I,Q,(Initialize t))) . a = (Initialize t) . a by A6, A11, A13
.= t . a by SCMPDS_5:15 ;
set l2 = (card I) + 2;
A35: 0 in dom (stop (for-down (a,i,n,I))) by COMPOS_1:36;
(card I) + 2 < (card I) + 3 by XREAL_1:6;
then A36: (card I) + 2 in dom (for-down (a,i,n,I)) by Th61;
A37: stop (for-down (a,i,n,I)) c= Q +* (stop (for-down (a,i,n,I))) by FUNCT_4:25;
for-down (a,i,n,I) c= stop (for-down (a,i,n,I)) by AFINSQ_1:74;
then A38: for-down (a,i,n,I) c= Q +* (stop (for-down (a,i,n,I))) by A37, XBOOLE_1:1;
Shift (I,1) c= for-down (a,i,n,I) by Lm5;
then A39: Shift (I,1) c= Q +* (stop (for-down (a,i,n,I))) by A38, XBOOLE_1:1;
I is_halting_on Initialize t,Q by A6, A11, A13;
then I is_halting_on t,Q by SCMPDS_6:126;
then A40: Q +* (stop I) halts_on Initialize t by SCMPDS_6:def 3;
(Q +* (stop I)) +* (stop I) halts_on Initialize (Initialize t) by A40, FUNCT_4:25, FUNCT_4:98;
then A42: 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 A12, FUNCT_4:11
.= t . (DataLoc ((s . a),i)) by A31, FUNCT_4:11 ;
then A43: IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),1)) = succ (IC (Initialize t)) by A14, A21, SCMPDS_2:56
.= 0 + 1 by A32 ;
then A44: IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) = (card I) + 1 by A17, A42, A30, A23, A39, Th36;
A45: (Q +* (stop (for-down (a,i,n,I)))) /. (IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) = (Q +* (stop (for-down (a,i,n,I)))) . (IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by PBOOLE:143;
A46: Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)) = Comput ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t)))) by EXTPRO_1:4;
then A47: CurInstr ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) = (Q +* (stop (for-down (a,i,n,I)))) . ((card I) + 1) by A17, A42, A30, A43, A23, A39, Th36, A45
.= (for-down (a,i,n,I)) . ((card I) + 1) by A33, A38, GRFUNC_1:2
.= AddTo (a,i,(- n)) by Th62 ;
A50: Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)) = Following ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (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-down (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by A47 ;
then A51: IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) = succ (IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by SCMPDS_2:48
.= ((card I) + 1) + 1 by A44, A46, NAT_1:38
.= (card I) + (1 + 1) ;
then A52: CurInstr ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)))) = (Q +* (stop (for-down (a,i,n,I)))) . ((card I) + 2) by PBOOLE:143
.= (for-down (a,i,n,I)) . ((card I) + 2) by A38, A36, GRFUNC_1:2
.= goto (- ((card I) + 2)) by Th62 ;
A54: Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1)) = Following ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (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-down (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)))) by A52 ;
then IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))),(0 - ((card I) + 2))) by SCMPDS_2:54
.= 0 by A51, Th1 ;
then A55: Initialize (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) = Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1)) by MEMSTR_0:46;
A57: DataPart (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) = DataPart (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) by A17, A42, A30, A43, A23, A39, Th36;
then A58: (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (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 A12, A34, A40, EXTPRO_1:23 ;
then DataLoc (((Comput ((Q +* (stop (for-down (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-down (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . a = (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1))) . a by A5, A50, SCMPDS_2:48
.= s . a by A58, EXTPRO_1:4 ;
then A59: (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) . a = s . a by A54, SCMPDS_2:54;

A60: now

let x be Int_position ; :: thesis:
assume A61: x in X ; :: thesis:
(Comput ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (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 A57, SCMPDS_4:8
.= (IExec (I,Q,(Initialize t))) . x by A40, EXTPRO_1:23
.= (Initialize t) . x by A6, A61, A11, A13
.= t . x by SCMPDS_5:15
.= s . x by A10, A61 ;
then (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . x = s . x by A2, A58, A46, A50, A61, SCMPDS_2:48;
hence (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) . x = s . x by A54, SCMPDS_2:54; :: thesis:

end;

A63: (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (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 A57, SCMPDS_4:8
.= t . (DataLoc ((s . a),i)) by A26, A40, EXTPRO_1:23 ;
A64: (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1))) . (DataLoc ((s . a),i)) = (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . (DataLoc ((s . a),i)) by A54, SCMPDS_2:54
.= (t . (DataLoc ((s . a),i))) + (- n) by A58, A63, A46, A50, SCMPDS_2:48 ;
then A65: for-down (a,i,n,I) is_closed_on Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1)),Q +* (stop (for-down (a,i,n,I))) by A8, A59, A60, A28;

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 A66: ( 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 A66, NAT_1:8;

suppose A67: k <= LifeSpan ((Q +* (stop I)),(Initialize t)) ; :: thesis:

hereby :: thesis:

per cases ;

suppose k = 0 ; :: thesis:

hence IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),k)) in dom (stop (for-down (a,i,n,I))) by A35, A32, EXTPRO_1:2; :: thesis:

end;

suppose k <> 0 ; :: thesis:

then consider kn being Nat such that
A68: 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 A68, XREAL_1:29;
then kn < LifeSpan ((Q +* (stop I)),(Initialize t)) by A67, XXREAL_0:2;
then (IC (Comput ((Q +* (stop I)),(Initialize t),kn))) + 1 = IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),1)),kn)) by A17, A42, A30, A43, A23, A39, Th34;
then A69: IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),k)) = lm + 1 by A68, EXTPRO_1:4;
IC (Comput ((Q +* (stop I)),(Initialize t),kn)) in dom (stop I) by A29, SCMPDS_6:def 2;
then lm < card (stop I) by AFINSQ_1:66;
then lm < (card I) + 1 by COMPOS_1:55;
then A70: 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 A70, XXREAL_0:2;
then lm + 1 < card (stop (for-down (a,i,n,I))) by Lm4;
hence IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),k)) in dom (stop (for-down (a,i,n,I))) by A69, AFINSQ_1:66; :: thesis:

end;

suppose A71: k = (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1 ; :: thesis:

(card I) + 1 in dom (stop (for-down (a,i,n,I))) by A33, COMPOS_1:62;
hence IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),k)) in dom (stop (for-down (a,i,n,I))) by A17, A42, A30, A43, A23, A39, A46, A71, Th36; :: thesis:

end;

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

hence IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),k)) in dom (stop (for-down (a,i,n,I))) by A51, A36, COMPOS_1:62; :: thesis:

end;

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

then consider nn being Nat such that
A72: 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-down (a,i,n,I))) = (Q +* (stop (for-down (a,i,n,I)))) +* (stop (for-down (a,i,n,I))) by FUNCT_4:93;
then Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),k) = Comput (((Q +* (stop (for-down (a,i,n,I)))) +* (stop (for-down (a,i,n,I)))),(Initialize (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),((((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + 1)))),nn) by A55, A72, EXTPRO_1:4;
hence IC (Comput ((Q +* (stop (for-down (a,i,n,I)))),(Initialize t),k)) in dom (stop (for-down (a,i,n,I))) by A65, SCMPDS_6:def 2; :: thesis:

end;

end;

reconsider n = s . (DataLoc ((s . a),i)) as Element of NAT by A1, INT_1:3;
A73: S₁[ 0 ] by Th63;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A73, A7);
then A74: 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,P & for-down (a,i,n,I) is_halting_on s,P ) by A74; :: thesis: