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:
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 - (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
( 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
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 & (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 x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ; :: thesis:
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis:
assume A3: S₁[k] ; :: thesis:

now :: thesis:

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
A4: Initialize t = t by MEMSTR_0:44;
assume A5: - (t . (DataLoc ((s . a),i))) <= k + 1 ; :: thesis:
assume A6: for x being Int_position st x in X holds
t . x = s . x ; :: thesis:
assume A7: 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 A7, Th7; :: thesis:

end;

suppose A8: t . (DataLoc ((s . a),i)) < 0 ; :: thesis:

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)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . x = (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . x by A31, SCMPDS_4:8
.= (Result ((Q +* (stop I)),(Initialize t))) . x by A24, EXTPRO_1:23
.= (IExec (I,Q,(Initialize t))) . x by SCMPDS_4:def 5
.= t . x by A1, A6, A7, A8, A36, A4
.= s . x by A6, A36 ;
hence (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . x = s . x by A29, A33, SCMPDS_2:54; :: thesis:

end;

(Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,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 A31, SCMPDS_4:8
.= (Result ((Q +* (stop I)),(Initialize t))) . (DataLoc ((s . a),i)) by A24, EXTPRO_1:23
.= (IExec (I,Q,(Initialize t))) . (DataLoc ((s . a),i)) by SCMPDS_4:def 5 ;
then A37: (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . (DataLoc ((s . a),i)) = (IExec (I,Q,(Initialize t))) . (DataLoc ((s . a),i)) by A29, A33, SCMPDS_2:54;

A38: now :: thesis:

- ((Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . (DataLoc ((s . a),i))) < - (t . (DataLoc ((s . a),i))) by A9, A37, XREAL_1:24;
then A39: - ((Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . (DataLoc ((s . a),i))) < k + 1 by A5, XXREAL_0:2;
assume - ((Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . (DataLoc ((s . a),i))) > k ; :: thesis:
hence contradiction by A39, INT_1:7; :: thesis:

end;

A40: (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . a = (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1))) . a by A33, SCMPDS_2:54
.= s . a by A32, EXTPRO_1:4 ;
then A41: while<0 (a,i,I) is_closed_on Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)),Q +* (stop (while<0 (a,i,I))) by A3, A35, A38, A34;

now :: thesis:

let k be Nat; :: thesis:

per cases ;

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

then A42: k <= (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1 by INT_1:7;

hereby :: thesis:

per cases by A42, NAT_1:8;

suppose A43: k <= LifeSpan ((Q +* (stop I)),(Initialize t)) ; :: thesis:

hereby :: thesis:

per cases ;

suppose k = 0 ; :: thesis:

hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize 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
A44: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Nat ;
reconsider lm = IC (Comput ((Q +* (stop I)),(Initialize t),kn)) as Element of NAT ;
kn < k by A44, XREAL_1:29;
then kn < LifeSpan ((Q +* (stop I)),(Initialize t)) by A43, XXREAL_0:2;
then (IC (Comput ((Q +* (stop I)),(Initialize t),kn))) + 1 = IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)),kn)) by A13, A25, A23, A26, A15, A18, SCMPDS_7:16;
then A45: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) = lm + 1 by A44, EXTPRO_1:4;
IC (Comput ((Q +* (stop I)),(Initialize t),kn)) in dom (stop I) by A22, SCMPDS_6:def 2;
then lm < card (stop I) by AFINSQ_1:66;
then lm < (card I) + 1 by COMPOS_1:55;
then A46: 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 A46, 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)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A45, AFINSQ_1:66; :: thesis:

end;

suppose A47: k = (LifeSpan ((Q +* (stop I)),(Initialize 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)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A13, A25, A23, A26, A15, A18, A29, A47, SCMPDS_7:18; :: thesis:

end;

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

then consider nn being Nat such that
A48: k = (((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Nat ;
Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k) = Comput (((Q +* (stop (while<0 (a,i,I)))) +* (stop (while<0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)))),nn) by A34, A48, EXTPRO_1:4;
hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A41, SCMPDS_6:def 2; :: thesis:

end;

end;

hence S₁[k + 1] ; :: thesis:

end;

A50: S₁[ 0 ]

proof

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
assume - (t . (DataLoc ((s . a),i))) <= 0 ; :: thesis:
then - (t . (DataLoc ((s . a),i))) <= - 0 ;
then A51: t . (DataLoc ((s . a),i)) >= 0 by XREAL_1:24;
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 A51, Th7; :: thesis:

end;

A52: for k being Nat holds S₁[k] from NAT_1:sch 2(A50, A2);

per cases ;

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

hence ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) by Th7; :: thesis:

end;

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

then reconsider n = - (s . (DataLoc ((s . a),i))) as Element of NAT by INT_1:3;
( S₁[n] & ( for x being Int_position st x in X holds
s . x = s . x ) ) by A52;
hence ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) ; :: thesis:

end;