let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: for s being 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 Object-Kind of SCMPDS),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(IExec (I,P,s)))
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 X being set
for f being Function of (product the Object-Kind of SCMPDS),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(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 Object-Kind of SCMPDS),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(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 Object-Kind of SCMPDS),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(IExec (I,P,s)))
let i be Integer; :: thesis: for X being set
for f being Function of (product the Object-Kind of SCMPDS),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(IExec (I,P,s)))
let X be set ; :: thesis: for f being Function of (product the Object-Kind of SCMPDS),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(IExec (I,P,s)))
let f be Function of (product the Object-Kind of SCMPDS),NAT; :: thesis: ( s . (DataLoc ((s . a),i)) < 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) implies IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(IExec (I,P,s))) )
set b = DataLoc ((s . a),i);
deffunc H₁( State of SCMPDS) -> Element of NAT = f . $1;
defpred S₁[ State of SCMPDS] means for x being Int_position st x in X holds
$1 . x = s . x;
assume A2: s . (DataLoc ((s . a),i)) < 0 ; :: thesis: ( ex t being State of SCMPDS st
( f . (Dstate t) = 0 & not t . (DataLoc ((s . a),i)) >= 0 ) or ex t being State of SCMPDS ex Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) or IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(IExec (I,P,s))) )
assume for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ; :: thesis: ( ex t being State of SCMPDS ex Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) or IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(IExec (I,P,s))) )
then A3: for t being State of SCMPDS st S₁[ Dstate t] & H₁( Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ;
assume A4: for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT 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 . (Dstate (IExec (I,Q,t))) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ; :: thesis: IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(IExec (I,P,s)))

A5: now

let t be State of SCMPDS; :: thesis: for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st S₁[ Dstate t] & 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 & H₁( Dstate (IExec (I,Q,t))) < H₁( Dstate t) & S₁[ Dstate (IExec (I,Q,t))] )
let Q be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: ( S₁[ Dstate t] & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 implies ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H₁( Dstate (IExec (I,Q,t))) < H₁( Dstate t) & S₁[ Dstate (IExec (I,Q,t))] ) )
set v = Dstate t;
assume that
A6: S₁[ Dstate t] and
A7: ( t . a = s . a & t . (DataLoc ((s . a),i)) < 0 ) ; :: thesis: ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H₁( Dstate (IExec (I,Q,t))) < H₁( Dstate t) & S₁[ Dstate (IExec (I,Q,t))] )
set It = IExec (I,Q,t);

A8: now

let x be Int_position ; :: thesis: ( x in X implies t . x = s . x )
assume x in X ; :: thesis: t . x = s . x
then (Dstate t) . x = s . x by A6;
hence t . x = s . x by Th4; :: thesis: verum

end;

hence ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H₁( Dstate (IExec (I,Q,t))) < H₁( Dstate t) ) by A4, A7; :: thesis: S₁[ Dstate (IExec (I,Q,t))]
thus S₁[ Dstate (IExec (I,Q,t))] :: thesis: verum

proof

set v = Dstate (IExec (I,Q,t));

hereby :: thesis: verum

let x be Int_position ; :: thesis: ( x in X implies (Dstate (IExec (I,Q,t))) . x = s . x )
assume A9: x in X ; :: thesis: (Dstate (IExec (I,Q,t))) . x = s . x
then (IExec (I,Q,t)) . x = t . x by A4, A7, A8;
then (Dstate (IExec (I,Q,t))) . x = t . x by Th4;
hence (Dstate (IExec (I,Q,t))) . x = s . x by A8, A9; :: thesis: verum

end;

end;

end;

A10: S₁[ Dstate s] by Th4;
( ( H₁(s) = H₁(s) or S₁[s] ) & IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(IExec (I,P,s))) ) from SCMPDS_8:sch 2(A2, A3, A10, A5);
hence IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(IExec (I,P,s))) ; :: thesis: verum