let s be State of SCMPDS; :: thesis: for I being halt-free shiftable Program of SCMPDS
for a, x1, x2, x3, x4 being Int_position
for i, c, md being Integer st card I > 0 & s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being State of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & 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)) & (IExec (I,t)) . x4 = (((IExec (I,t)) . x3) - c) + ((IExec (I,t)) . x1) & md <= ((IExec (I,t)) . x3) - c & (IExec (I,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s & while>0 (a,i,I) is_halting_on s & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),s) = IExec ((while>0 (a,i,I)),(IExec (I,s))) ) )
let I be halt-free shiftable Program of SCMPDS; :: thesis: for a, x1, x2, x3, x4 being Int_position
for i, c, md being Integer st card I > 0 & s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being State of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & 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)) & (IExec (I,t)) . x4 = (((IExec (I,t)) . x3) - c) + ((IExec (I,t)) . x1) & md <= ((IExec (I,t)) . x3) - c & (IExec (I,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s & while>0 (a,i,I) is_halting_on s & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),s) = IExec ((while>0 (a,i,I)),(IExec (I,s))) ) )
let a, x1, x2, x3, x4 be Int_position ; :: thesis: for i, c, md being Integer st card I > 0 & s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being State of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & 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)) & (IExec (I,t)) . x4 = (((IExec (I,t)) . x3) - c) + ((IExec (I,t)) . x1) & md <= ((IExec (I,t)) . x3) - c & (IExec (I,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s & while>0 (a,i,I) is_halting_on s & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),s) = IExec ((while>0 (a,i,I)),(IExec (I,s))) ) )
let i, c, md be Integer; :: thesis: ( card I > 0 & s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being State of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & 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)) & (IExec (I,t)) . x4 = (((IExec (I,t)) . x3) - c) + ((IExec (I,t)) . x1) & md <= ((IExec (I,t)) . x3) - c & (IExec (I,t)) . x2 = t . x2 ) ) implies ( while>0 (a,i,I) is_closed_on s & while>0 (a,i,I) is_halting_on s & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),s) = IExec ((while>0 (a,i,I)),(IExec (I,s))) ) ) )
set b = DataLoc ((s . a),i);
assume A1: card I > 0 ; :: thesis: ( not s . x4 = ((s . x3) - c) + (s . x1) or not md <= (s . x3) - c or ex t being State of SCMPDS st
( t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (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)) & (IExec (I,t)) . x4 = (((IExec (I,t)) . x3) - c) + ((IExec (I,t)) . x1) & md <= ((IExec (I,t)) . x3) - c & (IExec (I,t)) . x2 = t . x2 ) ) or ( while>0 (a,i,I) is_closed_on s & while>0 (a,i,I) is_halting_on s & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),s) = IExec ((while>0 (a,i,I)),(IExec (I,s))) ) ) )
defpred S1[ set ] means ex t being State of SCMPDS st
( t = $1 & t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 );
assume that
A2: s . x4 = ((s . x3) - c) + (s . x1) and
A3: md <= (s . x3) - c ; :: thesis: ( ex t being State of SCMPDS st
( t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (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)) & (IExec (I,t)) . x4 = (((IExec (I,t)) . x3) - c) + ((IExec (I,t)) . x1) & md <= ((IExec (I,t)) . x3) - c & (IExec (I,t)) . x2 = t . x2 ) ) or ( while>0 (a,i,I) is_closed_on s & while>0 (a,i,I) is_halting_on s & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),s) = IExec ((while>0 (a,i,I)),(IExec (I,s))) ) ) )
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 SCMPDS_8:5;
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: ( H1( Dstate t) = 0 iff t . (DataLoc ((s . a),i)) <= 0 )
thus ( H1( Dstate t) = 0 implies t . (DataLoc ((s . a),i)) <= 0 ) :: thesis: ( t . (DataLoc ((s . a),i)) <= 0 implies H1( Dstate t) = 0 )
proof
assume A6: H1( Dstate t) = 0 ; :: thesis: t . (DataLoc ((s . a),i)) <= 0
assume t . (DataLoc ((s . a),i)) > 0 ; :: thesis: contradiction
then (Dstate t) . (DataLoc ((s . a),i)) > 0 by SCMPDS_8:4;
hence contradiction by A4, A6; :: thesis: verum
end;
assume t . (DataLoc ((s . a),i)) <= 0 ; :: thesis: H1( Dstate t) = 0
then (Dstate t) . (DataLoc ((s . a),i)) <= 0 by SCMPDS_8:4;
hence H1( Dstate t) = 0 by A4; :: thesis: verum
end;
