theorem Th23: :: SCMPDS_8:25

for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence 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,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))