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 be Integer; :: thesis:
let X be set ; :: thesis:
let f be Function of (product the Object-Kind of SCMPDS),NAT; :: thesis:
set b = DataLoc ((s . a),i);
set WHL = while<0 (a,i,I);
set pWHL = stop (while<0 (a,i,I));
set pI = stop I;
set i1 = (a,i) >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
defpred S₁[ Element of NAT ] means for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st f . t <= $1 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a holds
( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q );
assume A2: for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ; :: thesis:
assume A3: 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 & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ; :: thesis:
A4: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis:
assume A5: S₁[k] ; :: thesis:

now

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
T: Initialize t = t by MEMSTR_0:44;
assume A6: f . t <= k + 1 ; :: thesis:
assume A7: for x being Int_position st x in X holds
t . x = s . x ; :: thesis:
assume A8: t . a = s . a ; :: thesis:

per cases ;

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

hence ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) by A8, Th9; :: thesis:

end;

suppose A9: t . (DataLoc ((s . a),i)) < 0 ; :: thesis:

A13: 0 in dom (stop (while<0 (a,i,I))) by COMPOS_1:36;
A15: while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
A16: f . (Initialize (IExec (I,Q,t))) < f . t by A3, A7, A8, A9;
set t2 = t;
set Q2 = Q +* (stop I);
set t3 = t;
set Q3 = Q +* (stop (while<0 (a,i,I)));
set t4 = Comput ((Q +* (stop (while<0 (a,i,I)))),t,1);
set Q4 = Q +* (stop (while<0 (a,i,I)));
A20: stop I c= Q +* (stop I) by FUNCT_4:25;
A21: Comput ((Q +* (stop (while<0 (a,i,I)))),t,(0 + 1)) = Following ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,0))) by EXTPRO_1:3
.= Following ((Q +* (stop (while<0 (a,i,I)))),t) by EXTPRO_1:2
.= Exec (((a,i) >=0_goto ((card I) + 2)),t) by A15, SCMPDS_6:11, T ;
for a being Int_position holds t . a = (Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)) . a by A21, SCMPDS_2:57;
then A23: DataPart t = DataPart (Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)) by SCMPDS_4:8;
XX: while<0 (a,i,I) c= stop (while<0 (a,i,I)) by AFINSQ_1:74;
stop (while<0 (a,i,I)) c= Q +* (stop (while<0 (a,i,I))) by FUNCT_4:25;
then A24: while<0 (a,i,I) c= Q +* (stop (while<0 (a,i,I))) by XX, XBOOLE_1:1;
Shift (I,1) c= while<0 (a,i,I) by Lm2;
then A25: Shift (I,1) c= Q +* (stop (while<0 (a,i,I))) by A24, XBOOLE_1:1;
set m2 = LifeSpan ((Q +* (stop I)),t);
set t5 = Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)));
set Q5 = Q +* (stop (while<0 (a,i,I)));
set l1 = (card I) + 1;
A26: IC t = 0 by MEMSTR_0:def 8;
set m3 = (LifeSpan ((Q +* (stop I)),t)) + 1;
set t6 = Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1));
set Q6 = Q +* (stop (while<0 (a,i,I)));
set t7 = Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1));
set Q7 = Q +* (stop (while<0 (a,i,I)));
(card I) + 1 < (card I) + 2 by XREAL_1:6;
then A27: (card I) + 1 in dom (while<0 (a,i,I)) by Th7;
A28: I is_closed_on t,Q by A3, A7, A8, A9;
A29: I is_closed_on t,Q +* (stop I) by A3, A7, A8, A9;
I is_halting_on t,Q by A3, A7, A8, A9;
then A30: Q +* (stop I) halts_on t by SCMPDS_6:def 3, T;
(Q +* (stop I)) +* (stop I) halts_on t by A30, FUNCT_4:25, FUNCT_4:98;
then A32: I is_halting_on t,Q +* (stop I) by SCMPDS_6:def 3, T;
A33: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)) = succ (IC t) by A9, A21, SCMPDS_2:57, A8
.= 0 + 1 by A26 ;
then A34: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) = (card I) + 1 by A20, A32, A29, A23, A25, SCMPDS_7:18;
A35: (Q +* (stop (while<0 (a,i,I)))) /. (IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) = (Q +* (stop (while<0 (a,i,I)))) . (IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) by PBOOLE:143;
A36: Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)) = Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t))) by EXTPRO_1:4;
then A37: CurInstr ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) = (Q +* (stop (while<0 (a,i,I)))) . ((card I) + 1) by A20, A32, A29, A33, A23, A25, A35, SCMPDS_7:18
.= (while<0 (a,i,I)) . ((card I) + 1) by A27, A24, GRFUNC_1:2
.= goto (- ((card I) + 1)) by Th8 ;
A38: DataPart (Comput ((Q +* (stop I)),t,(LifeSpan ((Q +* (stop I)),t)))) = DataPart (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) by A20, A32, A29, A33, A23, A25, SCMPDS_7:18;
then A39: DataPart (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) = DataPart (Result ((Q +* (stop I)),t)) by A30, EXTPRO_1:23
.= DataPart (IExec (I,Q,t)) by SCMPDS_4:def 5 ;
A41: Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)) = Following ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 1))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) by A37 ;
then IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1))),(0 - ((card I) + 1))) by SCMPDS_2:54
.= 0 by A34, A36, SCMPDS_7:1 ;
then A42: Initialize (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) = Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)) by MEMSTR_0:46;
A43: IExec (I,Q,t) = Result ((Q +* (stop I)),t) by SCMPDS_4:def 5;

A44: now

let x be Int_position ; :: thesis:
assume A45: x in X ; :: thesis:
(Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) . x = (Comput ((Q +* (stop I)),t,(LifeSpan ((Q +* (stop I)),t)))) . x by A38, SCMPDS_4:8
.= (Result ((Q +* (stop I)),t)) . x by A30, EXTPRO_1:23
.= (IExec (I,Q,t)) . x by SCMPDS_4:def 5
.= t . x by A3, A7, A8, A9, A45
.= s . x by A7, A45 ;
hence (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x = s . x by A36, A41, SCMPDS_2:54; :: thesis:

end;

InsCode (goto (- ((card I) + 1))) = 0 by SCMPDS_2:12;
then InsCode (goto (- ((card I) + 1))) in {0,4,5,6} by ENUMSET1:def 2;
then A47: Initialize (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) = Initialize (Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1))) by A41, Th3
.= Initialize (IExec (I,Q,t)) by A39, A36, MEMSTR_0:80 ;

A48: now

assume A49: f . (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) > k ; :: thesis:
f . (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) < k + 1 by A6, A16, A47, XXREAL_0:2, A42;
hence contradiction by A49, INT_1:7; :: thesis:

end;

A50: (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) . a = (Comput ((Q +* (stop I)),t,(LifeSpan ((Q +* (stop I)),t)))) . a by A38, SCMPDS_4:8
.= (Result ((Q +* (stop I)),t)) . a by A30, EXTPRO_1:23
.= s . a by A8, A3, A7, A9, A43 ;
A52: (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . a = (Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1))) . a by A41, SCMPDS_2:54
.= s . a by A50, EXTPRO_1:4 ;
then A53: while<0 (a,i,I) is_closed_on Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)),Q +* (stop (while<0 (a,i,I))) by A5, A44, A48, A42;

now

let k be Element of NAT ; :: thesis:

per cases ;

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

then A54: k <= (LifeSpan ((Q +* (stop I)),t)) + 1 by INT_1:7;

hereby :: thesis:

per cases by A54, NAT_1:8;

suppose A55: k <= LifeSpan ((Q +* (stop I)),t) ; :: thesis:

hereby :: thesis:

per cases ;

suppose k = 0 ; :: thesis:

hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I))) by A13, A26, EXTPRO_1:2; :: thesis:

end;

suppose k <> 0 ; :: thesis:

then consider kn being Nat such that
A56: 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)),t,kn)) as Element of NAT ;
kn < k by A56, XREAL_1:29;
then kn < LifeSpan ((Q +* (stop I)),t) by A55, XXREAL_0:2;
then (IC (Comput ((Q +* (stop I)),t,kn))) + 1 = IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),kn)) by A20, A32, A29, A33, A23, A25, SCMPDS_7:16;
then A58: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) = lm + 1 by A56, EXTPRO_1:4;
IC (Comput ((Q +* (stop I)),t,kn)) in dom (stop I) by A28, SCMPDS_6:def 2, T;
then lm < card (stop I) by AFINSQ_1:66;
then lm < (card I) + 1 by COMPOS_1:55;
then A59: lm + 1 <= (card I) + 1 by INT_1:7;
(card I) + 1 < (card I) + 3 by XREAL_1:6;
then lm + 1 < (card I) + 3 by A59, XXREAL_0:2;
then lm + 1 < card (stop (while<0 (a,i,I))) by Lm1;
hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I))) by A58, AFINSQ_1:66; :: thesis:

end;

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

(card I) + 1 in dom (stop (while<0 (a,i,I))) by A27, COMPOS_1:62;
hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I))) by A20, A32, A29, A33, A23, A25, A36, A60, SCMPDS_7:18; :: thesis:

end;

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

then consider nn being Nat such that
A61: k = (((LifeSpan ((Q +* (stop I)),t)) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 12;
Q +* (stop (while<0 (a,i,I))) = (Q +* (stop (while<0 (a,i,I)))) +* (stop (while<0 (a,i,I))) by FUNCT_4:93;
then Comput ((Q +* (stop (while<0 (a,i,I)))),t,k) = Comput (((Q +* (stop (while<0 (a,i,I)))) +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))),nn) by A61, EXTPRO_1:4;
hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I))) by A53, SCMPDS_6:def 2, A42; :: thesis:

end;

end;

hence S₁[k + 1] ; :: thesis:

end;

set n = f . s;
A62: S₁[ 0 ]

proof

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
assume f . t <= 0 ; :: thesis:
then f . t = 0 ;
then A63: t . (DataLoc ((s . a),i)) >= 0 by A2;
assume for x being Int_position st x in X holds
t . x = s . x ; :: thesis:
assume t . a = s . a ; :: thesis:
hence ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) by A63, Th9; :: thesis:

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A62, A4);
then A64: S₁[f . s] ;
for x being Int_position st x in X holds
s . x = s . x ;
hence ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) by A64; :: thesis: