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 SCMPDS; :: thesis:
let a be Int_position; :: thesis:
let i, c be Integer; :: thesis:
let X, Y be set ; :: thesis:
let f be Function of (product (the_Values_of SCMPDS)),NAT; :: thesis:
A1: Initialize s = s by MEMSTR_0:44;
set b = DataLoc ((s . a),i);
set WHL = while>0 (a,i,I);
set pWHL = stop (while>0 (a,i,I));
set P1 = P +* (stop (while>0 (a,i,I)));
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
assume A2: s . (DataLoc ((s . a),i)) > 0 ; :: thesis:
set s4 = Comput ((P +* (stop (while>0 (a,i,I)))),s,1);
set P4 = P +* (stop (while>0 (a,i,I)));
A3: IC s = 0 by MEMSTR_0:def 11;
A4: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
A5: Comput ((P +* (stop (while>0 (a,i,I)))),s,(0 + 1)) = Following ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (while>0 (a,i,I)))),s)
.= Exec (((a,i) <=0_goto ((card I) + 2)),s) by A4, A1, SCMPDS_6:11 ;
set m0 = LifeSpan ((P +* (stop (while>0 (a,i,I)))),s);
set Es = IExec (I,P,s);
set bj = DataLoc (((Initialize (IExec (I,P,s))) . a),i);
set EP = P;
assume A6: for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ; :: thesis:
assume A7: for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ; :: thesis:
assume A8: 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 >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ; :: thesis:
then while>0 (a,i,I) is_halting_on s,P by A6, A7, Th22;
then A9: P +* (stop (while>0 (a,i,I))) halts_on s by A1, SCMPDS_6:def 3;
A10: for x being Int_position st x in Y holds
s . x = s . x ;
A11: DataLoc (((Initialize (IExec (I,P,s))) . a),i) = DataLoc (((IExec (I,P,s)) . a),i) by SCMPDS_5:15
.= DataLoc ((s . a),i) by A2, A7, A8, A10 ;
set PI = P +* (stop I);
set m1 = (LifeSpan ((P +* (stop I)),s)) + 2;
set s2 = Initialize (IExec (I,P,s));
set P2 = P +* (stop (while>0 (a,i,I)));
set m2 = LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))));
A12: stop I c= P +* (stop I) by FUNCT_4:25;
A13: (Initialize (IExec (I,P,s))) . a = (IExec (I,P,s)) . a by SCMPDS_5:15
.= s . a by A2, A7, A8, A10 ;

A14: now :: thesis:

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
assume that
A15: for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) and
A16: for x being Int_position st x in Y holds
t . x = (Initialize (IExec (I,P,s))) . x and
A17: t . a = (Initialize (IExec (I,P,s))) . a and
A18: t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > 0 ; :: thesis:

A19: now :: thesis:

let x be Int_position; :: thesis:
assume A20: x in Y ; :: thesis:
hence t . x = (Initialize (IExec (I,P,s))) . x by A16
.= (IExec (I,P,s)) . x by SCMPDS_5:15
.= s . x by A2, A7, A8, A10, A20 ;
:: thesis:

end;

thus (Initialize (IExec (I,Q,t))) . a = (IExec (I,Q,t)) . a by SCMPDS_5:15
.= t . a by A8, A13, A15, A17, A18, A19 ; :: thesis:
A21: t . a = (IExec (I,P,s)) . a by A17, SCMPDS_5:15
.= s . a by A2, A7, A8, A10 ;
hence ( I is_closed_on t,Q & I is_halting_on t,Q ) by A8, A15, A17, A18, A19; :: thesis:
thus f . (Initialize (IExec (I,Q,t))) < f . t by A8, A15, A17, A18, A19, A21; :: thesis:
thus for x being Int_position st x in X holds
(Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) :: thesis:

proof

let x be Int_position; :: thesis:
( (Initialize (IExec (I,Q,t))) . x = (IExec (I,Q,t)) . x & (Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) = (IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) ) by SCMPDS_5:15;
hence ( x in X implies (Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) by A8, A13, A15, A17, A18, A19; :: thesis:

end;

thus for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x by A8, A15, A17, A18, A19, A21; :: thesis:

end;

A22: for x being Int_position st x in X holds
(Initialize (IExec (I,P,s))) . x >= c + ((Initialize (IExec (I,P,s))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)))

proof

let x be Int_position; :: thesis:
A23: (Initialize (IExec (I,P,s))) . x = (IExec (I,P,s)) . x by SCMPDS_5:15;
A24: (Initialize (IExec (I,P,s))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) = (IExec (I,P,s)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) by SCMPDS_5:15;
assume x in X ; :: thesis:
hence (Initialize (IExec (I,P,s))) . x >= c + ((Initialize (IExec (I,P,s))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) by A2, A7, A8, A10, A13, A23, A24; :: thesis:

end;

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 >= c + (t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = (Initialize (IExec (I,P,s))) . x ) & t . a = (Initialize (IExec (I,P,s))) . a & t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )

proof

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
assume that
A25: for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) and
A26: for x being Int_position st x in Y holds
t . x = (Initialize (IExec (I,P,s))) . x and
A27: t . a = (Initialize (IExec (I,P,s))) . a and
A28: t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > 0 ; :: thesis:
thus (IExec (I,Q,t)) . a = (Initialize (IExec (I,Q,t))) . a by SCMPDS_5:15
.= t . a by A14, A25, A26, A27, A28 ; :: thesis:
thus ( I is_closed_on t,Q & I is_halting_on t,Q ) by A14, A25, A26, A27, A28; :: thesis:
thus f . (Initialize (IExec (I,Q,t))) < f . t by A14, A25, A26, A27, A28; :: thesis:
thus for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) :: thesis:

proof

let x be Int_position; :: thesis:
( (IExec (I,Q,t)) . x = (Initialize (IExec (I,Q,t))) . x & (IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) = (Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) ) by SCMPDS_5:15;
hence ( x in X implies (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) by A14, A25, A26, A27, A28; :: thesis:

end;

thus for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x by A14, A25, A26, A27, A28; :: thesis:

end;

then while>0 (a,i,I) is_halting_on Initialize (IExec (I,P,s)),P by A6, A11, Th22, A22;
then A29: P +* (stop (while>0 (a,i,I))) halts_on Initialize (Initialize (IExec (I,P,s))) by SCMPDS_6:def 3;
for a being Int_position holds s . a = (Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) . a by A5, SCMPDS_2:56;
then A30: DataPart s = DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) by SCMPDS_4:8;
set mI = LifeSpan ((P +* (stop I)),s);
set s5 = Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)));
set P5 = P +* (stop (while>0 (a,i,I)));
set l1 = (card I) + 1;
A31: while>0 (a,i,I) c= stop (while>0 (a,i,I)) by AFINSQ_1:74;
stop (while>0 (a,i,I)) c= P +* (stop (while>0 (a,i,I))) by FUNCT_4:25;
then A32: while>0 (a,i,I) c= P +* (stop (while>0 (a,i,I))) by A31, XBOOLE_1:1;
set m3 = (LifeSpan ((P +* (stop I)),s)) + 1;
set s7 = Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1));
set P7 = P +* (stop (while>0 (a,i,I)));
set s6 = Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1));
set P6 = P +* (stop (while>0 (a,i,I)));
(card I) + 1 < (card I) + 2 by XREAL_1:6;
then A33: (card I) + 1 in dom (while>0 (a,i,I)) by Th16;
Shift (I,1) c= while>0 (a,i,I) by Lm4;
then A34: Shift (I,1) c= P +* (stop (while>0 (a,i,I))) by A32, XBOOLE_1:1;
I is_halting_on s,P by A2, A7, A8, A10;
then A35: P +* (stop I) halts_on s by A1, SCMPDS_6:def 3;
(P +* (stop I)) +* (stop I) halts_on s by A35;
then A36: I is_halting_on s,P +* (stop I) by A1, SCMPDS_6:def 3;
A37: IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) = (IC s) + 1 by A2, A5, SCMPDS_2:56
.= 0 + 1 by A3 ;
A38: I is_closed_on s,P +* (stop I) by A2, A7, A8, A10;
then A39: IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) = (card I) + 1 by A12, A36, A37, A30, A34, SCMPDS_7:18;
A40: (P +* (stop (while>0 (a,i,I)))) /. (IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = (P +* (stop (while>0 (a,i,I)))) . (IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by PBOOLE:143;
A41: Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)) = Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s))) by EXTPRO_1:4;
then A42: CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = (P +* (stop (while>0 (a,i,I)))) . ((card I) + 1) by A12, A36, A38, A37, A30, A34, A40, SCMPDS_7:18
.= (while>0 (a,i,I)) . ((card I) + 1) by A33, A32, GRFUNC_1:2
.= goto (- ((card I) + 1)) by Th17 ;
A43: Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1)) = Following ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 1))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by A42 ;
then IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) = ICplusConst ((Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1))),(0 - ((card I) + 1))) by SCMPDS_2:54
.= 0 by A39, A41, SCMPDS_7:1 ;
then A44: IC (Initialize (IExec (I,P,s))) = IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 2))) by MEMSTR_0:def 11;
A45: DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) by A12, A36, A38, A37, A30, A34, SCMPDS_7:18;

now :: thesis:

let x be Int_position; :: thesis:
A46: not x in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
(Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) . x = (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) . x by A45, SCMPDS_4:8
.= (Result ((P +* (stop I)),s)) . x by A35, EXTPRO_1:23
.= (IExec (I,P,s)) . x by SCMPDS_4:def 5 ;
hence (Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) . x = (IExec (I,P,s)) . x by A41, A43, SCMPDS_2:54
.= (Initialize (IExec (I,P,s))) . x by A46, FUNCT_4:11 ;
:: thesis:

end;

then A47: DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) = DataPart (Initialize (IExec (I,P,s))) by SCMPDS_4:8;
A48: Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 2)) = Initialize (IExec (I,P,s)) by A47, A44, MEMSTR_0:78;
then CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 2)))) = (a,i) <=0_goto ((card I) + 2) by A4, SCMPDS_6:11;
then LifeSpan ((P +* (stop (while>0 (a,i,I)))),s) > (LifeSpan ((P +* (stop I)),s)) + 2 by A9, EXTPRO_1:36, SCMPDS_6:17;
then consider nn being Nat such that
A49: LifeSpan ((P +* (stop (while>0 (a,i,I)))),s) = ((LifeSpan ((P +* (stop I)),s)) + 2) + nn by NAT_1:10;
reconsider nn = nn as Nat ;
Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 2) + (LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))))))) = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))),(LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))))) by A48, EXTPRO_1:4;
then CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 2) + (LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))))))))) = halt SCMPDS by A29, EXTPRO_1:def 15;
then ((LifeSpan ((P +* (stop I)),s)) + 2) + (LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))))) >= LifeSpan ((P +* (stop (while>0 (a,i,I)))),s) by A9, EXTPRO_1:def 15;
then A50: LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))) >= nn by A49, XREAL_1:6;
A51: Comput ((P +* (stop (while>0 (a,i,I)))),s,(LifeSpan ((P +* (stop (while>0 (a,i,I)))),s))) = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))),nn) by A48, A49, EXTPRO_1:4;
then CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))),nn))) = halt SCMPDS by A9, EXTPRO_1:def 15;
then nn >= LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))) by A29, EXTPRO_1:def 15;
then nn = LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))) by A50, XXREAL_0:1;
then Result ((P +* (stop (while>0 (a,i,I)))),s) = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))),(LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))))) by A9, A51, EXTPRO_1:23;
hence IExec ((while>0 (a,i,I)),P,s) = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))),(LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))))) by SCMPDS_4:def 5
.= Result ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))) by A29, EXTPRO_1:23
.= IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) by SCMPDS_4:def 5 ;
:: thesis: