let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
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_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( 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 = 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 = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let s be 0 -started 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_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( 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 = 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 = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,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_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( 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 = 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 = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let a be Int_position; :: thesis: for i being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( 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 = 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 = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let i be Integer; :: thesis: for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( 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 = 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 = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let X be set ; :: thesis: for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( 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 = 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 = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let f be Function of (product (the_Values_of SCMPDS)),NAT; :: thesis: ( ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( 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 = 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 = t . x ) ) ) implies ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
set b = DataLoc ((s . a),i);
assume A1: for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ; :: thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence 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,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 = t . x ) ) ) or ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
assume A2: 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 = 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 = t . x ) ) ; :: thesis: ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
then A3: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in {} holds
t . x >= 0 + (t . (DataLoc ((s . a),i))) ) & ( 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,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 {} holds
(IExec (I,Q,t)) . x >= 0 + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ;
A4: for x being Int_position st x in {} holds
s . x >= 0 + (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 {} holds
t . x >= 0 + (t . (DataLoc ((s . a),i))) ) & ( 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,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 {} holds
(IExec (I,Q,t)) . x >= 0 + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) by A2;
hence ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) by A1, A4, Th22; :: thesis: ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
assume s . (DataLoc ((s . a),i)) > 0 ; :: thesis: IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
hence IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) by A1, A4, A3, Th23; :: thesis: verum