then A7: for t being State of SCMPDS st S1[ Dstate t] & H1( Dstate t) = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ;
assume A8: for t being State of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & 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)) & (IExec (I,t)) . x4 = (((IExec (I,t)) . x3) - c) + ((IExec (I,t)) . x1) & md <= ((IExec (I,t)) . x3) - c & (IExec (I,t)) . x2 = t . x2 ) ; :: thesis: ( while>0 (a,i,I) is_closed_on s & while>0 (a,i,I) is_halting_on s & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),s) = IExec ((while>0 (a,i,I)),(IExec (I,s))) ) )
A9: now
let t be State of SCMPDS; :: thesis: ( S1[ Dstate t] & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 implies ( (IExec (I,t)) . a = t . a & I is_closed_on t & I is_halting_on t & H1( Dstate (IExec (I,t))) < H1( Dstate t) & S1[ Dstate (IExec (I,t))] ) )
assume that
A10: S1[ Dstate t] and
A11: t . a = s . a and
A12: t . (DataLoc ((s . a),i)) > 0 ; :: thesis: ( (IExec (I,t)) . a = t . a & I is_closed_on t & I is_halting_on t & H1( Dstate (IExec (I,t))) < H1( Dstate t) & S1[ Dstate (IExec (I,t))] )
set It = IExec (I,t);
set t2 = Dstate (IExec (I,t));
set t1 = Dstate t;
consider v being State of SCMPDS such that
A13: v = Dstate t and
A14: v . x4 = ((v . x3) - c) + (v . x1) and
A15: md <= (v . x3) - c and
A16: v . x2 = s . x2 by A10;
A17: t . x2 = s . x2 by A13, A16, SCMPDS_8:4;
t . x4 = ((v . x3) - c) + (v . x1) by A13, A14, SCMPDS_8:4;
then t . x4 = ((t . x3) - c) + (v . x1) by A13, SCMPDS_8:4;
then A18: t . x4 = ((t . x3) - c) + (t . x1) by A13, SCMPDS_8:4;
A19: md <= (t . x3) - c by A13, A15, SCMPDS_8:4;
hence ( (IExec (I,t)) . a = t . a & I is_closed_on t & I is_halting_on t ) by A8, A11, A12, A18, A17; :: thesis: ( H1( Dstate (IExec (I,t))) < H1( Dstate t) & S1[ Dstate (IExec (I,t))] )
thus H1( Dstate (IExec (I,t))) < H1( Dstate t) :: thesis: S1[ Dstate (IExec (I,t))]
proof
(Dstate t) . (DataLoc ((s . a),i)) > 0 by A12, SCMPDS_8:4;
then A20: H1( Dstate t) = (Dstate t) . (DataLoc ((s . a),i)) by A4
.= t . (DataLoc ((s . a),i)) by SCMPDS_8:4 ;
assume A21: H1( Dstate (IExec (I,t))) >= H1( Dstate t) ; :: thesis: contradiction
then (IExec (I,t)) . (DataLoc ((s . a),i)) > 0 by A5, A12, A20;
then (Dstate (IExec (I,t))) . (DataLoc ((s . a),i)) > 0 by SCMPDS_8:4;
then H1( Dstate (IExec (I,t))) = (Dstate (IExec (I,t))) . (DataLoc ((s . a),i)) by A4
.= (IExec (I,t)) . (DataLoc ((s . a),i)) by SCMPDS_8:4 ;
hence contradiction by A8, A11, A12, A18, A19, A17, A21, A20; :: thesis: verum
end;
thus S1[ Dstate (IExec (I,t))] :: thesis: verum
proof
take v = Dstate (IExec (I,t)); :: thesis: ( v = Dstate (IExec (I,t)) & v . x4 = ((v . x3) - c) + (v . x1) & md <= (v . x3) - c & v . x2 = s . x2 )
thus v = Dstate (IExec (I,t)) ; :: thesis: ( v . x4 = ((v . x3) - c) + (v . x1) & md <= (v . x3) - c & v . x2 = s . x2 )
(IExec (I,t)) . x4 = (((IExec (I,t)) . x3) - c) + ((IExec (I,t)) . x1) by A8, A11, A12, A18, A19, A17;
then v . x4 = (((IExec (I,t)) . x3) - c) + ((IExec (I,t)) . x1) by SCMPDS_8:4;
then v . x4 = ((v . x3) - c) + ((IExec (I,t)) . x1) by SCMPDS_8:4;
hence v . x4 = ((v . x3) - c) + (v . x1) by SCMPDS_8:4; :: thesis: ( md <= (v . x3) - c & v . x2 = s . x2 )
md <= ((IExec (I,t)) . x3) - c by A8, A11, A12, A18, A19, A17;
hence md <= (v . x3) - c by SCMPDS_8:4; :: thesis: v . x2 = s . x2
(IExec (I,t)) . x2 = t . x2 by A8, A11, A12, A18, A19, A17;
hence v . x2 = s . x2 by A17, SCMPDS_8:4; :: thesis: verum
end;
end;
A22: S1[ Dstate s]
proof
take t = Dstate s; :: thesis: ( t = Dstate s & t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 )
thus t = Dstate s ; :: thesis: ( t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 )
t . x4 = ((s . x3) - c) + (s . x1) by A2, SCMPDS_8:4;
then t . x4 = ((t . x3) - c) + (s . x1) by SCMPDS_8:4;
hence t . x4 = ((t . x3) - c) + (t . x1) by SCMPDS_8:4; :: thesis: ( md <= (t . x3) - c & t . x2 = s . x2 )
thus md <= (t . x3) - c by A3, SCMPDS_8:4; :: thesis: t . x2 = s . x2
thus t . x2 = s . x2 by SCMPDS_8:4; :: thesis: verum
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, A22, A9);
 hence ( while>0 (a,i,I) is_closed_on s & while>0 (a,i,I) is_halting_on s ) ; :: thesis: ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),s) = IExec ((while>0 (a,i,I)),(IExec (I,s))) )
assume A23: s . (DataLoc ((s . a),i)) > 0 ; :: thesis: IExec ((while>0 (a,i,I)),s) = IExec ((while>0 (a,i,I)),(IExec (I,s)))
( ( 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, A23, A7, A22, A9);
 hence IExec ((while>0 (a,i,I)),s) = IExec ((while>0 (a,i,I)),(IExec (I,s))) ; :: thesis: verum