let P be Instruction-Sequence of SCMPDS; :: thesis:
let s be 0 -started State of SCMPDS; :: thesis:
let I be halt-free shiftable Program of ; :: thesis:
let a be Int_position; :: thesis:
let i be Integer; :: thesis:
let n be 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 s1 = s;
set P1 = P +* (stop (for-up (a,i,n,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:
set s4 = Comput ((P +* (stop (for-up (a,i,n,I)))),s,1);
set P4 = P +* (stop (for-up (a,i,n,I)));
A2: IC s = 0 by MEMSTR_0:def 11;
set m0 = LifeSpan ((P +* (stop (for-up (a,i,n,I)))),s);
set l2 = (card I) + 2;
set sI = s;
set PI = P +* (stop I);
set m1 = (LifeSpan ((P +* (stop I)),s)) + 3;
set J = I ';' (AddTo (a,i,n));
set sJ = s;
set PJ = P +* (stop (I ';' (AddTo (a,i,n))));
set s2 = Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s));
set P2 = P +* (stop (for-up (a,i,n,I)));
set m2 = LifeSpan ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))));
set Es = IExec ((I ';' (AddTo (a,i,n))),P,s);
set bj = DataLoc (((IExec ((I ';' (AddTo (a,i,n))),P,s)) . a),i);
A3: stop I c= P +* (stop I) by FUNCT_4:25;
A4: for-up (a,i,n,I) = ((a,i) >=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,n))) ';' (goto (- ((card I) + 2)))) by Th2;
set mI = LifeSpan ((P +* (stop I)),s);
set s5 = Comput ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)));
set P5 = P +* (stop (for-up (a,i,n,I)));
set l1 = (card I) + 1;
set m3 = (LifeSpan ((P +* (stop I)),s)) + 1;
set s6 = Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1));
set P6 = P +* (stop (for-up (a,i,n,I)));
set s7 = Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1));
set P7 = P +* (stop (for-up (a,i,n,I)));
A5: stop (for-up (a,i,n,I)) c= P +* (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 A6: for-up (a,i,n,I) c= P +* (stop (for-up (a,i,n,I))) by A5, XBOOLE_1:1;
Shift (I,1) c= for-up (a,i,n,I) by Lm3;
then A7: Shift (I,1) c= P +* (stop (for-up (a,i,n,I))) by A6, XBOOLE_1:1;
(card I) + 2 < (card I) + 3 by XREAL_1:6;
then A8: (card I) + 2 in dom (for-up (a,i,n,I)) by Th31;
set m5 = (((LifeSpan ((P +* (stop I)),s)) + 1) + 1) + 1;
set s8 = Comput ((P +* (stop (for-up (a,i,n,I)))),s,((((LifeSpan ((P +* (stop I)),s)) + 1) + 1) + 1));
(card I) + 1 < (card I) + 3 by XREAL_1:6;
then A9: (card I) + 1 in dom (for-up (a,i,n,I)) by Th31;
assume A10: not DataLoc ((s . a),i) in X ; :: thesis:
assume A11: n > 0 ; :: thesis:
assume A12: a <> DataLoc ((s . a),i) ; :: thesis:
A13: Initialize s = s by MEMSTR_0:44;
assume A14: 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:
then for-up (a,i,n,I) is_halting_on s,P by A1, A10, A11, A12, Th37;
then A15: P +* (stop (for-up (a,i,n,I))) halts_on s by A13, SCMPDS_6:def 3;
A16: for x being Int_position st x in X holds
s . x = s . x ;
then A17: (IExec (I,P,s)) . (DataLoc ((s . a),i)) = s . (DataLoc ((s . a),i)) by A14, A13;
A18: (IExec (I,P,s)) . a = s . a by A14, A16, A13;
A19: DataLoc ((s . a),i) = DataLoc (((IExec (I,P,s)) . a),i) by A14, A16, A13;
A20: (IExec (I,P,s)) . a = s . a by A14, A16, A13;
A21: Comput ((P +* (stop (for-up (a,i,n,I)))),s,(0 + 1)) = Following ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,0))) by EXTPRO_1:3
.= Exec (((a,i) >=0_goto ((card I) + 3)),s) by A4, A13, SCMPDS_6:11 ;
A22: IC (Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)) = 0 + 1 by A2, A1, A21, SCMPDS_2:57;
for a being Int_position holds s . a = (Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)) . a by A21, SCMPDS_2:57;
then A23: DataPart s = DataPart (Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)) by SCMPDS_4:8;
A24: I is_halting_on s,P by A14, A16;
then A25: P +* (stop I) halts_on s by A13, SCMPDS_6:def 3;
(P +* (stop I)) +* (stop I) halts_on s by A25;
then A26: I is_halting_on s,P +* (stop I) by A13, SCMPDS_6:def 3;
A27: I is_closed_on s,P by A14, A16;
then A28: (IExec ((I ';' (AddTo (a,i,n))),P,s)) . (DataLoc ((s . a),i)) = (Exec ((AddTo (a,i,n)),(IExec (I,P,s)))) . (DataLoc ((s . a),i)) by A24, Th29
.= ((IExec (I,P,s)) . (DataLoc ((s . a),i))) + n by A18, SCMPDS_2:48
.= (s . (DataLoc ((s . a),i))) + n by A14, A16, A13 ;
A29: I is_closed_on s,P +* (stop I) by A14, A16;
then A30: IC (Comput ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) = (card I) + 1 by A3, A26, A22, A23, A7, Th16;
A31: (P +* (stop (for-up (a,i,n,I)))) /. (IC (Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = (P +* (stop (for-up (a,i,n,I)))) . (IC (Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by PBOOLE:143;
A32: Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)) = Comput ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)),(LifeSpan ((P +* (stop I)),s))) by EXTPRO_1:4;
then A33: CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = (P +* (stop (for-up (a,i,n,I)))) . ((card I) + 1) by A3, A26, A29, A22, A23, A7, Th16, A31
.= (for-up (a,i,n,I)) . ((card I) + 1) by A9, A6, GRFUNC_1:2
.= AddTo (a,i,n) by Th32 ;
A34: DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) by A3, A26, A29, A22, A23, A7, Th16;
then A35: (Comput ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) . a = (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) . a by SCMPDS_4:8
.= s . a by A20, A25, EXTPRO_1:23 ;
A36: (IExec ((I ';' (AddTo (a,i,n))),P,s)) . a = (Exec ((AddTo (a,i,n)),(IExec (I,P,s)))) . a by A27, A24, Th29
.= s . a by A12, A18, SCMPDS_2:48 ;

now :: thesis:

per cases ) . (DataLoc (((IExec ((I ';' (AddTo (a,i,n))),P,s)) . a),i)) >= 0 or (IExec ((I ';' (AddTo (a,i,n))),P,s)) . (DataLoc (((IExec ((I ';' (AddTo (a,i,n))),P,s)) . a),i)) < 0 ) ;

suppose (IExec ((I ';' (AddTo (a,i,n))),P,s)) . (DataLoc (((IExec ((I ';' (AddTo (a,i,n))),P,s)) . a),i)) >= 0 ; :: thesis:

hence for-up (a,i,n,I) is_halting_on IExec ((I ';' (AddTo (a,i,n))),P,s),P by Th33; :: thesis:

end;

suppose A37: (IExec ((I ';' (AddTo (a,i,n))),P,s)) . (DataLoc (((IExec ((I ';' (AddTo (a,i,n))),P,s)) . a),i)) < 0 ; :: thesis:

now :: thesis:

let t be State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
assume that
A38: for x being Int_position st x in X holds
t . x = (IExec ((I ';' (AddTo (a,i,n))),P,s)) . x and
A39: t . a = (IExec ((I ';' (AddTo (a,i,n))),P,s)) . a ; :: thesis:

A40: now :: thesis:

let x be Int_position; :: thesis:
assume A41: x in X ; :: thesis:
hence t . x = (IExec ((I ';' (AddTo (a,i,n))),P,s)) . x by A38
.= (Exec ((AddTo (a,i,n)),(IExec (I,P,s)))) . x by A27, A24, Th29
.= (IExec (I,P,s)) . x by A10, A18, A41, SCMPDS_2:48
.= s . x by A14, A16, A41, A13 ;
:: thesis:

end;

hence (IExec (I,Q,(Initialize t))) . a = t . a by A14, A36, A39; :: thesis:
thus (IExec (I,Q,(Initialize t))) . (DataLoc (((IExec ((I ';' (AddTo (a,i,n))),P,s)) . a),i)) = t . (DataLoc (((IExec ((I ';' (AddTo (a,i,n))),P,s)) . a),i)) by A14, A36, A39, A40; :: thesis:
thus ( 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 ) ) by A14, A36, A39, A40; :: thesis:

end;

hence for-up (a,i,n,I) is_halting_on IExec ((I ';' (AddTo (a,i,n))),P,s),P by A10, A11, A12, A36, A37, Th37; :: thesis:

end;

then A42: P +* (stop (for-up (a,i,n,I))) halts_on Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s)) by SCMPDS_6:def 3;
A43: Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1)) = Following ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by EXTPRO_1:3
.= Exec ((AddTo (a,i,n)),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by A33 ;
then A44: IC (Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) = (IC (Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) + 1 by SCMPDS_2:48
.= (card I) + (1 + 1) by A30, A32 ;
then A45: CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1)))) = (P +* (stop (for-up (a,i,n,I)))) . ((card I) + 2) by PBOOLE:143
.= (for-up (a,i,n,I)) . ((card I) + 2) by A6, A8, GRFUNC_1:2
.= goto (- ((card I) + 2)) by Th32 ;
A46: Comput ((P +* (stop (for-up (a,i,n,I)))),s,((((LifeSpan ((P +* (stop I)),s)) + 1) + 1) + 1)) = Following ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 2))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1)))) by A45 ;
then IC (Comput ((P +* (stop (for-up (a,i,n,I)))),s,((((LifeSpan ((P +* (stop I)),s)) + 1) + 1) + 1))) = ICplusConst ((Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))),(0 - ((card I) + 2))) by SCMPDS_2:54
.= 0 by A44, Th1 ;
then A47: IC (Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))) = IC (Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 3))) by MEMSTR_0:def 11;
A48: (Comput ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) . (DataLoc ((s . a),i)) = (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) . (DataLoc ((s . a),i)) by A34, SCMPDS_4:8
.= s . (DataLoc ((s . a),i)) by A17, A25, EXTPRO_1:23 ;
A49: (Comput ((P +* (stop (for-up (a,i,n,I)))),s,((((LifeSpan ((P +* (stop I)),s)) + 1) + 1) + 1))) . (DataLoc ((s . a),i)) = (Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) . (DataLoc ((s . a),i)) by A46, SCMPDS_2:54
.= (s . (DataLoc ((s . a),i))) + n by A35, A48, A32, A43, SCMPDS_2:48 ;

now :: thesis:

let x be Int_position; :: thesis:
A50: not x in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A51: (Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))) . x = (IExec ((I ';' (AddTo (a,i,n))),P,s)) . x by FUNCT_4:11;

per cases ;

suppose x = DataLoc ((s . a),i) ; :: thesis:

hence (Comput ((P +* (stop (for-up (a,i,n,I)))),s,((((LifeSpan ((P +* (stop I)),s)) + 1) + 1) + 1))) . x = (Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))) . x by A49, A28, A50, FUNCT_4:11; :: thesis:

end;

suppose A52: x <> DataLoc ((s . a),i) ; :: thesis:

A53: (Comput ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) . x = (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) . x by A34, SCMPDS_4:8
.= (IExec (I,P,s)) . x by A25, EXTPRO_1:23 ;
A54: (Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) . x = (Comput ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) . x by A35, A32, A43, A52, SCMPDS_2:48;
(IExec ((I ';' (AddTo (a,i,n))),P,s)) . x = (Exec ((AddTo (a,i,n)),(IExec (I,P,s)))) . x by A27, A24, Th29
.= (IExec (I,P,s)) . x by A19, A52, SCMPDS_2:48 ;
hence (Comput ((P +* (stop (for-up (a,i,n,I)))),s,((((LifeSpan ((P +* (stop I)),s)) + 1) + 1) + 1))) . x = (Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))) . x by A46, A51, A53, A54, SCMPDS_2:54; :: thesis:

end;

then A55: DataPart (Comput ((P +* (stop (for-up (a,i,n,I)))),s,((((LifeSpan ((P +* (stop I)),s)) + 1) + 1) + 1))) = DataPart (Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))) by SCMPDS_4:8;
A56: Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 3)) = Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s)) by A55, A47, MEMSTR_0:78;
then CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 3)))) = (a,i) >=0_goto ((card I) + 3) by A4, SCMPDS_6:11;
then LifeSpan ((P +* (stop (for-up (a,i,n,I)))),s) > (LifeSpan ((P +* (stop I)),s)) + 3 by A15, EXTPRO_1:36;
then consider nn being Nat such that
A57: LifeSpan ((P +* (stop (for-up (a,i,n,I)))),s) = ((LifeSpan ((P +* (stop I)),s)) + 3) + nn by NAT_1:10;
reconsider nn = nn as Nat ;
Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 3) + (LifeSpan ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))))))) = Comput ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))),(LifeSpan ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s)))))) by A56, EXTPRO_1:4;
then CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 3) + (LifeSpan ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))))))))) = halt SCMPDS by A42, EXTPRO_1:def 15;
then ((LifeSpan ((P +* (stop I)),s)) + 3) + (LifeSpan ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))))) >= LifeSpan ((P +* (stop (for-up (a,i,n,I)))),s) by A15, EXTPRO_1:def 15;
then A58: LifeSpan ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s)))) >= nn by A57, XREAL_1:6;
A59: Comput ((P +* (stop (for-up (a,i,n,I)))),s,(LifeSpan ((P +* (stop (for-up (a,i,n,I)))),s))) = Comput ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))),nn) by A56, A57, EXTPRO_1:4;
then CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))),nn))) = halt SCMPDS by A15, EXTPRO_1:def 15;
then nn >= LifeSpan ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s)))) by A42, EXTPRO_1:def 15;
then nn = LifeSpan ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s)))) by A58, XXREAL_0:1;
then Result ((P +* (stop (for-up (a,i,n,I)))),s) = Comput ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s))),(LifeSpan ((P +* (stop (for-up (a,i,n,I)))),(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s)))))) by A15, A59, EXTPRO_1:23;
hence IExec ((for-up (a,i,n,I)),P,s) = IExec ((for-up (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,n))),P,s)))) by A42, EXTPRO_1:23; :: thesis: