let s be State of SCMPDS ; :: thesis: for I being shiftable No-StopCode 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-down a,i,n,I is_closed_on s & for-down a,i,n,I is_halting_on s )
let I be shiftable No-StopCode 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-down a,i,n,I is_closed_on s & for-down 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-down a,i,n,I is_closed_on s & for-down 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-down a,i,n,I is_closed_on s & for-down 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-down a,i,n,I is_closed_on s & for-down 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-down a,i,n,I is_closed_on s & for-down a,i,n,I is_halting_on s ) )
set b = DataLoc (s . a),i;
set FOR = for-down a,i,n,I;
set pFOR = stop (for-down a,i,n,I);
set iFOR = Initialized (stop (for-down a,i,n,I));
set pI = stop I;
set IsI = Initialized (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-down a,i,n,I is_closed_on s & for-down a,i,n,I is_halting_on s ) )
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-down a,i,n,I is_closed_on s & for-down 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-down a,i,n,I is_closed_on s & for-down 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-down a,i,n,I is_closed_on s & for-down 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-down a,i,n,I is_closed_on s & for-down 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-down a,i,n,I is_closed_on s & for-down 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-down a,i,n,I is_closed_on t & for-down a,i,n,I is_halting_on t );
A7: S1[ 0 ] by Th63;
A8: 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 A9: 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-down a,i,n,I is_closed_on b1 & for-down a,i,n,I is_halting_on b1 ) )
assume A10: 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-down a,i,n,I is_closed_on b1 & for-down a,i,n,I is_halting_on b1 ) )
assume A11: for x being Int_position st x in X holds
t . x = s . x ; :: thesis: ( t . a = s . a implies ( for-down a,i,n,I is_closed_on b1 & for-down a,i,n,I is_halting_on b1 ) )
assume A12: t . a = s . a ; :: thesis: ( for-down a,i,n,I is_closed_on b1 & for-down 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-down a,i,n,I is_closed_on b1 & for-down a,i,n,I is_halting_on b1 )
hence ( for-down a,i,n,I is_closed_on t & for-down a,i,n,I is_halting_on t ) by A12, Th63; :: thesis: verum
end;
suppose A13: t . (DataLoc (s . a),i) > 0 ; :: thesis: ( for-down a,i,n,I is_closed_on b1 & for-down a,i,n,I is_halting_on b1 )
set t2 = t +* (Initialized (stop I));
set t3 = t +* (Initialized (stop (for-down a,i,n,I)));
set t4 = Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1;
A14: ( (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 ) ) by A6, A11, A12;
A15: Initialized (stop I) c= t +* (Initialized (stop I)) by FUNCT_4:26;
A16: t +* (Initialized (stop I)) is halting by A14, SCMPDS_6:def 3;
then (t +* (Initialized (stop I))) +* (Initialized (stop I)) is halting by A15, FUNCT_4:79;
then A17: I is_halting_on t +* (Initialized (stop I)) by SCMPDS_6:def 3;
A18: I is_closed_on t +* (Initialized (stop I)) by A14, SCMPDS_6:38;
A19: inspos 0 in dom (stop (for-down a,i,n,I)) by SCMPDS_4:75;
A20: IC (t +* (Initialized (stop (for-down a,i,n,I)))) = inspos 0 by SCMPDS_6:21;
A21: for-down a,i,n,I = (a,i <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo a,i,(- n))) ';' (goto (- ((card I) + 2)))) by Th15;
A22: Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(0 + 1) = Following (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),0 ) by AMI_1:14
.= Following (t +* (Initialized (stop (for-down a,i,n,I)))) by AMI_1:13
.= Exec (a,i <=0_goto ((card I) + 3)),(t +* (Initialized (stop (for-down a,i,n,I)))) by A21, SCMPDS_6:22 ;
A23: ( not a in dom (Initialized (stop (for-down a,i,n,I))) & a in dom t ) by SCMPDS_2:49, SCMPDS_4:31;
A24: ( not DataLoc (s . a),i in dom (Initialized (stop (for-down a,i,n,I))) & DataLoc (s . a),i in dom t ) by SCMPDS_2:49, SCMPDS_4:31;
(t +* (Initialized (stop (for-down a,i,n,I)))) . (DataLoc ((t +* (Initialized (stop (for-down a,i,n,I)))) . a),i) = (t +* (Initialized (stop (for-down a,i,n,I)))) . (DataLoc (s . a),i) by A12, A23, FUNCT_4:12
.= t . (DataLoc (s . a),i) by A24, FUNCT_4:12 ;
then A25: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1) = Next (IC (t +* (Initialized (stop (for-down a,i,n,I))))) by A13, A22, SCMPDS_2:68
.= inspos (0 + 1) by A20 ;
A26: DataPart (t +* (Initialized (stop I))) = DataPart (t +* (Initialized (stop (for-down a,i,n,I)))) by SCMPDS_4:24, SCMPDS_4:36;
now
let a be Int_position ; :: thesis: (t +* (Initialized (stop I))) . a = (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1) . a
thus (t +* (Initialized (stop I))) . a = (t +* (Initialized (stop (for-down a,i,n,I)))) . a by A26, SCMPDS_4:23
.= (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1) . a by A22, SCMPDS_2:68 ; :: thesis: verum
end;
then A27: DataPart (t +* (Initialized (stop I))) = DataPart (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1) by SCMPDS_4:23;
set m2 = LifeSpan (t +* (Initialized (stop I)));
set t5 = Computation (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1),(LifeSpan (t +* (Initialized (stop I))));
set l1 = inspos ((card I) + 1);
A28: dom (t | NAT ) = NAT by SCMPDS_6:1;
A29: now
assume a in dom (t | NAT ) ; :: thesis: contradiction
then reconsider l = a as Instruction-Location of SCMPDS by A28, AMI_1:def 4;
l = a ;
hence contradiction by SCMPDS_2:53; :: thesis: verum
end;
A30: now
assume DataLoc (s . a),i in dom (t | NAT ) ; :: thesis: contradiction
then reconsider l = DataLoc (s . a),i as Instruction-Location of SCMPDS by A28, AMI_1:def 4;
l = DataLoc (s . a),i ;
hence contradiction by SCMPDS_2:53; :: thesis: verum
end;
(card I) + 1 < (card I) + 3 by XREAL_1:8;
then A31: inspos ((card I) + 1) in dom (for-down a,i,n,I) by Th61;
A32: for-down a,i,n,I c= Initialized (stop (for-down a,i,n,I)) by SCMPDS_6:17;
Initialized (stop (for-down a,i,n,I)) c= t +* (Initialized (stop (for-down a,i,n,I))) by FUNCT_4:26;
then A33: for-down a,i,n,I c= t +* (Initialized (stop (for-down a,i,n,I))) by A32, XBOOLE_1:1;
Shift I,1 c= for-down a,i,n,I by Lm5;
then Shift I,1 c= t +* (Initialized (stop (for-down a,i,n,I))) by A33, XBOOLE_1:1;
then A34: Shift I,1 c= Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1 by AMI_1:81;
then A35: DataPart (Computation (t +* (Initialized (stop I))),(LifeSpan (t +* (Initialized (stop I))))) = DataPart (Computation (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1),(LifeSpan (t +* (Initialized (stop I))))) by A4, A15, A17, A18, A25, A27, Th36;
then A36: (Computation (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1),(LifeSpan (t +* (Initialized (stop I))))) . a = (Computation (t +* (Initialized (stop I))),(LifeSpan (t +* (Initialized (stop I))))) . a by SCMPDS_4:23
.= (Result (t +* (Initialized (stop I)))) . a by A16, AMI_1:122
.= s . a by A12, A14, A29, FUNCT_4:12 ;
A37: (Computation (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1),(LifeSpan (t +* (Initialized (stop I))))) . (DataLoc (s . a),i) = (Computation (t +* (Initialized (stop I))),(LifeSpan (t +* (Initialized (stop I))))) . (DataLoc (s . a),i) by A35, SCMPDS_4:23
.= (Result (t +* (Initialized (stop I)))) . (DataLoc (s . a),i) by A16, AMI_1:122
.= t . (DataLoc (s . a),i) by A14, A30, FUNCT_4:12 ;
set m3 = (LifeSpan (t +* (Initialized (stop I)))) + 1;
set t6 = Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((LifeSpan (t +* (Initialized (stop I)))) + 1);
A38: IC (Computation (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1),(LifeSpan (t +* (Initialized (stop I))))) = inspos ((card I) + 1) by A4, A15, A17, A18, A25, A27, A34, Th36;
A39: Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((LifeSpan (t +* (Initialized (stop I)))) + 1) = Computation (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1),(LifeSpan (t +* (Initialized (stop I)))) by AMI_1:51;
then A40: CurInstr (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((LifeSpan (t +* (Initialized (stop I)))) + 1)) = (Computation (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1),(LifeSpan (t +* (Initialized (stop I))))) . (inspos ((card I) + 1)) by A4, A15, A17, A18, A25, A27, A34, Th36
.= (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1) . (inspos ((card I) + 1)) by AMI_1:54
.= (t +* (Initialized (stop (for-down a,i,n,I)))) . (inspos ((card I) + 1)) by AMI_1:54
.= (for-down a,i,n,I) . (inspos ((card I) + 1)) by A31, A33, GRFUNC_1:8
.= AddTo a,i,(- n) by Th62 ;
set t7 = Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1);
A41: Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) = Following (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((LifeSpan (t +* (Initialized (stop I)))) + 1)) by AMI_1:14
.= Exec (AddTo a,i,(- n)),(Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((LifeSpan (t +* (Initialized (stop I)))) + 1)) by A40 ;
then A42: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) = Next (IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((LifeSpan (t +* (Initialized (stop I)))) + 1))) by SCMPDS_2:60
.= inspos (((card I) + 1) + 1) by A38, A39, NAT_1:39
.= inspos ((card I) + (1 + 1)) ;
DataLoc ((Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((LifeSpan (t +* (Initialized (stop I)))) + 1)) . a),i = DataLoc (s . a),i by A36, AMI_1:51;
then A43: (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . a = (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((LifeSpan (t +* (Initialized (stop I)))) + 1)) . a by A5, A41, SCMPDS_2:60
.= s . a by A36, AMI_1:51 ;
set l2 = inspos ((card I) + 2);
(card I) + 2 < (card I) + 3 by XREAL_1:8;
then A44: inspos ((card I) + 2) in dom (for-down a,i,n,I) by Th61;
A45: CurInstr (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) = (t +* (Initialized (stop (for-down a,i,n,I)))) . (inspos ((card I) + 2)) by A42, AMI_1:54
.= (for-down a,i,n,I) . (inspos ((card I) + 2)) by A33, A44, GRFUNC_1:8
.= goto (- ((card I) + 2)) by Th62 ;
set m5 = (((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1;
set t8 = Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1);
A46: Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1) = Following (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) by AMI_1:14
.= Exec (goto (- ((card I) + 2))),(Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) by A45 ;
then A47: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1)) = ICplusConst (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)),(0 - ((card I) + 2)) by SCMPDS_2:66
.= inspos 0 by A42, Th1 ;
A48: (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1)) . a = s . a by A43, A46, SCMPDS_2:66;
A49: now
let x be Int_position ; :: thesis: ( x in X implies (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1)) . x = s . x )
assume A50: x in X ; :: thesis: (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1)) . x = s . x
A51: now
assume x in dom (t | NAT ) ; :: thesis: contradiction
then reconsider l = x as Instruction-Location of SCMPDS by A28, AMI_1:def 4;
l = x ;
hence contradiction by SCMPDS_2:53; :: thesis: verum
end;
(Computation (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1),(LifeSpan (t +* (Initialized (stop I))))) . x = (Computation (t +* (Initialized (stop I))),(LifeSpan (t +* (Initialized (stop I))))) . x by A35, SCMPDS_4:23
.= (Result (t +* (Initialized (stop I)))) . x by A16, AMI_1:122
.= (IExec I,t) . x by A51, FUNCT_4:12
.= t . x by A6, A11, A12, A50
.= s . x by A11, A50 ;
then (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . x = s . x by A2, A36, A39, A41, A50, SCMPDS_2:60;
hence (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1)) . x = s . x by A46, SCMPDS_2:66; :: thesis: verum
end;
A52: (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1)) . (DataLoc (s . a),i) = (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),(((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1)) . (DataLoc (s . a),i) by A46, SCMPDS_2:66
.= (t . (DataLoc (s . a),i)) + (- n) by A36, A37, A39, A41, SCMPDS_2:60 ;
- (- n) > 0 by A3;
then - n < 0 ;
then - n <= - 1 by INT_1:21;
then A53: (- n) + (t . (DataLoc (s . a),i)) <= (- 1) + (t . (DataLoc (s . a),i)) by XREAL_1:8;
(t . (DataLoc (s . a),i)) - 1 <= k by A10, XREAL_1:22;
then (- n) + (t . (DataLoc (s . a),i)) <= k by A53, XXREAL_0:2;
then A54: ( for-down a,i,n,I is_closed_on Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1) & for-down a,i,n,I is_halting_on Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1) ) by A9, A48, A49, A52;
A55: (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1)) +* (Initialized (stop (for-down a,i,n,I))) = Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1) by A47, Th37;
now
let k be Element of NAT ; :: thesis: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),b1) in dom (stop (for-down a,i,n,I))
per cases ( k < (((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1 or k >= (((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1 ) ;
suppose k < (((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1 ; :: thesis: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),b1) in dom (stop (for-down a,i,n,I))
then k <= ((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1 by INT_1:20;
then A56: ( k <= (LifeSpan (t +* (Initialized (stop I)))) + 1 or k = ((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1 ) by NAT_1:8;
hereby :: thesis: verum
per cases ( k <= LifeSpan (t +* (Initialized (stop I))) or k = (LifeSpan (t +* (Initialized (stop I)))) + 1 or k = ((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1 ) by A56, NAT_1:8;
suppose A57: k <= LifeSpan (t +* (Initialized (stop I))) ; :: thesis: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I))
hereby :: thesis: verum
per cases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; :: thesis: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I))
hence IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I)) by A19, A20, AMI_1:13; :: thesis: verum
end;
suppose k <> 0 ; :: thesis: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I))
then consider kn being Nat such that
A58: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def 13;
kn < k by A58, XREAL_1:31;
then kn < LifeSpan (t +* (Initialized (stop I))) by A57, XXREAL_0:2;
then A59: (IC (Computation (t +* (Initialized (stop I))),kn)) + 1 = IC (Computation (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),1),kn) by A4, A15, A17, A18, A25, A27, A34, Th34;
A60: IC (Computation (t +* (Initialized (stop I))),kn) in dom (stop I) by A14, SCMPDS_6:def 2;
reconsider lm = IC (Computation (t +* (Initialized (stop I))),kn) as Element of NAT by ORDINAL1:def 13;
lm < card (stop I) by A60, SCMPDS_4:1;
then lm < (card I) + 1 by SCMPDS_5:7;
then A62: 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 A62, XXREAL_0:2;
then A63: lm + 1 < card (stop (for-down a,i,n,I)) by Lm4;
IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) = (inspos lm) + 1 by A58, A59, AMI_1:51
.= inspos (lm + 1) ;
hence IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I)) by A63, SCMPDS_4:1; :: thesis: verum
end;
end;
end;
end;
suppose A64: k = (LifeSpan (t +* (Initialized (stop I)))) + 1 ; :: thesis: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I))
inspos ((card I) + 1) in dom (stop (for-down a,i,n,I)) by A31, SCMPDS_6:18;
hence IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I)) by A4, A15, A17, A18, A25, A27, A34, A39, A64, Th36; :: thesis: verum
end;
suppose k = ((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1 ; :: thesis: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I))
hence IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I)) by A42, A44, SCMPDS_6:18; :: thesis: verum
end;
end;
end;
end;
suppose k >= (((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1 ; :: thesis: IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),b1) in dom (stop (for-down a,i,n,I))
then consider nn being Nat such that
A65: k = ((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 13;
Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k = Computation ((Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1)) +* (Initialized (stop (for-down a,i,n,I)))),nn by A55, A65, AMI_1:51;
hence IC (Computation (t +* (Initialized (stop (for-down a,i,n,I)))),k) in dom (stop (for-down a,i,n,I)) by A54, SCMPDS_6:def 2; :: thesis: verum
end;
end;
end;
hence for-down a,i,n,I is_closed_on t by SCMPDS_6:def 2; :: thesis: for-down a,i,n,I is_halting_on t
Computation (t +* (Initialized (stop (for-down a,i,n,I)))),((((LifeSpan (t +* (Initialized (stop I)))) + 1) + 1) + 1) is halting by A54, A55, SCMPDS_6:def 3;
then t +* (Initialized (stop (for-down a,i,n,I))) is halting by AMI_1:93;
hence for-down 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;
A66: for k being Element of NAT holds S1[k] from NAT_1:sch 1(A7, A8);
 reconsider n = s . (DataLoc (s . a),i) as Element of NAT by A1, INT_1:16;
A67: S1[n] by A66;
for x being Int_position st x in X holds
s . x = s . x ;
hence ( for-down a,i,n,I is_closed_on s & for-down a,i,n,I is_halting_on s ) by A67; :: thesis: verum