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);
assume card I > 0 ; :: thesis:
consider f being Function of (product (the_Values_of SCMPDS)),NAT such that
A1: for s being State of SCMPDS holds
( ( s . b = s . c implies f . s = 0 ) & ( s . b <> s . c implies f . s = max (|.(s . b).|,|.(s . c).|) ) ) by Th2;
deffunc H₁( State of SCMPDS) -> Element of NAT = f . $1;
defpred S₁[ set ] means ex t being State of SCMPDS st
( t = $1 & t . b > 0 & t . c > 0 & t . (DataLoc (d,i)) = (t . b) - (t . c) );
assume that
A2: s . a = d and
A3: s . b > 0 and
A4: s . c > 0 and
A5: s . (DataLoc (d,i)) = (s . b) - (s . c) ; :: thesis:
assume A6: 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:

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
assume that
A8: S₁[t] and
A9: t . a = s . a and
A10: 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
A11: v = t and
A12: v . b > 0 and
A13: v . c > 0 and
A14: v . (DataLoc (d,i)) = (v . b) - (v . c) by A8;
A15: t . b > 0 by A11, A12;
A16: t . c > 0 by A11, A13;
A17: t . (DataLoc (d,i)) = (v . b) - (v . c) by A11, A14
.= (t . b) - (v . c) by A11
.= (t . b) - (t . c) by A11 ;
then A18: t . b <> t . c by A2, A10;
hence (IExec (I,Q,t)) . a = t . a by A2, A6, A9, A15, A16, A17; :: thesis:
thus ( I is_closed_on t,Q & I is_halting_on t,Q ) by A2, A6, A9, A15, A16, A17, A18; :: thesis:

then (t . b) - (t . c) > 0 by XREAL_1:50;
hence (IExec (I,Q,t)) . b > 0 by A2, A6, A9, A16, A17, A20; :: thesis:
thus (IExec (I,Q,t)) . c > 0 by A2, A6, A9, A16, A17, A20; :: thesis:
A21: (IExec (I,Q,t)) . b = (t . b) - (t . c) by A2, A6, A9, A16, A17, A20;

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 A16, A21, A22, 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 A2, A6, A9, A16, A17, A20, A22; :: thesis:

end;

hence (IExec (I,Q,t)) . b > 0 by A2, A6, A9, A15, A17, A18; :: thesis:
t . b < t . c by A18, A23, XXREAL_0:1;
then (t . c) - (t . b) > 0 by XREAL_1:50;
hence (IExec (I,Q,t)) . c > 0 by A2, A6, A9, A15, A17, A18, A23; :: thesis:
A24: (IExec (I,Q,t)) . c = (t . c) - (t . b) by A2, A6, A9, A15, A17, A18, A23;
A25: (IExec (I,Q,t)) . b = t . b by A2, A6, A9, A15, A17, A18, A23;

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 A15, A24, A26, 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 A18, A23, A25, A26, XXREAL_0:1; :: thesis:

end;

t . b <> t . c by A18;
then t . b <> t . c ;
then A27: H₁(t) = max (|.(t . b).|,|.(t . c).|) by A1
.= max (|.(t . b).|,|.(t . c).|)
.= max (|.(t . b).|,|.(t . c).|)
.= max ((t . b),|.(t . c).|) by A15, ABSVALUE:def 1
.= max ((t . b),(t . c)) by A16, ABSVALUE:def 1 ;
then H₁(t) >= t . b by XXREAL_0:25;
then A28: H₁(t) > 0 by A15;

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

hence H₁( Initialize (IExec (I,Q,t))) < H₁(t) by A1, A28; :: thesis:

end;

end;

A29: (IExec (I,Q,t)) . (DataLoc (d,i)) = ((IExec (I,Q,t)) . b) - ((IExec (I,Q,t)) . c) by A2, A6, A9, A15, A16, A17, A18;
thus S₁[ Initialize (IExec (I,Q,t))] :: thesis:

take v = Initialize (IExec (I,Q,t)); :: thesis:
thus v = Initialize (IExec (I,Q,t)) ; :: thesis:
thus ( v . b > 0 & v . c > 0 ) by A19, SCMPDS_5:15; :: thesis:
thus v . (DataLoc (d,i)) = ((IExec (I,Q,t)) . b) - ((IExec (I,Q,t)) . c) by A29, SCMPDS_5:15
.= (v . b) - ((IExec (I,Q,t)) . c) by SCMPDS_5:15
.= (v . b) - (v . c) by SCMPDS_5:15 ; :: thesis:

end;

A30: for t being 0 -started State of SCMPDS st S₁[t] & H₁(t) = 0 holds
t . (DataLoc ((s . a),i)) = 0

let t be 0 -started State of SCMPDS; :: thesis:
assume that
A31: S₁[t] and
A32: H₁(t) = 0 ; :: thesis:
consider v being State of SCMPDS such that
A33: v = t and
A34: v . b > 0 and
v . c > 0 and
A35: v . (DataLoc (d,i)) = (v . b) - (v . c) by A31;

assume t . b <> t . c ; :: thesis:
then t . b <> t . c ;
then t . b <> t . c ;
then A37: H₁(t) = max (|.(t . b).|,|.(t . c).|) by A1
.= max (|.(t . b).|,|.(t . c).|)
.= max (|.(t . b).|,|.(t . c).|) ;
t . b > 0 by A33, A34;
then |.(t . b).| > 0 by COMPLEX1:47;
hence contradiction by A32, A37, XXREAL_0:25; :: thesis:

end;

thus t . (DataLoc ((s . a),i)) = (v . b) - (v . c) by A2, A33, A35
.= (t . b) - (v . c) by A33
.= (t . b) - (t . c) by A33
.= 0 by A36 ; :: thesis:

end;