let s be State of SCMPDS ; :: thesis:
let I be shiftable No-StopCode Program of SCMPDS ; :: thesis:
let a be Int_position ; :: thesis:
let i, c be Integer; :: thesis:
let X, Y be set ; :: thesis:
set b = DataLoc (s . a),i;
assume A1: card I > 0 ; :: thesis:
assume A2: for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc (s . a),i)) ; :: thesis:
assume A3: for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (s . a),i)) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a & t . (DataLoc (s . a),i) > 0 holds
( (IExec I,t) . a = t . a & I is_closed_on t & I is_halting_on t & (IExec I,t) . (DataLoc (s . a),i) < t . (DataLoc (s . a),i) & ( for x being Int_position st x in X holds
(IExec I,t) . x >= c + ((IExec I,t) . (DataLoc (s . a),i)) ) & ( for x being Int_position st x in Y holds
(IExec I,t) . x = t . x ) ) ; :: thesis:
defpred S₁[ State of SCMPDS ] means ( ( for x being Int_position st x in X holds
$1 . x >= c + ($1 . (DataLoc (s . a),i)) ) & ( for x being Int_position st x in Y holds
$1 . x = s . x ) );
consider f being Function of (product the Object-Kind of SCMPDS ),NAT such that
A4: for s being State of SCMPDS holds
( ( s . (DataLoc (s . a),i) <= 0 implies f . s = 0 ) & ( s . (DataLoc (s . a),i) > 0 implies f . s = s . (DataLoc (s . a),i) ) ) by Th5;
deffunc H₁( State of SCMPDS ) -> Element of NAT = f . $1;
A5: for t being State of SCMPDS holds
( H₁( Dstate t) = 0 iff t . (DataLoc (s . a),i) <= 0 )

proof

let t be State of SCMPDS ; :: thesis:
thus ( H₁( Dstate t) = 0 implies t . (DataLoc (s . a),i) <= 0 ) :: thesis:

proof

assume A6: H₁( Dstate t) = 0 ; :: thesis:
assume t . (DataLoc (s . a),i) > 0 ; :: thesis:
then (Dstate t) . (DataLoc (s . a),i) > 0 by Th4;
hence contradiction by A4, A6; :: thesis:

end;

assume t . (DataLoc (s . a),i) <= 0 ; :: thesis:
then (Dstate t) . (DataLoc (s . a),i) <= 0 by Th4;
hence H₁( Dstate t) = 0 by A4; :: thesis:

end;

then A7: for t being State of SCMPDS st S₁[ Dstate t] & H₁( Dstate t) = 0 holds
t . (DataLoc (s . a),i) <= 0 ;
A8: S₁[ Dstate s]

proof

set t = Dstate s;

hereby :: thesis:

let x be Int_position ; :: thesis:
assume x in X ; :: thesis:
then s . x >= c + (s . (DataLoc (s . a),i)) by A2;
then (Dstate s) . x >= c + (s . (DataLoc (s . a),i)) by Th4;
hence (Dstate s) . x >= c + ((Dstate s) . (DataLoc (s . a),i)) by Th4; :: thesis:

end;

thus for x being Int_position st x in Y holds
(Dstate s) . x = s . x by Th4; :: thesis:

end;

A9: now

let t be State of SCMPDS ; :: thesis:
assume A10: ( S₁[ Dstate t] & t . a = s . a & t . (DataLoc (s . a),i) > 0 ) ; :: thesis:
then consider v being State of SCMPDS such that
A11: ( v = Dstate t & ( for x being Int_position st x in X holds
v . x >= c + (v . (DataLoc (s . a),i)) ) & ( for x being Int_position st x in Y holds
v . x = s . x ) ) ;

A12: now

let x be Int_position ; :: thesis:
assume x in X ; :: thesis:
then v . x >= c + (v . (DataLoc (s . a),i)) by A11;
then t . x >= c + (v . (DataLoc (s . a),i)) by A11, Th4;
hence t . x >= c + (t . (DataLoc (s . a),i)) by A11, Th4; :: thesis:

end;

A13: now

let x be Int_position ; :: thesis:
assume x in Y ; :: thesis:
then v . x = s . x by A11;
hence t . x = s . x by A11, Th4; :: thesis:

end;

hence ( (IExec I,t) . a = t . a & I is_closed_on t & I is_halting_on t ) by A3, A10, A12; :: thesis:
set It = IExec I,t;
set t2 = Dstate (IExec I,t);
set t1 = Dstate t;
thus H₁( Dstate (IExec I,t)) < H₁( Dstate t) :: thesis:

proof

assume A14: H₁( Dstate (IExec I,t)) >= H₁( Dstate t) ; :: thesis:
(Dstate t) . (DataLoc (s . a),i) > 0 by A10, Th4;
then A15: H₁( Dstate t) = (Dstate t) . (DataLoc (s . a),i) by A4
.= t . (DataLoc (s . a),i) by Th4 ;
then (IExec I,t) . (DataLoc (s . a),i) > 0 by A5, A10, A14;
then (Dstate (IExec I,t)) . (DataLoc (s . a),i) > 0 by Th4;
then H₁( Dstate (IExec I,t)) = (Dstate (IExec I,t)) . (DataLoc (s . a),i) by A4
.= (IExec I,t) . (DataLoc (s . a),i) by Th4 ;
hence contradiction by A3, A10, A12, A13, A14, A15; :: thesis:

end;

thus S₁[ Dstate (IExec I,t)] :: thesis:

proof

set v = Dstate (IExec I,t);

hereby :: thesis:

let x be Int_position ; :: thesis:
assume x in X ; :: thesis:
then (IExec I,t) . x >= c + ((IExec I,t) . (DataLoc (s . a),i)) by A3, A10, A12, A13;
then (Dstate (IExec I,t)) . x >= c + ((IExec I,t) . (DataLoc (s . a),i)) by Th4;
hence (Dstate (IExec I,t)) . x >= c + ((Dstate (IExec I,t)) . (DataLoc (s . a),i)) by Th4; :: thesis:

end;

hereby :: thesis:

let x be Int_position ; :: thesis:
assume A16: x in Y ; :: thesis:
then (IExec I,t) . x = t . x by A3, A10, A12, A13;
then (Dstate (IExec I,t)) . x = t . x by Th4;
hence (Dstate (IExec I,t)) . x = s . x by A13, A16; :: thesis:

end;

( ( H₁(s) = H₁(s) or S₁[s] ) & while>0 a,i,I is_closed_on s & while>0 a,i,I is_halting_on s ) from SCMPDS_8:sch 3(A1, A7, A8, A9);
hence ( while>0 a,i,I is_closed_on s & while>0 a,i,I is_halting_on s ) ; :: thesis:
assume A17: s . (DataLoc (s . a),i) > 0 ; :: thesis:
( ( H₁(s) = H₁(s) or S₁[s] ) & IExec (while>0 a,i,I),s = IExec (while>0 a,i,I),(IExec I,s) ) from SCMPDS_8:sch 4(A1, A17, A7, A8, A9);
hence IExec (while>0 a,i,I),s = IExec (while>0 a,i,I),(IExec I,s) ; :: thesis: