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 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,t) . a = t . a & f . (Dstate (IExec I,t)) < f . (Dstate t) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s )
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 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,t) . a = t . a & f . (Dstate (IExec I,t)) < f . (Dstate t) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s )
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 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,t) . a = t . a & f . (Dstate (IExec I,t)) < f . (Dstate t) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s )
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 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,t) . a = t . a & f . (Dstate (IExec I,t)) < f . (Dstate t) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s )
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 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,t) . a = t . a & f . (Dstate (IExec I,t)) < f . (Dstate t) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s )
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 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,t) . a = t . a & f . (Dstate (IExec I,t)) < f . (Dstate t) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) implies ( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s ) )
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 A1: 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 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,t) . a = t . a & f . (Dstate (IExec I,t)) < f . (Dstate t) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) or ( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s ) )
defpred S₁[ Element of NAT ] means for t being State of SCMPDS 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 & while<0 a,i,I is_halting_on t );
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 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,t) . a = t . a & f . (Dstate (IExec I,t)) < f . (Dstate t) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) or ( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s ) )
assume A3: for t being State 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,t) . a = t . a & f . (Dstate (IExec I,t)) < f . (Dstate t) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ; :: thesis: ( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s )
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: ( 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 & while<0 a,i,I is_halting_on b₁ ) )
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 & while<0 a,i,I is_halting_on b₁ ) )
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 & while<0 a,i,I is_halting_on b₁ ) )
assume A8: t . a = s . a ; :: thesis: ( while<0 a,i,I is_closed_on b₁ & while<0 a,i,I is_halting_on b₁ )

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 & while<0 a,i,I is_halting_on b₁ )

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

end;

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

A10: dom (ProgramPart t) = NAT by COMPOS_1:34;
A11: not a in dom (t | NAT ) by A10, SCMPDS_2:53;
A12: (IExec I,t) . a = t . a by A3, A7, A8, A9;
A13: 0 in dom (stop (while<0 a,i,I)) by SCMPDS_4:75;
A14: not DataLoc (s . a),i in dom (Initialize (stop (while<0 a,i,I))) by SCMPDS_4:31;
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,t)) < f . (Dstate t) by A3, A7, A8, A9;
A17: dom (ProgramPart t) = NAT by COMPOS_1:34;
set t2 = (Initialize t) +* (stop I);
set t3 = (Initialize t) +* (stop (while<0 a,i,I));
set t4 = Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1;
I1: (Initialize t) +* (stop I) = t +* (Initialize (stop I)) by SCMPDS_4:5;
I2: (Initialize t) +* (stop (while<0 a,i,I)) = t +* (Initialize (stop (while<0 a,i,I))) by SCMPDS_4:5;
A18: Initialize (stop I) c= (Initialize t) +* (stop I) by I1, FUNCT_4:26;
A19: Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(0 + 1) = Following (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),0 ) by AMI_1:14
.= Following (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))) by AMI_1:13
.= Exec (a,i >=0_goto ((card I) + 2)),((Initialize t) +* (stop (while<0 a,i,I))) by A15, I2, SCMPDS_6:22 ;
A20: DataPart ((Initialize t) +* (stop I)) = DataPart ((Initialize t) +* (stop (while<0 a,i,I))) by FUNCT_7:134, SCMPDS_4:24;

now

let a be Int_position ; :: thesis: ((Initialize t) +* (stop I)) . a = (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1) . a
thus ((Initialize t) +* (stop I)) . a = ((Initialize t) +* (stop (while<0 a,i,I))) . a by A20, SCMPDS_4:23
.= (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1) . a by A19, SCMPDS_2:69 ; :: thesis: verum

end;

then A21: DataPart ((Initialize t) +* (stop I)) = DataPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1) by SCMPDS_4:23;
( while<0 a,i,I c= Initialize (stop (while<0 a,i,I)) & Initialize (stop (while<0 a,i,I)) c= (Initialize t) +* (stop (while<0 a,i,I)) ) by I2, FUNCT_4:26, SCMPDS_6:17;
then A22: while<0 a,i,I c= (Initialize t) +* (stop (while<0 a,i,I)) by XBOOLE_1:1;
Shift I,1 c= while<0 a,i,I by Lm2;
then Shift I,1 c= (Initialize t) +* (stop (while<0 a,i,I)) by A22, XBOOLE_1:1;
then A23: Shift I,1 c= Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1 by AMI_1:81;
set m2 = LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I));
set t5 = Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)));
set l1 = (card I) + 1;
A24: IC ((Initialize t) +* (stop (while<0 a,i,I))) = 0 by SCMPDS_6:21;
set m3 = (LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1;
set t6 = Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1);
set t7 = Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1);
(card I) + 1 < (card I) + 2 by XREAL_1:8;
then A25: (card I) + 1 in dom (while<0 a,i,I) by Th7;
A26: I is_closed_on t by A3, A7, A8, A9;
then A27: I is_closed_on (Initialize t) +* (stop I) by SCMPDS_6:38;
I is_halting_on t by A3, A7, A8, A9;
then A28: ProgramPart ((Initialize t) +* (stop I)) halts_on (Initialize t) +* (stop I) by SCMPDS_6:def 3;
I3: (Initialize ((Initialize t) +* (stop I))) +* (stop I) = ((Initialize t) +* (stop I)) +* (Initialize (stop I)) by SCMPDS_4:5;
(Initialize t) +* (stop I) = (Initialize ((Initialize t) +* (stop I))) +* (stop I) by A18, I3, FUNCT_4:79;
then ProgramPart ((Initialize ((Initialize t) +* (stop I))) +* (stop I)) halts_on (Initialize ((Initialize t) +* (stop I))) +* (stop I) by A28;
then A29: I is_halting_on (Initialize t) +* (stop I) by SCMPDS_6:def 3;
not a in dom (Initialize (stop (while<0 a,i,I))) by SCMPDS_4:31;
then ((Initialize t) +* (stop (while<0 a,i,I))) . (DataLoc (((Initialize t) +* (stop (while<0 a,i,I))) . a),i) = ((Initialize t) +* (stop (while<0 a,i,I))) . (DataLoc (s . a),i) by A8, I2, FUNCT_4:12
.= t . (DataLoc (s . a),i) by A14, I2, FUNCT_4:12 ;
then A30: IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1) = succ (IC ((Initialize t) +* (stop (while<0 a,i,I)))) by A9, A19, SCMPDS_2:69
.= 0 + 1 by A24 ;
then A31: IC (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) = (card I) + 1 by A1, A18, A29, A27, A21, A23, SCMPDS_7:36;
Y: (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1))) /. (IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1))) = (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) . (IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1))) by COMPOS_1:38;
ProgramPart ((Initialize t) +* (stop (while<0 a,i,I))) = ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1) by AMI_1:123;
then A32: Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) = Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) by AMI_1:51;
then A33: CurInstr (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1))),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) = (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . ((card I) + 1) by A1, A18, A29, A27, A30, A21, A23, Y, SCMPDS_7:36
.= (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1) . ((card I) + 1) by AMI_1:54
.= ((Initialize t) +* (stop (while<0 a,i,I))) . ((card I) + 1) by AMI_1:54
.= (while<0 a,i,I) . ((card I) + 1) by A25, A22, GRFUNC_1:8
.= goto (- ((card I) + 1)) by Th8 ;
A34: DataPart (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) = DataPart (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) by A1, A18, A29, A27, A30, A21, A23, SCMPDS_7:36;
then A35: DataPart (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) = DataPart (Result (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) by A28, AMI_1:122
.= DataPart ((Result (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) +* (t | NAT )) by A17, AMI_2:29, FUNCT_4:76, SCMPDS_2:100
.= DataPart (IExec I,t) by SCMPDS_4:def 8 ;
T: ProgramPart ((Initialize t) +* (stop (while<0 a,i,I))) = ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) by AMI_1:123;
A36: Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) = Following (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) by AMI_1:14
.= Exec (goto (- ((card I) + 1))),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) by A33, T ;
then IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) = ICplusConst (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)),(0 - ((card I) + 1)) by SCMPDS_2:66
.= 0 by A31, A32, SCMPDS_7:1 ;
then A37: (Initialize (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1))) +* (stop (while<0 a,i,I)) = Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) by SCMPDS_7:37;
A38: IExec I,t = (Result (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) +* (t | NAT ) by SCMPDS_4:def 8;

A39: now

let x be Int_position ; :: thesis: ( x in X implies (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) . x = s . x )
assume A40: x in X ; :: thesis: (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) . x = s . x
A41: not x in dom (t | NAT ) by A10, SCMPDS_2:53;
(Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . x = (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . x by A34, SCMPDS_4:23
.= (Result (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) . x by A28, AMI_1:122
.= (IExec I,t) . x by A38, A41, FUNCT_4:12
.= t . x by A3, A7, A8, A9, A40
.= s . x by A7, A40 ;
hence (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) . x = s . x by A32, A36, 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 A42: Dstate (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) = Dstate (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) by A36, Th3
.= Dstate (IExec I,t) by A35, A32, Th2 ;

A43: now

assume A44: f . (Dstate (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1))) > k ; :: thesis: contradiction
f . (Dstate (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1))) < k + 1 by A6, A16, A42, XXREAL_0:2;
hence contradiction by A44, INT_1:20; :: thesis: verum

end;

A45: (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . a = (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),(LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . a by A34, SCMPDS_4:23
.= (Result (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) . a by A28, AMI_1:122
.= s . a by A8, A12, A38, A11, FUNCT_4:12 ;
T1: ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1) = ProgramPart ((Initialize t) +* (stop (while<0 a,i,I))) by AMI_1:123;
A46: (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) . a = (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1)) . a by A36, SCMPDS_2:66
.= s . a by A45, T1, AMI_1:51 ;
then A47: while<0 a,i,I is_closed_on Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) by A5, A39, A43;

now

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

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

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

then A48: k <= (LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1 by INT_1:20;

hereby :: thesis: verum

per cases ( k <= LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)) or k = (LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1 ) by A48, NAT_1:8;

suppose A49: k <= LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)) ; :: thesis: IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),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 (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),k) in dom (stop (while<0 a,i,I))

hence IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),k) in dom (stop (while<0 a,i,I)) by A13, A24, AMI_1:13; :: thesis: verum

end;

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

then consider kn being Nat such that
A50: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def 13;
reconsider lm = IC (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),kn) as Element of NAT ;
t: ProgramPart ((Initialize t) +* (stop (while<0 a,i,I))) = ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1) by AMI_1:123;
kn < k by A50, XREAL_1:31;
then kn < LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)) by A49, XXREAL_0:2;
then (IC (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),kn)) + 1 = IC (Comput (ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1)),(Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),1),kn) by A1, A18, A29, A27, A30, A21, A23, SCMPDS_7:34;
then A51: IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),k) = lm + 1 by A50, t, AMI_1:51;
IC (Comput (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),kn) in dom (stop I) by A26, SCMPDS_6:def 2;
then lm < card (stop I) by AFINSQ_1:70;
then lm < (card I) + 1 by SCMPDS_5:7;
then A52: 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 A52, XXREAL_0:2;
then lm + 1 < card (stop (while<0 a,i,I)) by Lm1;
hence IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),k) in dom (stop (while<0 a,i,I)) by A51, AFINSQ_1:70; :: thesis: verum

end;

end;

end;

end;

suppose A53: k = (LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1 ; :: thesis: IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),k) in dom (stop (while<0 a,i,I))

(card I) + 1 in dom (stop (while<0 a,i,I)) by A25, SCMPDS_6:18;
hence IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),k) in dom (stop (while<0 a,i,I)) by A1, A18, A29, A27, A30, A21, A23, A32, A53, SCMPDS_7:36; :: thesis: verum

end;

end;

end;

end;

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

then consider nn being Nat such that
A54: k = (((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 13;
ProgramPart ((Initialize t) +* (stop (while<0 a,i,I))) = ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) by AMI_1:123;
then Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),k = Comput (ProgramPart ((Initialize (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1))) +* (stop (while<0 a,i,I)))),((Initialize (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1))) +* (stop (while<0 a,i,I))),nn by A37, A54, AMI_1:51;
hence IC (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),k) in dom (stop (while<0 a,i,I)) by A47, SCMPDS_6:def 2; :: thesis: verum

end;

end;

end;

hence while<0 a,i,I is_closed_on t by SCMPDS_6:def 2; :: thesis: while<0 a,i,I is_halting_on t
while<0 a,i,I is_halting_on Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) by A5, A46, A39, A43;
then ProgramPart (Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1)) halts_on Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) by A37, SCMPDS_6:def 3;
then ProgramPart ((Initialize t) +* (stop (while<0 a,i,I))) halts_on Comput (ProgramPart ((Initialize t) +* (stop (while<0 a,i,I)))),((Initialize t) +* (stop (while<0 a,i,I))),(((LifeSpan (ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) + 1) + 1) by AMI_1:123;
then ProgramPart ((Initialize t) +* (stop (while<0 a,i,I))) halts_on (Initialize t) +* (stop (while<0 a,i,I)) by AMI_1:93;
hence while<0 a,i,I is_halting_on t by SCMPDS_6:def 3; :: thesis: verum

end;

end;

end;

hence S₁[k + 1] ; :: thesis: verum

end;

set n = f . (Dstate s);
A55: S₁[ 0 ]

proof

let t be State of SCMPDS ; :: 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 & while<0 a,i,I is_halting_on t ) )
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 & while<0 a,i,I is_halting_on t ) )
then f . (Dstate t) = 0 ;
then A56: 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 & while<0 a,i,I is_halting_on t ) )
assume t . a = s . a ; :: thesis: ( while<0 a,i,I is_closed_on t & while<0 a,i,I is_halting_on t )
hence ( while<0 a,i,I is_closed_on t & while<0 a,i,I is_halting_on t ) by A56, Th9; :: thesis: verum

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A55, A4);
then A57: 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 & while<0 a,i,I is_halting_on s ) by A57; :: thesis: verum