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_Values_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[ 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 A1: for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ; :: thesis:
assume A2: 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:
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis:
assume A4: S₁[k] ; :: thesis:

now :: thesis:

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
A5: 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, Th7; :: thesis:

end;

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

A10: 0 in dom (stop (while<0 (a,i,I))) by COMPOS_1:36;
A11: while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
A12: f . (Initialize (IExec (I,Q,t))) < f . t by A2, 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)));
A13: stop I c= Q +* (stop I) by FUNCT_4:25;
A14: 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)
.= Exec (((a,i) >=0_goto ((card I) + 2)),t) by A11, A5, SCMPDS_6:11 ;
for a being Int_position holds t . a = (Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)) . a by A14, SCMPDS_2:57;
then A15: DataPart t = DataPart (Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)) by SCMPDS_4:8;
A16: 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 A17: while<0 (a,i,I) c= Q +* (stop (while<0 (a,i,I))) by A16, XBOOLE_1:1;
Shift (I,1) c= while<0 (a,i,I) by Lm2;
then A18: Shift (I,1) c= Q +* (stop (while<0 (a,i,I))) by A17, 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;
A19: IC t = 0 by MEMSTR_0:def 11;
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 A20: (card I) + 1 in dom (while<0 (a,i,I)) by Th5;
A21: I is_closed_on t,Q by A2, A7, A8, A9;
A22: I is_closed_on t,Q +* (stop I) by A2, A7, A8, A9;
I is_halting_on t,Q by A2, A7, A8, A9;
then A23: Q +* (stop I) halts_on t by A5, SCMPDS_6:def 3;
(Q +* (stop I)) +* (stop I) halts_on t by A23;
then A24: I is_halting_on t,Q +* (stop I) by A5, SCMPDS_6:def 3;
A25: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)) = (IC t) + 1 by A9, A14, A8, SCMPDS_2:57
.= 0 + 1 by A19 ;
then A26: 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 A13, A24, A22, A15, A18, SCMPDS_7:18;
A27: (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;
A28: 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 A29: 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 A13, A24, A22, A25, A15, A18, A27, SCMPDS_7:18
.= (while<0 (a,i,I)) . ((card I) + 1) by A20, A17, GRFUNC_1:2
.= goto (- ((card I) + 1)) by Th6 ;
A30: 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 A13, A24, A22, A25, A15, A18, SCMPDS_7:18;
then A31: 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 A23, EXTPRO_1:23
.= DataPart (IExec (I,Q,t)) by SCMPDS_4:def 5 ;
A32: 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 A29 ;
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 A26, A28, SCMPDS_7:1 ;
then A33: 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;
A34: IExec (I,Q,t) = Result ((Q +* (stop I)),t) by SCMPDS_4:def 5;

A35: now :: thesis:

let x be Int_position; :: thesis:
assume A36: 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 A30, SCMPDS_4:8
.= (Result ((Q +* (stop I)),t)) . x by A23, EXTPRO_1:23
.= (IExec (I,Q,t)) . x by SCMPDS_4:def 5
.= t . x by A2, A7, A8, A9, A36
.= s . x by A7, A36 ;
hence (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x = s . x by A28, A32, SCMPDS_2:54; :: thesis:

end;

InsCode (goto (- ((card I) + 1))) = 14 by SCMPDS_2:12;
then InsCode (goto (- ((card I) + 1))) in {0,4,5,6,14} by ENUMSET1:def 3;
then A37: 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 A32, Th2
.= Initialize (IExec (I,Q,t)) by A31, A28, MEMSTR_0:80 ;

A38: now :: thesis:

assume A39: 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, A12, A37, A33, XXREAL_0:2;
hence contradiction by A39, INT_1:7; :: thesis:

end;

A40: (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 A30, SCMPDS_4:8
.= (Result ((Q +* (stop I)),t)) . a by A23, EXTPRO_1:23
.= s . a by A8, A2, A7, A9, A34 ;
A41: (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 A32, SCMPDS_2:54
.= s . a by A40, EXTPRO_1:4 ;
then A42: 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 A4, A35, A38, A33;

now :: thesis:

let k be Nat; :: thesis:

per cases ;

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

then A43: k <= (LifeSpan ((Q +* (stop I)),t)) + 1 by INT_1:7;

hereby :: thesis:

per cases by A43, NAT_1:8;

suppose A44: 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 A10, A19; :: thesis:

end;

suppose k <> 0 ; :: thesis:

then consider kn being Nat such that
A45: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Nat ;
reconsider lm = IC (Comput ((Q +* (stop I)),t,kn)) as Element of NAT ;
kn < k by A45, XREAL_1:29;
then kn < LifeSpan ((Q +* (stop I)),t) by A44, 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 A13, A24, A22, A25, A15, A18, SCMPDS_7:16;
then A46: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) = lm + 1 by A45, EXTPRO_1:4;
IC (Comput ((Q +* (stop I)),t,kn)) in dom (stop I) by A21, A5, SCMPDS_6:def 2;
then lm < card (stop I) by AFINSQ_1:66;
then lm < (card I) + 1 by COMPOS_1:55;
then A47: 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 A47, 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 A46, AFINSQ_1:66; :: thesis:

end;

suppose A48: k = (LifeSpan ((Q +* (stop I)),t)) + 1 ; :: thesis:

(card I) + 1 in dom (stop (while<0 (a,i,I))) by A20, COMPOS_1:62;
hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I))) by A13, A24, A22, A25, A15, A18, A28, A48, SCMPDS_7:18; :: thesis:

end;

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

then consider nn being Nat such that
A49: k = (((LifeSpan ((Q +* (stop I)),t)) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Nat ;
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 A49, EXTPRO_1:4;
hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I))) by A42, A33, SCMPDS_6:def 2; :: thesis:

end;

end;

hence S₁[k + 1] ; :: thesis:

end;

set n = f . s;
A51: 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 A52: t . (DataLoc ((s . a),i)) >= 0 by A1;
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 A52, Th7; :: thesis:

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A51, A3);
then A53: 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 A53; :: thesis: