let s be State of SCMPDS ; :: thesis:
let I be halt-free shiftable Program of SCMPDS ; :: thesis:
let a, b, c be Int_position ; :: thesis:
let i, d be Integer; :: thesis:
set ci = DataLoc (s . a),i;
assume A1: card I > 0 ; :: thesis:
consider f being Function of (product the Object-Kind of SCMPDS ),NAT such that
A2: for s being State of SCMPDS holds
( ( s . b = s . c implies f . s = 0 ) & ( s . b <> s . c implies f . s = max (abs (s . b)),(abs (s . c)) ) ) by Th6;
deffunc H₁( State of SCMPDS ) -> Element of NAT = f . $1;
set s1 = IExec (while<>0 a,i,I),s;
set ss = Dstate (IExec (while<>0 a,i,I),s);
defpred S₁[ set ] means ex t being State of SCMPDS st
( t = $1 & t . b > 0 & t . c > 0 & (t . b) gcd (t . c) = (s . b) gcd (s . c) & t . (DataLoc d,i) = (t . b) - (t . c) );
assume that
A3: s . a = d and
A4: s . b > 0 and
A5: s . c > 0 and
A6: s . (DataLoc d,i) = (s . b) - (s . c) ; :: thesis:
assume A7: for t being State of SCMPDS st t . b > 0 & t . c > 0 & t . a = d & t . (DataLoc d,i) = (t . b) - (t . c) & t . b <> t . c holds
( (IExec I,t) . a = d & I is_closed_on t & I is_halting_on t & ( t . b > t . c implies ( (IExec I,t) . b = (t . b) - (t . c) & (IExec I,t) . c = t . c ) ) & ( t . b <= t . c implies ( (IExec I,t) . c = (t . c) - (t . b) & (IExec I,t) . b = t . b ) ) & (IExec I,t) . (DataLoc d,i) = ((IExec I,t) . b) - ((IExec I,t) . c) ) ; :: thesis:

A8: now

let t be State of SCMPDS ; :: thesis:
assume that
A9: S₁[ Dstate t] and
A10: t . a = s . a and
A11: t . (DataLoc (s . a),i) <> 0 ; :: thesis:
set It = IExec I,t;
set t2 = Dstate (IExec I,t);
set t1 = Dstate t;
set x = (IExec I,t) . b;
set y = (IExec I,t) . c;
consider v being State of SCMPDS such that
A12: v = Dstate t and
A13: v . b > 0 and
A14: v . c > 0 and
A15: (v . b) gcd (v . c) = (s . b) gcd (s . c) and
A16: v . (DataLoc d,i) = (v . b) - (v . c) by A9;
A17: t . b > 0 by A12, A13, SCMPDS_8:4;
A18: t . c > 0 by A12, A14, SCMPDS_8:4;
A19: t . (DataLoc d,i) = (v . b) - (v . c) by A12, A16, SCMPDS_8:4
.= (t . b) - (v . c) by A12, SCMPDS_8:4
.= (t . b) - (t . c) by A12, SCMPDS_8:4 ;
then A20: ( t . b > t . c implies ( (IExec I,t) . b = (t . b) - (t . c) & (IExec I,t) . c = t . c ) ) by A3, A7, A10, A17, A18;
A21: t . b <> t . c by A3, A11, A19;
hence (IExec I,t) . a = t . a by A3, A7, A10, A17, A18, A19; :: thesis:
thus ( I is_closed_on t & I is_halting_on t ) by A3, A7, A10, A17, A18, A19, A21; :: thesis:
A22: ( t . b <= t . c implies ( (IExec I,t) . c = (t . c) - (t . b) & (IExec I,t) . b = t . b ) ) by A3, A7, A10, A17, A18, A19, A21;

A23: now

per cases ;

suppose A24: t . b > t . c ; :: thesis:

then (t . b) - (t . c) > 0 by XREAL_1:52;
hence (IExec I,t) . b > 0 by A3, A7, A10, A17, A18, A19, A24; :: thesis:
thus (IExec I,t) . c > 0 by A3, A7, A10, A17, A18, A19, A24; :: thesis:
thus ((IExec I,t) . b) gcd ((IExec I,t) . c) = (t . b) gcd (t . c) by A17, A18, A20, A22, PREPOWER:111; :: thesis:
A25: (IExec I,t) . b = (t . b) - (t . c) by A3, A7, A10, A17, A18, A19, A24;

hereby :: thesis:

A26: max (t . b),(t . c) = t . b by A24, XXREAL_0:def 10;

per cases by XXREAL_0:16;

suppose max ((IExec I,t) . b),((IExec I,t) . c) = (IExec I,t) . b ; :: thesis:

hence max ((IExec I,t) . b),((IExec I,t) . c) < max (t . b),(t . c) by A18, A25, A26, XREAL_1:46; :: thesis:

end;

suppose max ((IExec I,t) . b),((IExec I,t) . c) = (IExec I,t) . c ; :: thesis:

hence max ((IExec I,t) . b),((IExec I,t) . c) < max (t . b),(t . c) by A3, A7, A10, A17, A18, A19, A24, A26; :: thesis:

end;

suppose A27: t . b <= t . c ; :: thesis:

hence (IExec I,t) . b > 0 by A3, A7, A10, A17, A18, A19, A21; :: thesis:
t . b < t . c by A21, A27, XXREAL_0:1;
then (t . c) - (t . b) > 0 by XREAL_1:52;
hence (IExec I,t) . c > 0 by A3, A7, A10, A17, A18, A19, A21, A27; :: thesis:
thus ((IExec I,t) . b) gcd ((IExec I,t) . c) = (t . b) gcd (t . c) by A17, A18, A20, A22, PREPOWER:111; :: thesis:
A28: (IExec I,t) . c = (t . c) - (t . b) by A3, A7, A10, A17, A18, A19, A21, A27;
A29: (IExec I,t) . b = t . b by A3, A7, A10, A17, A18, A19, A21, A27;

hereby :: thesis:

A30: max (t . b),(t . c) = t . c by A27, XXREAL_0:def 10;

per cases by XXREAL_0:16;

suppose max ((IExec I,t) . b),((IExec I,t) . c) = (IExec I,t) . c ; :: thesis:

hence max ((IExec I,t) . b),((IExec I,t) . c) < max (t . b),(t . c) by A17, A28, A30, XREAL_1:46; :: thesis:

end;

suppose max ((IExec I,t) . b),((IExec I,t) . c) = (IExec I,t) . b ; :: thesis:

hence max ((IExec I,t) . b),((IExec I,t) . c) < max (t . b),(t . c) by A21, A27, A29, A30, XXREAL_0:1; :: thesis:

end;

thus H₁( Dstate (IExec I,t)) < H₁( Dstate t) :: thesis:

proof

(Dstate t) . b <> t . c by A21, SCMPDS_8:4;
then (Dstate t) . b <> (Dstate t) . c by SCMPDS_8:4;
then A31: H₁( Dstate t) = max (abs ((Dstate t) . b)),(abs ((Dstate t) . c)) by A2
.= max (abs (t . b)),(abs ((Dstate t) . c)) by SCMPDS_8:4
.= max (abs (t . b)),(abs (t . c)) by SCMPDS_8:4
.= max (t . b),(abs (t . c)) by A17, ABSVALUE:def 1
.= max (t . b),(t . c) by A18, ABSVALUE:def 1 ;
then H₁( Dstate t) >= t . b by XXREAL_0:25;
then A32: H₁( Dstate t) > 0 by A17, XXREAL_0:2;

per cases ) . b = (Dstate (IExec I,t)) . c or (Dstate (IExec I,t)) . b <> (Dstate (IExec I,t)) . c ) ;

suppose (Dstate (IExec I,t)) . b = (Dstate (IExec I,t)) . c ; :: thesis:

hence H₁( Dstate (IExec I,t)) < H₁( Dstate t) by A2, A32; :: thesis:

end;

suppose (Dstate (IExec I,t)) . b <> (Dstate (IExec I,t)) . c ; :: thesis:

then H₁( Dstate (IExec I,t)) = max (abs ((Dstate (IExec I,t)) . b)),(abs ((Dstate (IExec I,t)) . c)) by A2
.= max (abs ((IExec I,t) . b)),(abs ((Dstate (IExec I,t)) . c)) by SCMPDS_8:4
.= max (abs ((IExec I,t) . b)),(abs ((IExec I,t) . c)) by SCMPDS_8:4
.= max ((IExec I,t) . b),(abs ((IExec I,t) . c)) by A23, ABSVALUE:def 1
.= max ((IExec I,t) . b),((IExec I,t) . c) by A23, ABSVALUE:def 1 ;
hence H₁( Dstate (IExec I,t)) < H₁( Dstate t) by A23, A31; :: thesis:

end;

A33: (IExec I,t) . (DataLoc d,i) = ((IExec I,t) . b) - ((IExec I,t) . c) by A3, A7, A10, A17, A18, A19, A21;
thus S₁[ Dstate (IExec I,t)] :: thesis:

proof

take u = Dstate (IExec I,t); :: thesis:
thus u = Dstate (IExec I,t) ; :: thesis:
thus ( u . b > 0 & u . c > 0 ) by A23, SCMPDS_8:4; :: thesis:
thus (u . b) gcd (u . c) = ((IExec I,t) . b) gcd (u . c) by SCMPDS_8:4
.= (t . b) gcd (t . c) by A23, SCMPDS_8:4
.= (v . b) gcd (t . c) by A12, SCMPDS_8:4
.= (s . b) gcd (s . c) by A12, A15, SCMPDS_8:4 ; :: thesis:
thus u . (DataLoc d,i) = ((IExec I,t) . b) - ((IExec I,t) . c) by A33, SCMPDS_8:4
.= (u . b) - ((IExec I,t) . c) by SCMPDS_8:4
.= (u . b) - (u . c) by SCMPDS_8:4 ; :: thesis:

end;

A34: for t being State of SCMPDS st S₁[ Dstate t] holds
( H₁( Dstate t) = 0 iff t . (DataLoc (s . a),i) = 0 )

proof

let t be State of SCMPDS ; :: thesis:
assume S₁[ Dstate t] ; :: thesis:
then consider v being State of SCMPDS such that
A35: v = Dstate t and
A36: v . b > 0 and
v . c > 0 and
(v . b) gcd (v . c) = (s . b) gcd (s . c) and
A37: v . (DataLoc d,i) = (v . b) - (v . c) ;
A38: t . (DataLoc (s . a),i) = (v . b) - (v . c) by A3, A35, A37, SCMPDS_8:4
.= (t . b) - (v . c) by A35, SCMPDS_8:4
.= (t . b) - (t . c) by A35, SCMPDS_8:4 ;

hereby :: thesis:

assume A39: H₁( Dstate t) = 0 ; :: thesis:

now

assume t . b <> t . c ; :: thesis:
then (Dstate t) . b <> t . c by SCMPDS_8:4;
then (Dstate t) . b <> (Dstate t) . c by SCMPDS_8:4;
then A40: H₁( Dstate t) = max (abs ((Dstate t) . b)),(abs ((Dstate t) . c)) by A2
.= max (abs (t . b)),(abs ((Dstate t) . c)) by SCMPDS_8:4
.= max (abs (t . b)),(abs (t . c)) by SCMPDS_8:4 ;
t . b > 0 by A35, A36, SCMPDS_8:4;
then abs (t . b) > 0 by COMPLEX1:133;
hence contradiction by A39, A40, XXREAL_0:25; :: thesis:

end;

hence t . (DataLoc (s . a),i) = 0 by A38; :: thesis:

end;

hereby :: thesis:

assume t . (DataLoc (s . a),i) = 0 ; :: thesis:
then (Dstate t) . b = t . c by A38, SCMPDS_8:4;
then (Dstate t) . b = (Dstate t) . c by SCMPDS_8:4;
hence H₁( Dstate t) = 0 by A2; :: thesis:

end;

A41: S₁[ Dstate s]

proof

take t = Dstate s; :: thesis:
thus t = Dstate s ; :: thesis:
thus t . b > 0 by A4, SCMPDS_8:4; :: thesis:
thus t . c > 0 by A5, SCMPDS_8:4; :: thesis:
thus (t . b) gcd (t . c) = (s . b) gcd (t . c) by SCMPDS_8:4
.= (s . b) gcd (s . c) by SCMPDS_8:4 ; :: thesis:
thus t . (DataLoc d,i) = (s . b) - (s . c) by A6, SCMPDS_8:4
.= (t . b) - (s . c) by SCMPDS_8:4
.= (t . b) - (t . c) by SCMPDS_8:4 ; :: thesis:

end;

A42: ( H₁( Dstate (IExec (while<>0 a,i,I),s)) = 0 & S₁[ Dstate (IExec (while<>0 a,i,I),s)] ) from SCPINVAR:sch 5(A1, A34, A41, A8);
then consider w being State of SCMPDS such that
A43: w = Dstate (IExec (while<>0 a,i,I),s) and
A44: w . b > 0 and
w . c > 0 and
A45: (w . b) gcd (w . c) = (s . b) gcd (s . c) and
A46: w . (DataLoc d,i) = (w . b) - (w . c) ;
A47: (w . b) - (w . c) = (IExec (while<>0 a,i,I),s) . (DataLoc (s . a),i) by A3, A43, A46, SCMPDS_8:4
.= 0 by A34, A42 ;
then A48: abs (w . b) = (abs (w . b)) gcd (abs (w . c)) by NAT_D:32
.= (s . b) gcd (s . c) by A45, INT_2:51 ;
thus (IExec (while<>0 a,i,I),s) . b = (Dstate (IExec (while<>0 a,i,I),s)) . b by SCMPDS_8:4
.= (s . b) gcd (s . c) by A43, A44, A48, ABSVALUE:def 1 ; :: thesis:
thus (IExec (while<>0 a,i,I),s) . c = (Dstate (IExec (while<>0 a,i,I),s)) . c by SCMPDS_8:4
.= (s . b) gcd (s . c) by A43, A44, A47, A48, ABSVALUE:def 1 ; :: thesis: