set a = GBP ;
let s be 0 -started State of SCMPDS; :: thesis:
let m4, md be Element of NAT ; :: thesis:
assume that
A1: s . GBP = 0 and
A2: s . (intpos 5) > 0 and
A3: s . (intpos 4) = m4 + (s . (intpos 5)) and
A4: m4 >= 8 and
A5: s . (intpos 2) = md and
A6: md >= 8 ; :: thesis:
s . (intpos 5) >= 1 + 0 by A2, INT_1:20;
then reconsider m5 = (s . (intpos 5)) - 1 as Element of NAT by INT_1:16, XREAL_1:50;
A7: s . (intpos 5) = m5 + 1 ;
defpred S₁[ Element of NAT ] means for t being State of SCMPDS st t . GBP = 0 & t . (intpos 5) = $1 + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE 0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos (m4 + mE)) ) ) );

A8: now

let k be Element of NAT ; :: thesis:
assume A9: S₁[k] ; :: thesis:

now

set me = m4 + ((k + 1) + 1);
let t be State of SCMPDS; :: thesis:
set b = DataLoc ((t . GBP),5);
assume that
A10: t . GBP = 0 and
A11: t . (intpos 5) = (k + 1) + 1 and
A12: t . (intpos 4) = m4 + (t . (intpos 5)) and
A13: t . (intpos 2) = md ; :: thesis:
set It = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t);
X1: IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t) = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),(Initialize t)) by SCMPDS_5:48;
X0: t . GBP = (Initialize t) . GBP by SCMPDS_5:40;
X2: t . (intpos 1) = (Initialize t) . (intpos 1) by SCMPDS_5:40;
X3: t . (intpos 2) = (Initialize t) . (intpos 2) by SCMPDS_5:40;
X8: t . (intpos 3) = (Initialize t) . (intpos 3) by SCMPDS_5:40;
X4: t . (intpos 4) = (Initialize t) . (intpos 4) by SCMPDS_5:40;
X5: t . (intpos 5) = (Initialize t) . (intpos 5) by SCMPDS_5:40;
X6: t . (intpos md) = (Initialize t) . (intpos md) by SCMPDS_5:40;
X7: t . (intpos (m4 + ((k + 1) + 1))) = (Initialize t) . (intpos (m4 + ((k + 1) + 1))) by SCMPDS_5:40;
A14: m4 + ((k + 1) + 1) >= m4 by NAT_1:11;
then A15: m4 + ((k + 1) + 1) >= 8 by A4, XXREAL_0:2;
then A16: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 1) = t . (intpos 1) by A6, A10, A11, A12, A13, Lm8, X1, X2, X3, X4, X0;
A17: ( t . (intpos md) >= t . (intpos (m4 + ((k + 1) + 1))) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 4) = t . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 7) = t . (intpos 5) ) ) by A6, A10, A11, A12, A13, A15, Lm8, X1, X3, X4, X5, X0, X6, X7;
A18: DataLoc ((t . GBP),5) = intpos (0 + 5) by A10, SCMP_GCD:5;

A19: now

let v be State of SCMPDS; :: thesis:
assume that
A20: v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) and
A21: v . (intpos 2) = t . (intpos 2) and
A22: v . GBP = t . GBP and
A23: v . (DataLoc ((t . GBP),5)) > 0 ; :: thesis:
A24: m4 + (v . (DataLoc ((t . GBP),5))) > m4 + 0 by A23, XREAL_1:8;
then reconsider ME = v . (intpos 4) as Element of NAT by A20, INT_1:16;
Y1: IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v) = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),(Initialize v)) by SCMPDS_5:48;
Y2: v . (intpos md) = (Initialize v) . (intpos md) by SCMPDS_5:40;
Y3: v . (intpos ME) = (Initialize v) . (intpos ME) by SCMPDS_5:40;
Y4: v . (intpos 4) = (Initialize v) . (intpos 4) by SCMPDS_5:40;
Y8: v . (intpos 2) = (Initialize v) . (intpos 2) by SCMPDS_5:40;
Y5: v . (intpos 5) = (Initialize v) . (intpos 5) by SCMPDS_5:40;
Y6: v . GBP = (Initialize v) . GBP by SCMPDS_5:40;
v . (intpos 4) >= m4 by A20, A24, XXREAL_0:2;
then A25: ME >= 8 by A4, XXREAL_0:2;
then A26: ( v . (intpos md) < v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 5) = (v . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 4) = (v . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 7) = (v . (intpos 5)) - 1 ) ) by A6, A10, A13, A21, A22, Lm8, Y1, Y2, Y3, Y4, Y5, Y6, Y8;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . GBP = v . GBP by A6, A10, A13, A21, A22, A25, Lm8, Y1, Y4, Y6, Y8; :: thesis:
thus ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v ) by SCMPDS_6:34, SCMPDS_6:35; :: thesis:
A27: ( v . (intpos md) >= v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 4) = v . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 7) = v . (intpos 5) ) ) by A6, A10, A13, A21, A22, A25, Lm8, Y1, Y2, Y3, Y4, Y5, Y6, Y8;

hereby :: thesis:

per cases ;

suppose A28: v . (intpos md) < v . (intpos ME) ; :: thesis:

hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A18, A26, XREAL_1:148; :: thesis:
(m4 + (v . (DataLoc ((t . GBP),5)))) - 1 = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5))) by A18, A26, A28;
hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5))) by A20, A26, A28, XREAL_1:11; :: thesis:

end;

suppose A29: v . (intpos md) >= v . (intpos ME) ; :: thesis:

hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A6, A10, A13, A18, A21, A22, A23, A25, Lm8, Y1, Y2, Y3, Y4, Y6, Y8; :: thesis:
m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5))) < m4 + (v . (DataLoc ((t . GBP),5))) by A18, A23, A27, A29, XREAL_1:8;
hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5))) by A20, A27, A29, XXREAL_0:2; :: thesis:

end;

thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 2) = v . (intpos 2) by A6, A10, A13, A21, A22, A25, Lm8, Y1, Y4, Y6, Y8; :: thesis:

end;

A30: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 3) = t . (intpos 3) by A6, A10, A11, A12, A13, A15, Lm8, X1, X3, X4, X0, X8;
A31: ( t . (intpos md) < t . (intpos (m4 + ((k + 1) + 1))) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 5) = (t . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 4) = (t . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 7) = (t . (intpos 5)) - 1 ) ) by A6, A10, A11, A12, A13, A15, Lm8, X1, X3, X4, X5, X0, X6, X7;
A32: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . GBP = 0 by A6, A10, A11, A12, A13, A15, Lm8, X1, X3, X4, X0;
then A33: DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . GBP),5) = intpos (0 + 5) by SCMP_GCD:5;
A34: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 2) = t . (intpos 2) by A6, A10, A11, A12, A13, A15, Lm8, X1, X3, X4, X0;

per cases ;

suppose A35: t . (intpos md) < t . (intpos (m4 + ((k + 1) + 1))) ; :: thesis:

then A36: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 4) = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 5)) by A12, A31;
then consider mE being Element of NAT such that
A37: mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)))) . (intpos 7) and
A38: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)))) . (intpos 4) = m4 + mE and
A39: mE <= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 5) and
A40: for i being Element of NAT st m4 + mE 0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)))) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)))) . (intpos i) and
A41: ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)))) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)))) . (intpos (m4 + mE)) ) by A9, A11, A12, A13, A32, A34, A31, A17;
A42: IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t) = IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t))) by A11, A12, A18, A19, Th5;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . GBP = 0 by A9, A11, A12, A13, A32, A34, A31, A17, A36; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 1) = t . (intpos 1) by A9, A11, A12, A13, A32, A16, A34, A31, A17, A36, A42; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 5) = 0 by A9, A11, A12, A13, A32, A34, A31, A17, A36, A42; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 2) = t . (intpos 2) by A9, A11, A12, A13, A32, A34, A31, A17, A36, A42; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 3) = t . (intpos 3) by A9, A11, A12, A13, A32, A34, A30, A31, A17, A36, A42; :: thesis:

A43: now

let i be Element of NAT ; :: thesis:
U1: t . (intpos i) = (Initialize t) . (intpos i) by SCMPDS_5:40;
assume A44: i >= 8 ; :: thesis:
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos i) by A9, A11, A12, A13, A32, A34, A31, A17, A36, A42
.= t . (intpos i) by A6, A10, A11, A12, A13, A15, A44, Lm8, X1, X3, X4, X0, U1 ;
:: thesis:

end;

hereby :: thesis:

let i be Element of NAT ; :: thesis:
assume that
A46: m4 + mE < i and
A47: i <= t . (intpos 4) ; :: thesis:

per cases ;

suppose A48: i = t . (intpos 4) ; :: thesis:

(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos (m4 + ((k + 1) + 1))) = t . (intpos (m4 + ((k + 1) + 1))) by A4, A14, A43, XXREAL_0:2;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) by A6, A11, A12, A35, A43, A48; :: thesis:

end;

suppose i <> t . (intpos 4) ; :: thesis:

then i < t . (intpos 4) by A47, XXREAL_0:1;
then i + 1 <= t . (intpos 4) by INT_1:20;
then i <= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 4) by A45, XREAL_1:21;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) by A42, A40, A46; :: thesis:

end;

thus ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos (m4 + mE)) ) by A42, A41; :: thesis:

end;

suppose A49: t . (intpos md) >= t . (intpos (m4 + ((k + 1) + 1))) ; :: thesis:

A50: now

let x be Int_position ; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . x = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)))) . x by A11, A12, A18, A19, Th5
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . x by A17, A33, A49, SCMPDS_8:23 ; :: thesis:

end;

A51: now

let i be Element of NAT ; :: thesis:
U1: t . (intpos i) = (Initialize t) . (intpos i) by SCMPDS_5:40;
assume A52: i >= 8 ; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos i) by A50
.= t . (intpos i) by A6, A10, A11, A12, A13, A15, A52, Lm8, X1, X3, X4, X0, U1 ; :: thesis:

end;

hence for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) = t . (intpos i) ; :: thesis:
take ME = (k + 1) + 1; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 7) = ME by A11, A17, A49, A50; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 4) = m4 + ME by A11, A12, A17, A49, A50; :: thesis:
thus ME <= t . (intpos 5) by A11; :: thesis:
thus for i being Element of NAT st m4 + ME 0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) by A11, A12; :: thesis:
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos (m4 + ((k + 1) + 1))) = t . (intpos (m4 + ((k + 1) + 1))) by A4, A14, A51, XXREAL_0:2;
hence ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos (m4 + ME)) ) by A6, A49, A51; :: thesis:

end;

hence S₁[k + 1] ; :: thesis:

end;

now

set me = m4 + 1;
let t be State of SCMPDS; :: thesis:
set It = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t);
X1: IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t) = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),(Initialize t)) by SCMPDS_5:48;
X0: t . GBP = (Initialize t) . GBP by SCMPDS_5:40;
X2: t . (intpos 1) = (Initialize t) . (intpos 1) by SCMPDS_5:40;
X3: t . (intpos 2) = (Initialize t) . (intpos 2) by SCMPDS_5:40;
X8: t . (intpos 3) = (Initialize t) . (intpos 3) by SCMPDS_5:40;
X4: t . (intpos 4) = (Initialize t) . (intpos 4) by SCMPDS_5:40;
X5: t . (intpos 5) = (Initialize t) . (intpos 5) by SCMPDS_5:40;
X6: t . (intpos md) = (Initialize t) . (intpos md) by SCMPDS_5:40;
X7: t . (intpos (m4 + 1)) = (Initialize t) . (intpos (m4 + 1)) by SCMPDS_5:40;
set b = DataLoc ((t . GBP),5);
assume that
A53: t . GBP = 0 and
A54: t . (intpos 5) = 0 + 1 and
A55: t . (intpos 4) = m4 + (t . (intpos 5)) and
A56: t . (intpos 2) = md ; :: thesis:
A57: m4 + 1 >= m4 by NAT_1:11;
then A58: m4 + 1 >= 8 by A4, XXREAL_0:2;
then A59: ( t . (intpos md) < t . (intpos (m4 + 1)) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 5) = (t . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 4) = (t . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 7) = (t . (intpos 5)) - 1 ) ) by A6, A53, A54, A55, A56, Lm8, X1, X3, X4, X5, X0, X6, X7;
A60: ( t . (intpos md) >= t . (intpos (m4 + 1)) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 4) = t . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 7) = t . (intpos 5) ) ) by A6, A53, A54, A55, A56, A58, Lm8, X1, X3, X4, X5, X0, X6, X7;
A61: DataLoc ((t . GBP),5) = intpos (0 + 5) by A53, SCMP_GCD:5;

A62: now

let v be State of SCMPDS; :: thesis:
assume that
A63: v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) and
A64: v . (intpos 2) = t . (intpos 2) and
A65: v . GBP = t . GBP and
A66: v . (DataLoc ((t . GBP),5)) > 0 ; :: thesis:
A67: m4 + (v . (DataLoc ((t . GBP),5))) > m4 + 0 by A66, XREAL_1:8;
then reconsider ME = v . (intpos 4) as Element of NAT by A63, INT_1:16;
Y1: IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v) = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),(Initialize v)) by SCMPDS_5:48;
Y2: v . (intpos md) = (Initialize v) . (intpos md) by SCMPDS_5:40;
Y3: v . (intpos ME) = (Initialize v) . (intpos ME) by SCMPDS_5:40;
Y4: v . (intpos 4) = (Initialize v) . (intpos 4) by SCMPDS_5:40;
Y8: v . (intpos 2) = (Initialize v) . (intpos 2) by SCMPDS_5:40;
Y5: v . (intpos 5) = (Initialize v) . (intpos 5) by SCMPDS_5:40;
Y6: v . GBP = (Initialize v) . GBP by SCMPDS_5:40;
v . (intpos 4) >= m4 by A63, A67, XXREAL_0:2;
then A68: ME >= 8 by A4, XXREAL_0:2;
then A69: ( v . (intpos md) < v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 5) = (v . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 4) = (v . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 7) = (v . (intpos 5)) - 1 ) ) by A6, A53, A56, A64, A65, Lm8, Y1, Y2, Y3, Y4, Y5, Y6, Y8;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . GBP = v . GBP by A6, A53, A56, A64, A65, A68, Lm8, Y1, Y4, Y6, Y8; :: thesis:
thus ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v ) by SCMPDS_6:34, SCMPDS_6:35; :: thesis:
A70: ( v . (intpos md) >= v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 4) = v . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 7) = v . (intpos 5) ) ) by A6, A53, A56, A64, A65, A68, Lm8, Y1, Y2, Y3, Y4, Y5, Y6, Y8;

hereby :: thesis:

per cases ;

suppose A71: v . (intpos md) < v . (intpos ME) ; :: thesis:

hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A61, A69, XREAL_1:148; :: thesis:
(m4 + (v . (DataLoc ((t . GBP),5)))) - 1 = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5))) by A61, A69, A71;
hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5))) by A63, A69, A71, XREAL_1:11; :: thesis:

end;

suppose A72: v . (intpos md) >= v . (intpos ME) ; :: thesis:

hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A6, A53, A56, A61, A64, A65, A66, A68, Lm8, Y1, Y2, Y3, Y4, Y6, Y8; :: thesis:
m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5))) < m4 + (v . (DataLoc ((t . GBP),5))) by A61, A66, A70, A72, XREAL_1:8;
hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (DataLoc ((t . GBP),5))) by A63, A70, A72, XXREAL_0:2; :: thesis:

end;

thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),v)) . (intpos 2) = v . (intpos 2) by A6, A53, A56, A64, A65, A68, Lm8, Y1, Y4, Y6, Y8; :: thesis:

end;

A73: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . GBP = 0 by A6, A53, A54, A55, A56, A58, Lm8, X1, X3, X4, X0;
then A74: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . GBP),5)) = 0 by A54, A59, A60, SCMP_GCD:5;

A75: now

let x be Int_position ; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . x = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)))) . x by A54, A55, A61, A62, Th5
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . x by A74, SCMPDS_8:23 ; :: thesis:

end;

hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . GBP = 0 by A73; :: thesis:
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 1) = t . (intpos 1) by A6, A53, A54, A55, A56, A58, Lm8, X1, X2, X3, X4, X0;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 1) = t . (intpos 1) by A75; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 5) = 0 by A54, A59, A60, A75; :: thesis:
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 2) = t . (intpos 2) by A6, A53, A54, A55, A56, A58, Lm8, X1, X3, X4, X0;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 2) = t . (intpos 2) by A75; :: thesis:
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos 3) = t . (intpos 3) by A6, A53, A54, A55, A56, A58, Lm8, X1, X3, X4, X0, X8;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 3) = t . (intpos 3) by A75; :: thesis:

A76: now

let i be Element of NAT ; :: thesis:
assume A77: i >= 8 ; :: thesis:
U1: t . (intpos i) = (Initialize t) . (intpos i) by SCMPDS_5:40;
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),t)) . (intpos i) by A75
.= t . (intpos i) by A6, A53, A54, A55, A56, A58, A77, Lm8, X1, X3, X4, X0, U1 ; :: thesis:

end;

hereby :: thesis:

per cases ;

suppose A79: t . (intpos md) < t . (intpos (m4 + 1)) ; :: thesis:

take ME = 0 ; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 7) = ME by A54, A59, A75, A79; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 4) = m4 + ME by A54, A55, A59, A75, A79; :: thesis:
thus ME <= t . (intpos 5) by A54; :: thesis:

hereby :: thesis:

let i be Element of NAT ; :: thesis:
assume that
A80: m4 + ME 0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) by A6, A76, A78, A79; :: thesis:

end;

thus ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos (m4 + ME)) ) ; :: thesis:

end;

suppose A82: t . (intpos md) >= t . (intpos (m4 + 1)) ; :: thesis:

take ME = 1; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 7) = ME by A54, A60, A75, A82; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos 4) = m4 + ME by A54, A55, A60, A75, A82; :: thesis:
thus ME <= t . (intpos 5) by A54; :: thesis:
thus for i being Element of NAT st m4 + ME 0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos i) by A54, A55; :: thesis:
thus ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),t)) . (intpos (m4 + ME)) ) by A6, A76, A78, A82; :: thesis:

end;

then A83: S₁[ 0 ] ;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A83, A8);
hence ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE 0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),s)) . (intpos (m4 + mE)) ) ) ) by A1, A3, A5, A7; :: thesis: