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 n being Element of NAT
for X being set st s . (DataLoc ((s . a),i)) < 0 & not DataLoc ((s . a),i) in X & n > 0 & card I > 0 & a <> DataLoc ((s . a),i) & ( 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 holds
( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) holds
( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,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 n being Element of NAT
for X being set st s . (DataLoc ((s . a),i)) < 0 & not DataLoc ((s . a),i) in X & n > 0 & card I > 0 & a <> DataLoc ((s . a),i) & ( 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 holds
( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) holds
( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s )
let a be Int_position ; :: thesis: for i being Integer
for n being Element of NAT
for X being set st s . (DataLoc ((s . a),i)) < 0 & not DataLoc ((s . a),i) in X & n > 0 & card I > 0 & a <> DataLoc ((s . a),i) & ( 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 holds
( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) holds
( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s )
let i be Integer; :: thesis: for n being Element of NAT
for X being set st s . (DataLoc ((s . a),i)) < 0 & not DataLoc ((s . a),i) in X & n > 0 & card I > 0 & a <> DataLoc ((s . a),i) & ( 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 holds
( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) holds
( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s )
let n be Element of NAT ; :: thesis: for X being set st s . (DataLoc ((s . a),i)) < 0 & not DataLoc ((s . a),i) in X & n > 0 & card I > 0 & a <> DataLoc ((s . a),i) & ( 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 holds
( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) holds
( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s )
let X be set ; :: thesis: ( s . (DataLoc ((s . a),i)) < 0 & not DataLoc ((s . a),i) in X & n > 0 & card I > 0 & a <> DataLoc ((s . a),i) & ( 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 holds
( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) implies ( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s ) )
set b = DataLoc ((s . a),i);
set FOR = for-up (a,i,n,I);
set pFOR = stop (for-up (a,i,n,I));
set pI = stop I;
set i1 = (a,i) >=0_goto ((card I) + 3);
set i2 = AddTo (a,i,n);
set i3 = goto (- ((card I) + 2));
assume A1: s . (DataLoc ((s . a),i)) < 0 ; :: thesis: ( DataLoc ((s . a),i) in X or not n > 0 or not card I > 0 or not a <> DataLoc ((s . a),i) 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 & not ( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) or ( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s ) )
defpred S₁[ Element of NAT ] means for t being State 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
( for-up (a,i,n,I) is_closed_on t & for-up (a,i,n,I) is_halting_on t );
assume A2: not DataLoc ((s . a),i) in X ; :: thesis: ( not n > 0 or not card I > 0 or not a <> DataLoc ((s . a),i) 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 & not ( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) or ( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s ) )
assume A3: n > 0 ; :: thesis: ( not card I > 0 or not a <> DataLoc ((s . a),i) 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 & not ( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) or ( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s ) )
assume A4: card I > 0 ; :: thesis: ( not a <> DataLoc ((s . a),i) 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 & not ( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) or ( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s ) )
assume A5: a <> DataLoc ((s . a),i) ; :: 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 & not ( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ) or ( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s ) )
assume A6: 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 holds
( (IExec (I,t)) . a = t . a & (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) & I is_closed_on t & I is_halting_on t & ( for y being Int_position st y in X holds
(IExec (I,t)) . y = t . y ) ) ; :: thesis: ( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s )
A7: 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 A8: S₁[k] ; :: thesis: S₁[k + 1]

now

let t be State of SCMPDS; :: thesis: ( - (t . (DataLoc ((s . a),i))) <= k + 1 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( for-up (a,i,n,I) is_closed_on b₁ & for-up (a,i,n,I) is_halting_on b₁ ) )
assume A9: - (t . (DataLoc ((s . a),i))) <= k + 1 ; :: thesis: ( ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( for-up (a,i,n,I) is_closed_on b₁ & for-up (a,i,n,I) is_halting_on b₁ ) )
assume A10: for x being Int_position st x in X holds
t . x = s . x ; :: thesis: ( t . a = s . a implies ( for-up (a,i,n,I) is_closed_on b₁ & for-up (a,i,n,I) is_halting_on b₁ ) )
assume A11: t . a = s . a ; :: thesis: ( for-up (a,i,n,I) is_closed_on b₁ & for-up (a,i,n,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: ( for-up (a,i,n,I) is_closed_on b₁ & for-up (a,i,n,I) is_halting_on b₁ )

hence ( for-up (a,i,n,I) is_closed_on t & for-up (a,i,n,I) is_halting_on t ) by A11, Th54; :: thesis: verum

end;

suppose A12: t . (DataLoc ((s . a),i)) < 0 ; :: thesis: ( for-up (a,i,n,I) is_closed_on b₁ & for-up (a,i,n,I) is_halting_on b₁ )

set t2 = (Initialize t) +* (stop I);
set t3 = (Initialize t) +* (stop (for-up (a,i,n,I)));
set t4 = Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1);
I1: t +* (Initialize (stop I)) = (Initialize t) +* (stop I) by COMPOS_1:125;
I2: t +* (Initialize (stop (for-up (a,i,n,I)))) = (Initialize t) +* (stop (for-up (a,i,n,I))) by COMPOS_1:125;
A13: Initialize (stop I) c= (Initialize t) +* (stop I) by I1, FUNCT_4:26;
A14: dom (ProgramPart t) = NAT by COMPOS_1:34;
A15: not a in dom (t | NAT) by A14, SCMPDS_2:53;
A16: for-up (a,i,n,I) = ((a,i) >=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,n))) ';' (goto (- ((card I) + 2)))) by Th15;
A17: Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(0 + 1)) = Following ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),0))) by EXTPRO_1:4
.= Following ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I))))) by EXTPRO_1:3
.= Exec (((a,i) >=0_goto ((card I) + 3)),((Initialize t) +* (stop (for-up (a,i,n,I))))) by A16, I2, SCMPDS_6:22 ;
A18: DataPart ((Initialize t) +* (stop I)) = DataPart ((Initialize t) +* (stop (for-up (a,i,n,I)))) by COMPOS_1:138, FUNCT_7:134;

now

let a be Int_position ; :: thesis: ((Initialize t) +* (stop I)) . a = (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)) . a
thus ((Initialize t) +* (stop I)) . a = ((Initialize t) +* (stop (for-up (a,i,n,I)))) . a by A18, SCMPDS_4:23
.= (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)) . a by A17, SCMPDS_2:69 ; :: thesis: verum

end;

then A19: DataPart ((Initialize t) +* (stop I)) = DataPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)) by SCMPDS_4:23;
A20: not DataLoc ((s . a),i) in dom (t | NAT) by A14, SCMPDS_2:53;
A21: (IExec (I,t)) . (DataLoc ((s . a),i)) = t . (DataLoc ((s . a),i)) by A6, A10, A11;
- (- n) > 0 by A3;
then - n < 0 ;
then - n <= - 1 by INT_1:21;
then A22: (- n) - (t . (DataLoc ((s . a),i))) <= (- 1) - (t . (DataLoc ((s . a),i))) by XREAL_1:11;
(- (t . (DataLoc ((s . a),i)))) - 1 <= k by A9, XREAL_1:22;
then A23: (- n) - (t . (DataLoc ((s . a),i))) <= k by A22, XXREAL_0:2;
A24: I is_closed_on t by A6, A10, A11;
then A25: I is_closed_on (Initialize t) +* (stop I) by SCMPDS_6:38;
A26: not DataLoc ((s . a),i) in dom (Initialize (stop (for-up (a,i,n,I)))) by SCMPDS_4:31;
set m2 = LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)));
set t5 = Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))));
set l1 = (card I) + 1;
A27: IC ((Initialize t) +* (stop (for-up (a,i,n,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 (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1));
(card I) + 1 < (card I) + 3 by XREAL_1:8;
then A28: (card I) + 1 in dom (for-up (a,i,n,I)) by Th52;
set m5 = (((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1;
set t8 = Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1));
set t7 = Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1));
A29: (IExec (I,t)) . a = t . a by A6, A10, A11;
set l2 = (card I) + 2;
A30: 0 in dom (stop (for-up (a,i,n,I))) by COMPOS_1:135;
(card I) + 2 < (card I) + 3 by XREAL_1:8;
then A31: (card I) + 2 in dom (for-up (a,i,n,I)) by Th52;
A32: Initialize (stop (for-up (a,i,n,I))) c= (Initialize t) +* (stop (for-up (a,i,n,I))) by I2, FUNCT_4:26;
for-up (a,i,n,I) c= Initialize (stop (for-up (a,i,n,I))) by SCMPDS_6:17;
then A33: for-up (a,i,n,I) c= (Initialize t) +* (stop (for-up (a,i,n,I))) by A32, XBOOLE_1:1;
Shift (I,1) c= for-up (a,i,n,I) by Lm3;
then Shift (I,1) c= (Initialize t) +* (stop (for-up (a,i,n,I))) by A33, XBOOLE_1:1;
then A34: Shift (I,1) c= Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1) by AMI_1:81;
I is_halting_on t by A6, A10, A11;
then A35: 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 COMPOS_1:125;
(Initialize t) +* (stop I) = (Initialize ((Initialize t) +* (stop I))) +* (stop I) by A13, I3, FUNCT_4:79;
then ProgramPart ((Initialize ((Initialize t) +* (stop I))) +* (stop I)) halts_on (Initialize ((Initialize t) +* (stop I))) +* (stop I) by A35;
then A36: I is_halting_on (Initialize t) +* (stop I) by SCMPDS_6:def 3;
not a in dom (Initialize (stop (for-up (a,i,n,I)))) by SCMPDS_4:31;
then ((Initialize t) +* (stop (for-up (a,i,n,I)))) . (DataLoc ((((Initialize t) +* (stop (for-up (a,i,n,I)))) . a),i)) = ((Initialize t) +* (stop (for-up (a,i,n,I)))) . (DataLoc ((s . a),i)) by A11, I2, FUNCT_4:12
.= t . (DataLoc ((s . a),i)) by A26, I2, FUNCT_4:12 ;
then A37: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)) = succ (IC ((Initialize t) +* (stop (for-up (a,i,n,I))))) by A12, A17, SCMPDS_2:69
.= 0 + 1 by A27 ;
then A38: IC (Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))))) = (card I) + 1 by A4, A13, A36, A25, A19, A34, Th36;
Y: (ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1)))) /. (IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1)))) = (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1))) . (IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1)))) by COMPOS_1:38;
ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I)))) = ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)) by AMI_1:123;
then A39: Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1)) = Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))))) by EXTPRO_1:5;
then A40: CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1)))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1)))) = (Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))))) . ((card I) + 1) by A4, A13, A36, A25, A37, A19, A34, Th36, Y
.= (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)) . ((card I) + 1) by AMI_1:54
.= ((Initialize t) +* (stop (for-up (a,i,n,I)))) . ((card I) + 1) by AMI_1:54
.= (for-up (a,i,n,I)) . ((card I) + 1) by A28, A33, GRFUNC_1:8
.= AddTo (a,i,n) by Th53 ;
Y: (ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1)))) /. (IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1)))) = (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1))) . (IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1)))) by COMPOS_1:38;
T: ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I)))) = ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1))) by AMI_1:123;
A41: Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1)) = Following ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1)))) by EXTPRO_1:4
.= Exec ((AddTo (a,i,n)),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1)))) by A40, T ;
then A42: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1))) = succ (IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1)))) by SCMPDS_2:60
.= ((card I) + 1) + 1 by A38, A39, NAT_1:39
.= (card I) + (1 + 1) ;
then A43: CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1)))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1)))) = ((Initialize t) +* (stop (for-up (a,i,n,I)))) . ((card I) + 2) by Y, AMI_1:54
.= (for-up (a,i,n,I)) . ((card I) + 2) by A33, A31, GRFUNC_1:8
.= goto (- ((card I) + 2)) by Th53 ;
S: ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I)))) = ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1))) by AMI_1:123;
A44: Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1)) = Following ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1)))) by EXTPRO_1:4
.= Exec ((goto (- ((card I) + 2))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1)))) by A43, S ;
then IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1))) = ICplusConst ((Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1))),(0 - ((card I) + 2))) by SCMPDS_2:66
.= 0 by A42, Th1 ;
then A45: (Initialize (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1)))) +* (stop (for-up (a,i,n,I))) = Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1)) by Th37;
t: ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I)))) = ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)) by AMI_1:123;
A46: 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 (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))))) by A4, A13, A36, A25, A37, A19, A34, Th36;
then A47: (Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,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 SCMPDS_4:23
.= (Result ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . a by A35, EXTPRO_1:23
.= s . a by A11, A29, A15, FUNCT_4:12 ;
then DataLoc (((Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1))) . a),i) = DataLoc ((s . a),i) by t, EXTPRO_1:5;
then (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1))) . a = (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1))) . a by A5, A41, SCMPDS_2:60
.= s . a by A47, t, EXTPRO_1:5 ;
then A48: (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1))) . a = s . a by A44, SCMPDS_2:66;

A49: now

let x be Int_position ; :: thesis: ( x in X implies (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1))) . x = s . x )
assume A50: x in X ; :: thesis: (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1))) . x = s . x
A51: not x in dom (t | NAT) by A14, SCMPDS_2:53;
(Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,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 A46, SCMPDS_4:23
.= (Result ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . x by A35, EXTPRO_1:23
.= (IExec (I,t)) . x by A51, FUNCT_4:12
.= t . x by A6, A10, A11, A50
.= s . x by A10, A50 ;
then (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1))) . x = s . x by A2, A47, A39, A41, A50, SCMPDS_2:60;
hence (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1))) . x = s . x by A44, SCMPDS_2:66; :: thesis: verum

end;

A52: (Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))))) . (DataLoc ((s . a),i)) = (Comput ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))))) . (DataLoc ((s . a),i)) by A46, SCMPDS_4:23
.= (Result ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) . (DataLoc ((s . a),i)) by A35, EXTPRO_1:23
.= t . (DataLoc ((s . a),i)) by A21, A20, FUNCT_4:12 ;
(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1))) . (DataLoc ((s . a),i)) = (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1))) . (DataLoc ((s . a),i)) by A44, SCMPDS_2:66
.= (t . (DataLoc ((s . a),i))) + n by A47, A52, A39, A41, SCMPDS_2:60 ;
then A53: - ((Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1))) . (DataLoc ((s . a),i))) = (- n) - (t . (DataLoc ((s . a),i))) ;
then A54: for-up (a,i,n,I) is_closed_on Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1)) by A8, A48, A49, A23;

now

let k be Element of NAT ; :: thesis: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),b₁)) in dom (stop (for-up (a,i,n,I)))

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

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

then k <= ((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1 by INT_1:20;
then A55: ( k <= (LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1 or k = ((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1 ) by NAT_1:8;

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 or k = ((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1 ) by A55, NAT_1:8;

suppose A56: k <= LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) ; :: thesis: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k)) in dom (stop (for-up (a,i,n,I)))

hereby :: thesis: verum

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

suppose k = 0 ; :: thesis: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k)) in dom (stop (for-up (a,i,n,I)))

hence IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k)) in dom (stop (for-up (a,i,n,I))) by A30, A27, EXTPRO_1:3; :: thesis: verum

end;

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

then consider kn being Nat such that
A57: 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 ;
kn < k by A57, XREAL_1:31;
then kn < LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I))) by A56, 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 (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),1)),kn)) by A4, A13, A36, A25, A37, A19, A34, Th34;
then A58: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k)) = lm + 1 by A57, t, EXTPRO_1:5
.= lm + 1 ;
IC (Comput ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)),kn)) in dom (stop I) by A24, 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) + 4 by XREAL_1:8;
then lm + 1 < (card I) + 4 by A59, XXREAL_0:2;
then lm + 1 < card (stop (for-up (a,i,n,I))) by Lm2;
hence IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k)) in dom (stop (for-up (a,i,n,I))) by A58, AFINSQ_1:70; :: thesis: verum

end;

end;

end;

end;

suppose A60: k = (LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1 ; :: thesis: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k)) in dom (stop (for-up (a,i,n,I)))

(card I) + 1 in dom (stop (for-up (a,i,n,I))) by A28, SCMPDS_6:18;
hence IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k)) in dom (stop (for-up (a,i,n,I))) by A4, A13, A36, A25, A37, A19, A34, A39, A60, Th36; :: thesis: verum

end;

suppose k = ((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1 ; :: thesis: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k)) in dom (stop (for-up (a,i,n,I)))

hence IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k)) in dom (stop (for-up (a,i,n,I))) by A42, A31, SCMPDS_6:18; :: thesis: verum

end;

end;

end;

end;

suppose k >= (((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1 ; :: thesis: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),b₁)) in dom (stop (for-up (a,i,n,I)))

then consider nn being Nat such that
A61: k = ((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 13;
ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I)))) = ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1))) by AMI_1:123;
then Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k) = Comput ((ProgramPart ((Initialize (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1)))) +* (stop (for-up (a,i,n,I))))),((Initialize (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1)))) +* (stop (for-up (a,i,n,I)))),nn) by A45, A61, EXTPRO_1:5;
hence IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),k)) in dom (stop (for-up (a,i,n,I))) by A54, SCMPDS_6:def 2; :: thesis: verum

end;

end;

end;

hence for-up (a,i,n,I) is_closed_on t by SCMPDS_6:def 2; :: thesis: for-up (a,i,n,I) is_halting_on t
for-up (a,i,n,I) is_halting_on Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1)) by A8, A48, A49, A23, A53;
then ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1))) halts_on Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1)) by A45, SCMPDS_6:def 3;
then ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I)))) halts_on Comput ((ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I))))),((Initialize t) +* (stop (for-up (a,i,n,I)))),((((LifeSpan ((ProgramPart ((Initialize t) +* (stop I))),((Initialize t) +* (stop I)))) + 1) + 1) + 1)) by AMI_1:123;
then ProgramPart ((Initialize t) +* (stop (for-up (a,i,n,I)))) halts_on (Initialize t) +* (stop (for-up (a,i,n,I))) by EXTPRO_1:22;
hence for-up (a,i,n,I) is_halting_on t by SCMPDS_6:def 3; :: thesis: verum

end;

end;

end;

hence S₁[k + 1] ; :: thesis: verum

end;

reconsider n = - (s . (DataLoc ((s . a),i))) as Element of NAT by A1, INT_1:16;
A62: S₁[ 0 ]

proof

let t be State of SCMPDS; :: thesis: ( - (t . (DataLoc ((s . a),i))) <= 0 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( for-up (a,i,n,I) is_closed_on t & for-up (a,i,n,I) is_halting_on t ) )
assume - (t . (DataLoc ((s . a),i))) <= 0 ; :: thesis: ( ex x being Int_position st
( x in X & not t . x = s . x ) or not t . a = s . a or ( for-up (a,i,n,I) is_closed_on t & for-up (a,i,n,I) is_halting_on t ) )
then - (t . (DataLoc ((s . a),i))) <= - 0 ;
then A63: t . (DataLoc ((s . a),i)) >= 0 by XREAL_1:26;
assume for x being Int_position st x in X holds
t . x = s . x ; :: thesis: ( not t . a = s . a or ( for-up (a,i,n,I) is_closed_on t & for-up (a,i,n,I) is_halting_on t ) )
assume t . a = s . a ; :: thesis: ( for-up (a,i,n,I) is_closed_on t & for-up (a,i,n,I) is_halting_on t )
hence ( for-up (a,i,n,I) is_closed_on t & for-up (a,i,n,I) is_halting_on t ) by A63, Th54; :: thesis: verum

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A62, A7);
then A64: S₁[n] ;
for x being Int_position st x in X holds
s . x = s . x ;
hence ( for-up (a,i,n,I) is_closed_on s & for-up (a,i,n,I) is_halting_on s ) by A64; :: thesis: verum