let P be Instruction-Sequence of SCMPDS; :: thesis:
let s be 0 -started 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);
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)),P,s);
set ss = IExec ((while<>0 (a,i,I)),P,s);
defpred S₁[ set ] means ex t being 0 -started 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 0 -started State of SCMPDS
for Q being Instruction-Sequence 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,Q,t)) . a = d & I is_closed_on t,Q & I is_halting_on t,Q & ( t . b > t . c implies ( (IExec (I,Q,t)) . b = (t . b) - (t . c) & (IExec (I,Q,t)) . c = t . c ) ) & ( t . b <= t . c implies ( (IExec (I,Q,t)) . c = (t . c) - (t . b) & (IExec (I,Q,t)) . b = t . b ) ) & (IExec (I,Q,t)) . (DataLoc (d,i)) = ((IExec (I,Q,t)) . b) - ((IExec (I,Q,t)) . c) ) ; :: thesis:

A8: now

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
assume that
A9: S₁[t] and
A10: t . a = s . a and
A11: t . (DataLoc ((s . a),i)) <> 0 ; :: thesis:
set It = IExec (I,Q,t);
set t2 = Initialize (IExec (I,Q,t));
set t1 = t;
set x = (IExec (I,Q,t)) . b;
set y = (IExec (I,Q,t)) . c;
consider v being State of SCMPDS such that
A12: v = 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;
A18: t . c > 0 by A12, A14;
A19: t . (DataLoc (d,i)) = (v . b) - (v . c) by A12, A16
.= (t . b) - (v . c) by A12
.= (t . b) - (t . c) by A12 ;
then A20: ( t . b > t . c implies ( (IExec (I,Q,t)) . b = (t . b) - (t . c) & (IExec (I,Q,t)) . c = t . c ) ) by A3, A7, A10, A17, A18;
A21: t . b <> t . c by A3, A11, A19;
hence (IExec (I,Q,t)) . a = t . a by A3, A7, A10, A17, A18, A19; :: thesis:
thus ( I is_closed_on t,Q & I is_halting_on t,Q ) by A3, A7, A10, A17, A18, A19, A21; :: thesis:
A22: ( t . b <= t . c implies ( (IExec (I,Q,t)) . c = (t . c) - (t . b) & (IExec (I,Q,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_1:50;
hence (IExec (I,Q,t)) . b > 0 by A3, A7, A10, A17, A18, A19, A24; :: thesis:
thus (IExec (I,Q,t)) . c > 0 by A3, A7, A10, A17, A18, A19, A24; :: thesis:
thus ((IExec (I,Q,t)) . b) gcd ((IExec (I,Q,t)) . c) = (t . b) gcd (t . c) by A17, A18, A20, A22, PREPOWER:97; :: thesis:
A25: (IExec (I,Q,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,Q,t)) . b),((IExec (I,Q,t)) . c)) = (IExec (I,Q,t)) . b ; :: thesis:

hence max (((IExec (I,Q,t)) . b),((IExec (I,Q,t)) . c)) < max ((t . b),(t . c)) by A18, A25, A26, XREAL_1:44; :: thesis:

end;

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

hence max (((IExec (I,Q,t)) . b),((IExec (I,Q,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,Q,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_1:50;
hence (IExec (I,Q,t)) . c > 0 by A3, A7, A10, A17, A18, A19, A21, A27; :: thesis:
thus ((IExec (I,Q,t)) . b) gcd ((IExec (I,Q,t)) . c) = (t . b) gcd (t . c) by A17, A18, A20, A22, PREPOWER:97; :: thesis:
A28: (IExec (I,Q,t)) . c = (t . c) - (t . b) by A3, A7, A10, A17, A18, A19, A21, A27;
A29: (IExec (I,Q,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,Q,t)) . b),((IExec (I,Q,t)) . c)) = (IExec (I,Q,t)) . c ; :: thesis:

hence max (((IExec (I,Q,t)) . b),((IExec (I,Q,t)) . c)) < max ((t . b),(t . c)) by A17, A28, A30, XREAL_1:44; :: thesis:

end;

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

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

end;

thus H₁( Initialize (IExec (I,Q,t))) < H₁(t) :: thesis:

proof

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

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

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

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

end;

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

then H₁( Initialize (IExec (I,Q,t))) = max ((abs ((Initialize (IExec (I,Q,t))) . b)),(abs ((Initialize (IExec (I,Q,t))) . c))) by A2
.= max ((abs ((IExec (I,Q,t)) . b)),(abs ((Initialize (IExec (I,Q,t))) . c))) by SCMPDS_5:15
.= max ((abs ((IExec (I,Q,t)) . b)),(abs ((IExec (I,Q,t)) . c))) by SCMPDS_5:15
.= max (((IExec (I,Q,t)) . b),(abs ((IExec (I,Q,t)) . c))) by A23, ABSVALUE:def 1
.= max (((IExec (I,Q,t)) . b),((IExec (I,Q,t)) . c)) by A23, ABSVALUE:def 1 ;
hence H₁( Initialize (IExec (I,Q,t))) < H₁(t) by A23, A31; :: thesis:

end;

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

proof

take u = Initialize (IExec (I,Q,t)); :: thesis:
thus u = Initialize (IExec (I,Q,t)) ; :: thesis:
thus ( u . b > 0 & u . c > 0 ) by A23, SCMPDS_5:15; :: thesis:
thus (u . b) gcd (u . c) = ((IExec (I,Q,t)) . b) gcd (u . c) by SCMPDS_5:15
.= (t . b) gcd (t . c) by A23, SCMPDS_5:15
.= (v . b) gcd (t . c) by A12
.= (s . b) gcd (s . c) by A12, A15 ; :: thesis:
thus u . (DataLoc (d,i)) = ((IExec (I,Q,t)) . b) - ((IExec (I,Q,t)) . c) by A33, SCMPDS_5:15
.= (u . b) - ((IExec (I,Q,t)) . c) by SCMPDS_5:15
.= (u . b) - (u . c) by SCMPDS_5:15 ; :: thesis:

end;

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

proof

let t be 0 -started State of SCMPDS; :: thesis:
assume S₁[t] ; :: thesis:
then consider v being State of SCMPDS such that
A35: v = 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
.= (t . b) - (v . c) by A35
.= (t . b) - (t . c) by A35 ;

hereby :: thesis:

assume A39: H₁(t) = 0 ; :: thesis:

now

assume t . b <> t . c ; :: thesis:
then t . b <> t . c ;
then t . b <> t . c ;
then A40: H₁(t) = max ((abs (t . b)),(abs (t . c))) by A2
.= max ((abs (t . b)),(abs (t . c)))
.= max ((abs (t . b)),(abs (t . c))) ;
t . b > 0 by A35, A36;
then abs (t . b) > 0 by COMPLEX1:47;
hence contradiction by A39, A40, XXREAL_0:25; :: thesis:

end;

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

end;

thus ( t . (DataLoc ((s . a),i)) = 0 implies H₁(t) = 0 ) by A2, A38; :: thesis:

end;

A41: S₁[s] by A4, A5, A6;
A42: ( H₁( Initialize (IExec ((while<>0 (a,i,I)),P,s))) = 0 & S₁[ Initialize (IExec ((while<>0 (a,i,I)),P,s))] ) from SCPINVAR:sch 5(A34, A41, A8);
then consider w being 0 -started State of SCMPDS such that
A43: w = Initialize (IExec ((while<>0 (a,i,I)),P,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) ;
CI: (Initialize (IExec ((while<>0 (a,i,I)),P,s))) . (DataLoc ((s . a),i)) = (IExec ((while<>0 (a,i,I)),P,s)) . (DataLoc ((s . a),i)) by SCMPDS_5:15;
B: (Initialize (IExec ((while<>0 (a,i,I)),P,s))) . b = (IExec ((while<>0 (a,i,I)),P,s)) . b by SCMPDS_5:15;
C: (Initialize (IExec ((while<>0 (a,i,I)),P,s))) . c = (IExec ((while<>0 (a,i,I)),P,s)) . c by SCMPDS_5:15;
A47: (w . b) - (w . c) = (IExec ((while<>0 (a,i,I)),P,s)) . (DataLoc ((s . a),i)) by A3, A43, A46, SCMPDS_5:15
.= 0 by A34, A42, CI ;
then A48: abs (w . b) = (abs (w . b)) gcd (abs (w . c)) by NAT_D:32
.= (s . b) gcd (s . c) by A45, INT_2:34 ;
thus (IExec ((while<>0 (a,i,I)),P,s)) . b = (IExec ((while<>0 (a,i,I)),P,s)) . b
.= (s . b) gcd (s . c) by A43, A44, A48, B, ABSVALUE:def 1 ; :: thesis:
thus (IExec ((while<>0 (a,i,I)),P,s)) . c = (IExec ((while<>0 (a,i,I)),P,s)) . c
.= (s . b) gcd (s . c) by A43, A44, A47, A48, C, ABSVALUE:def 1 ; :: thesis: