let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for X being set
for f being Function of (product the Object-Kind of SCMPDS),NAT st card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate 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 ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let s be State of SCMPDS; :: thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for X being set
for f being Function of (product the Object-Kind of SCMPDS),NAT st card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate 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 ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let I be halt-free shiftable Program of SCMPDS; :: thesis: for a being Int_position
for i being Integer
for X being set
for f being Function of (product the Object-Kind of SCMPDS),NAT st card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate 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 ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let a be Int_position ; :: thesis: for i being Integer
for X being set
for f being Function of (product the Object-Kind of SCMPDS),NAT st card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate 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 ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let i be Integer; :: thesis: for X being set
for f being Function of (product the Object-Kind of SCMPDS),NAT st card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate 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 ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let X be set ; :: thesis: for f being Function of (product the Object-Kind of SCMPDS),NAT st card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate 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 ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let f be Function of (product the Object-Kind of SCMPDS),NAT; :: thesis: ( card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate 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 ) ) ) implies ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) )
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));
assume card I > 0 ; :: thesis: ( ex t being State of SCMPDS st
( f . (Dstate t) = 0 & not t . (DataLoc ((s . a),i)) >= 0 ) or ex t being State of SCMPDS ex Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 & not ( (IExec (I,Q,t)) . a = t . a & f . (Dstate (IExec (I,Q,t))) < f . (Dstate 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 ) ) ) or ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) )
defpred S₁[ Element of NAT ] means for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st f . (Dstate 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 State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ; :: thesis: ( ex t being State of SCMPDS ex Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 & not ( (IExec (I,Q,t)) . a = t . a & f . (Dstate (IExec (I,Q,t))) < f . (Dstate 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 ) ) ) or ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) )
assume A3: for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate 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: ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
A4: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: S₁[k] ; :: thesis: S₁[k + 1]

now

let t be State of SCMPDS; :: thesis: for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st f . (Dstate t) <= k + 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 b₂,b₃ & while<0 (a,i,I) is_halting_on b₂,b₃ )
let Q be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: ( f . (Dstate t) <= k + 1 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( while<0 (a,i,I) is_closed_on b₁,b₂ & while<0 (a,i,I) is_halting_on b₁,b₂ ) )
assume A6: f . (Dstate t) <= k + 1 ; :: thesis: ( ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( while<0 (a,i,I) is_closed_on b₁,b₂ & while<0 (a,i,I) is_halting_on b₁,b₂ ) )
assume A7: for x being Int_position st x in X holds
t . x = s . x ; :: thesis: ( t . a = s . a implies ( while<0 (a,i,I) is_closed_on b₁,b₂ & while<0 (a,i,I) is_halting_on b₁,b₂ ) )
assume A8: t . a = s . a ; :: thesis: ( while<0 (a,i,I) is_closed_on b₁,b₂ & while<0 (a,i,I) is_halting_on b₁,b₂ )

per cases ( t . (DataLoc ((s . a),i)) >= 0 or t . (DataLoc ((s . a),i)) < 0 ) ;

suppose t . (DataLoc ((s . a),i)) >= 0 ; :: thesis: ( while<0 (a,i,I) is_closed_on b₁,b₂ & while<0 (a,i,I) is_halting_on b₁,b₂ )

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: verum

end;

suppose A9: t . (DataLoc ((s . a),i)) < 0 ; :: thesis: ( while<0 (a,i,I) is_closed_on b₁,b₂ & while<0 (a,i,I) is_halting_on b₁,b₂ )

A10: dom (ProgramPart t) = NAT by COMPOS_1:34;
A11: not a in dom (t | NAT) by A10, SCMPDS_2:53;
A12: (IExec (I,Q,t)) . a = t . a by A3, A7, A8, A9;
A13: 0 in dom (stop (while<0 (a,i,I))) by COMPOS_1:135;
A14: not DataLoc ((s . a),i) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:59;
A15: while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:51;
A16: f . (Dstate (IExec (I,Q,t))) < f . (Dstate t) by A3, A7, A8, A9;
A17: dom (ProgramPart t) = NAT by COMPOS_1:34;
set t2 = Initialize t;
set Q2 = Q +* (stop I);
set t3 = Initialize t;
set Q3 = Q +* (stop (while<0 (a,i,I)));
set t4 = Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1);
set Q4 = Q +* (stop (while<0 (a,i,I)));
A20: stop I c= Q +* (stop I) by FUNCT_4:26;
B20: Start-At (0,SCMPDS) c= Initialize t by FUNCT_4:26;
A21: Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(0 + 1)) = Following ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),0))) by EXTPRO_1:4
.= Following ((Q +* (stop (while<0 (a,i,I)))),(Initialize t)) by EXTPRO_1:3
.= Exec (((a,i) >=0_goto ((card I) + 2)),(Initialize t)) by A15, SCMPDS_6:22 ;

now

let a be Int_position ; :: thesis: (Initialize t) . a = (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)) . a
thus (Initialize t) . a = (Initialize t) . a
.= (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)) . a by A21, SCMPDS_2:69 ; :: thesis: verum

end;

then A23: DataPart (Initialize t) = DataPart (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)) by SCMPDS_4:23;
XX: while<0 (a,i,I) c= stop (while<0 (a,i,I)) by AFINSQ_1:78;
stop (while<0 (a,i,I)) c= Q +* (stop (while<0 (a,i,I))) by FUNCT_4:26;
then A24: while<0 (a,i,I) c= Q +* (stop (while<0 (a,i,I))) by XBOOLE_1:1, XX;
Shift (I,1) c= while<0 (a,i,I) by Lm2;
then Shift (I,1) c= Q +* (stop (while<0 (a,i,I))) by A24, XBOOLE_1:1;
then A25: Shift (I,1) c= Q +* (stop (while<0 (a,i,I))) ;
set m2 = LifeSpan ((Q +* (stop I)),(Initialize t));
set t5 = 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))));
set Q5 = Q +* (stop (while<0 (a,i,I)));
set l1 = (card I) + 1;
A26: IC (Initialize t) = 0 by COMPOS_1:def 16;
set m3 = (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1;
set t6 = Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1));
set Q6 = Q +* (stop (while<0 (a,i,I)));
set t7 = Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1));
set Q7 = Q +* (stop (while<0 (a,i,I)));
(card I) + 1 < (card I) + 2 by XREAL_1:8;
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;
then A29: I is_closed_on Initialize t,Q +* (stop I) by SCMPDS_6:38;
I is_halting_on t,Q by A3, A7, A8, A9;
then A30: Q +* (stop I) halts_on Initialize t by SCMPDS_6:def 3;
Q +* (stop I) = (Q +* (stop I)) +* (stop I) by A20, FUNCT_4:104;
then (Q +* (stop I)) +* (stop I) halts_on Initialize (Initialize t) by A30;
then A32: I is_halting_on Initialize t,Q +* (stop I) by SCMPDS_6:def 3;
not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:59;
then (Initialize t) . (DataLoc (((Initialize t) . a),i)) = (Initialize t) . (DataLoc ((s . a),i)) by A8, FUNCT_4:12
.= t . (DataLoc ((s . a),i)) by A14, FUNCT_4:12 ;
then A33: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)) = succ (IC (Initialize t)) by A9, A21, SCMPDS_2:69
.= 0 + 1 by A26 ;
then A34: IC (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))))) = (card I) + 1 by A20, B20, A32, A29, A23, A25, SCMPDS_7:36;
A35: (Q +* (stop (while<0 (a,i,I)))) /. (IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) = (Q +* (stop (while<0 (a,i,I)))) . (IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by PBOOLE:158;
A36: Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)) = 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)))) by EXTPRO_1:5;
then A37: CurInstr ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) = (Q +* (stop (while<0 (a,i,I)))) . ((card I) + 1) by A20, B20, A32, A29, A33, A23, A25, A35, SCMPDS_7:36
.= (Q +* (stop (while<0 (a,i,I)))) . ((card I) + 1)
.= (Q +* (stop (while<0 (a,i,I)))) . ((card I) + 1)
.= (while<0 (a,i,I)) . ((card I) + 1) by A27, A24, GRFUNC_1:8
.= goto (- ((card I) + 1)) by Th8 ;
A38: DataPart (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) = DataPart (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))))) by A20, B20, A32, A29, A33, A23, A25, SCMPDS_7:36;
then A39: DataPart (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))))) = DataPart (Result ((Q +* (stop I)),(Initialize t))) by A30, EXTPRO_1:23
.= DataPart ((Result ((Q +* (stop I)),(Initialize t))) +* (t | NAT)) by A17, AMI_2:29, FUNCT_4:76, SCMPDS_2:100
.= DataPart (IExec (I,Q,t)) by SCMPDS_4:def 8 ;
A41: Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)) = Following ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by EXTPRO_1:4
.= Exec ((goto (- ((card I) + 1))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by A37 ;
then IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1))),(0 - ((card I) + 1))) by SCMPDS_2:66
.= 0 by A34, A36, SCMPDS_7:1 ;
then A42: Initialize (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) = Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)) by COMPOS_1:84;
A43: IExec (I,Q,t) = (Result ((Q +* (stop I)),(Initialize t))) +* (t | NAT) by SCMPDS_4:def 8;

A44: now

let x be Int_position ; :: thesis: ( x in X implies (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . x = s . x )
assume A45: x in X ; :: thesis: (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . x = s . x
A46: not x in dom (t | NAT) by A10, SCMPDS_2:53;
(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 A38, SCMPDS_4:23
.= (Result ((Q +* (stop I)),(Initialize t))) . x by A30, EXTPRO_1:23
.= (IExec (I,Q,t)) . x by A43, A46, FUNCT_4:12
.= t . x by A3, A7, A8, A9, A45
.= s . x by A7, A45 ;
hence (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . x = s . x by A36, A41, SCMPDS_2:66; :: thesis: verum

end;

InsCode (goto (- ((card I) + 1))) = 0 by SCMPDS_2:21;
then InsCode (goto (- ((card I) + 1))) in {0,4,5,6} by ENUMSET1:def 2;
then A47: Dstate (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) = Dstate (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1))) by A41, Th3
.= Dstate (IExec (I,Q,t)) by A39, A36, Th2 ;

A48: now

assume A49: f . (Dstate (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)))) > k ; :: thesis: contradiction
f . (Dstate (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)))) < k + 1 by A6, A16, A47, XXREAL_0:2;
hence contradiction by A49, INT_1:20; :: thesis: verum

end;

A50: (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))))) . a = (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . a by A38, SCMPDS_4:23
.= (Result ((Q +* (stop I)),(Initialize t))) . a by A30, EXTPRO_1:23
.= s . a by A8, A12, A43, A11, FUNCT_4:12 ;
A52: (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 A41, SCMPDS_2:66
.= s . a by A50, EXTPRO_1:5 ;
then A53: 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 A5, A44, A48;

now

let k be Element of NAT ; :: thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),b₁)) in dom (stop (while<0 (a,i,I)))

per cases ( k < ((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1 or k >= ((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1 ) ;

suppose k < ((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1 ; :: thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),b₁)) in dom (stop (while<0 (a,i,I)))

then A54: k <= (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1 by INT_1:20;

hereby :: thesis: verum

per cases ( k <= LifeSpan ((Q +* (stop I)),(Initialize t)) or k = (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1 ) by A54, NAT_1:8;

suppose A55: k <= LifeSpan ((Q +* (stop I)),(Initialize t)) ; :: thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I)))

hereby :: thesis: verum

per cases ( k = 0 or k <> 0 ) ;

suppose k = 0 ; :: thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I)))

hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A13, A26, EXTPRO_1:3; :: thesis: verum

end;

suppose k <> 0 ; :: thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I)))

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 13;
reconsider lm = IC (Comput ((Q +* (stop I)),(Initialize t),kn)) as Element of NAT ;
kn < k by A56, XREAL_1:31;
then kn < LifeSpan ((Q +* (stop I)),(Initialize t)) by A55, 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 A20, B20, A32, A29, A33, A23, A25, SCMPDS_7:34;
then A58: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) = lm + 1 by A56, EXTPRO_1:5;
IC (Comput ((Q +* (stop I)),(Initialize t),kn)) in dom (stop I) by A28, SCMPDS_6:def 2;
then lm < card (stop I) by AFINSQ_1:70;
then lm < (card I) + 1 by SCMPDS_5:7;
then A59: lm + 1 <= (card I) + 1 by INT_1:20;
(card I) + 1 < (card I) + 3 by XREAL_1:8;
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)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A58, AFINSQ_1:70; :: thesis: verum

end;

end;

end;

end;

suppose A60: k = (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1 ; :: thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I)))

(card I) + 1 in dom (stop (while<0 (a,i,I))) by A27, SCMPDS_6:18;
hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A20, B20, A32, A29, A33, A23, A25, A36, A60, SCMPDS_7:36; :: thesis: verum

end;

end;

end;

end;

suppose k >= ((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1 ; :: thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),b₁)) in dom (stop (while<0 (a,i,I)))

then consider nn being Nat such that
A61: k = (((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 13;
Q +* (stop (while<0 (a,i,I))) = (Q +* (stop (while<0 (a,i,I)))) +* (stop (while<0 (a,i,I))) by FUNCT_4:99;
then 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 A42, A61, EXTPRO_1:5;
hence IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A53, SCMPDS_6:def 2; :: thesis: verum

end;

end;

end;

hence while<0 (a,i,I) is_closed_on t,Q by SCMPDS_6:def 2; :: thesis: while<0 (a,i,I) is_halting_on t,Q
RR: Q +* (stop (while<0 (a,i,I))) = (Q +* (stop (while<0 (a,i,I)))) +* (stop (while<0 (a,i,I))) by FUNCT_4:99;
while<0 (a,i,I) is_halting_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 A5, A52, A44, A48;
then Q +* (stop (while<0 (a,i,I))) halts_on Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)) by A42, SCMPDS_6:def 3, RR;
then Q +* (stop (while<0 (a,i,I))) halts_on Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)) ;
then Q +* (stop (while<0 (a,i,I))) halts_on Initialize t by EXTPRO_1:22;
hence while<0 (a,i,I) is_halting_on t,Q by SCMPDS_6:def 3; :: thesis: verum

end;

end;

end;

hence S₁[k + 1] ; :: thesis: verum

end;

set n = f . (Dstate s);
A62: S₁[ 0 ]

proof

let t be State of SCMPDS; :: thesis: for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st f . (Dstate t) <= 0 & ( 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 )
let Q be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: ( f . (Dstate t) <= 0 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) )
assume f . (Dstate t) <= 0 ; :: thesis: ( ex x being Int_position st
( x in X & not t . x = s . x ) or not t . a = s . a or ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) )
then f . (Dstate 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: ( not t . a = s . a or ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) )
assume t . a = s . a ; :: thesis: ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q )
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: verum

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A62, A4);
then A64: S₁[f . (Dstate 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: verum