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, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let s be State of SCMPDS; :: thesis: for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let I be halt-free shiftable Program of SCMPDS; :: thesis: for a, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let a, x, y be Int_position ; :: thesis: for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let i, c be Integer; :: thesis: for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let n be Element of NAT ; :: thesis: ( n > 0 & s . x >= (s . y) + c & ( for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) implies ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),i);
set J = I ';' (AddTo (a,i,(- n)));
assume A1: n > 0 ; :: thesis: ( not s . x >= (s . y) + c or ex t being State of SCMPDS ex Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st
( t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) or ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) )
defpred S₁[ set ] means ex t being State of SCMPDS st
( t = $1 & t . x >= (t . y) + c );
assume A2: s . x >= (s . y) + c ; :: thesis: ( ex t being State of SCMPDS ex Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st
( t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) or ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) )
A3: S₁[ Dstate s]

proof

take t = Dstate s; :: thesis: ( t = Dstate s & t . x >= (t . y) + c )
thus t = Dstate s ; :: thesis: t . x >= (t . y) + c
t . x >= (s . y) + c by A2, SCMPDS_8:4;
hence t . x >= (t . y) + c by SCMPDS_8:4; :: thesis: verum

end;

assume A4: for t being State of SCMPDS
for Q being the Instructions of SCMPDS -valued ManySortedSet of NAT st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ; :: thesis: ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )

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 ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & S₁[ Dstate (IExec ((I ';' (AddTo (a,i,(- n)))),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 ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & S₁[ Dstate (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))] ) )
assume that
A6: S₁[ Dstate t] and
A7: ( t . a = s . a & t . (DataLoc ((s . a),i)) > 0 ) ; :: thesis: ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & S₁[ Dstate (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))] )
consider v being State of SCMPDS such that
A8: v = Dstate t and
A9: v . x >= (v . y) + c by A6;
t . x = v . x by A8, SCMPDS_8:4;
then A10: t . x >= (t . y) + c by A8, A9, SCMPDS_8:4;
hence ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q ) by A4, A7; :: thesis: S₁[ Dstate (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))]
thus S₁[ Dstate (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))] :: thesis: verum

proof

take v = Dstate (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)); :: thesis: ( v = Dstate (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) & v . x >= (v . y) + c )
thus v = Dstate (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) ; :: thesis: v . x >= (v . y) + c
v . x = (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x by SCMPDS_8:4;
then v . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c by A4, A7, A10;
hence v . x >= (v . y) + c by SCMPDS_8:4; :: thesis: verum

end;

end;

( ( S₁[s] or not S₁[s] ) & for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) from SCPISORT:sch 1(A1, A3, A5);
hence ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) ; :: thesis: verum