let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; for s being State of SCMPDS st s . GBP = 0 holds
( ( s . (intpos 3) > 0 implies ( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = (s . (intpos 1)) - (s . (intpos 2)) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = s . (intpos 2) ) ) & ( s . (intpos 3) <= 0 implies ( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = (s . (intpos 2)) - (s . (intpos 1)) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = s . (intpos 1) ) ) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . GBP = 0 & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) = ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2)) )
let s be State of SCMPDS; ( s . GBP = 0 implies ( ( s . (intpos 3) > 0 implies ( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = (s . (intpos 1)) - (s . (intpos 2)) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = s . (intpos 2) ) ) & ( s . (intpos 3) <= 0 implies ( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = (s . (intpos 2)) - (s . (intpos 1)) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = s . (intpos 1) ) ) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . GBP = 0 & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) = ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2)) ) )
set s1 = IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s);
set s2 = IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s);
set a = GBP ;
set t0 = Initialize s;
set Q0 = P;
A1:
IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s) = IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,(Initialize s))
by SCMPDS_5:48;
A2:
IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s) = IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,(Initialize s))
by SCMPDS_5:48;
A3:
IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s) = IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,(Initialize s))
by SCMPDS_5:48;
A4:
now assume A5:
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP = 0
;
( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . GBP = 0 & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) = ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2)) )then A6:
DataLoc (
((IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP),3)
= intpos (0 + 3)
by SCMP_GCD:5;
A7:
(IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . (intpos 3) =
(Exec (((GBP,3) := (GBP,1)),(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)))) . (intpos 3)
by A1, A3, SCMPDS_5:46
.=
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (DataLoc (((IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP),1))
by A6, SCMPDS_2:59
.=
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos (0 + 1))
by A5, SCMP_GCD:5
;
1
<> abs (((IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP) + 3)
by A5, ABSVALUE:def 1;
then A8:
intpos 1
<> DataLoc (
((IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP),3)
by ZFMISC_1:33;
A9:
(IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . (intpos 1) =
(Exec (((GBP,3) := (GBP,1)),(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)))) . (intpos 1)
by A1, A3, SCMPDS_5:46
.=
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1)
by A8, SCMPDS_2:59
;
2
<> abs (((IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP) + 3)
by A5, ABSVALUE:def 1;
then A10:
intpos 2
<> DataLoc (
((IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP),3)
by ZFMISC_1:33;
A11:
(IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . (intpos 2) =
(Exec (((GBP,3) := (GBP,1)),(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)))) . (intpos 2)
by A1, A3, SCMPDS_5:46
.=
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2)
by A10, SCMPDS_2:59
;
0 <> abs (((IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP) + 3)
by A5, ABSVALUE:def 1;
then A12:
GBP <> DataLoc (
((IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP),3)
by ZFMISC_1:33;
A13:
(IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . GBP =
(Exec (((GBP,3) := (GBP,1)),(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)))) . GBP
by A1, A3, SCMPDS_5:46
.=
0
by A5, A12, SCMPDS_2:59
;
then A14:
DataLoc (
((IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . GBP),3)
= intpos (0 + 3)
by SCMP_GCD:5;
0 <> abs (((IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . GBP) + 3)
by A13, ABSVALUE:def 1;
then A15:
GBP <> DataLoc (
((IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . GBP),3)
by ZFMISC_1:33;
thus (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . GBP =
(Exec ((SubFrom (GBP,3,GBP,2)),(IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)))) . GBP
by A3, A2, SCMPDS_5:46
.=
0
by A13, A15, SCMPDS_2:62
;
( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) = ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2)) )
1
<> abs (((IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . GBP) + 3)
by A13, ABSVALUE:def 1;
then A16:
intpos 1
<> DataLoc (
((IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . GBP),3)
by ZFMISC_1:33;
thus A17:
(IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) =
(Exec ((SubFrom (GBP,3,GBP,2)),(IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)))) . (intpos 1)
by A3, A2, SCMPDS_5:46
.=
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1)
by A9, A16, SCMPDS_2:62
;
( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) = ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2)) )
2
<> abs (((IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . GBP) + 3)
by A13, ABSVALUE:def 1;
then A18:
intpos 2
<> DataLoc (
((IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . GBP),3)
by ZFMISC_1:33;
thus A19:
(IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) =
(Exec ((SubFrom (GBP,3,GBP,2)),(IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)))) . (intpos 2)
by A3, A2, SCMPDS_5:46
.=
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2)
by A11, A18, SCMPDS_2:62
;
(IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) = ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2))thus (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) =
(Exec ((SubFrom (GBP,3,GBP,2)),(IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)))) . (intpos 3)
by A3, A2, SCMPDS_5:46
.=
((IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . (intpos 3)) - ((IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . (DataLoc (((IExec (((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))),P,s)) . GBP),2)))
by A14, SCMPDS_2:62
.=
((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2))
by A13, A11, A7, A17, A19, SCMP_GCD:5
;
verum end;
set s0 = Initialize s;
set m1 = SubFrom (GBP,1,GBP,2);
set m2 = SubFrom (GBP,2,GBP,1);
assume A20:
s . GBP = 0
; ( ( s . (intpos 3) > 0 implies ( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = (s . (intpos 1)) - (s . (intpos 2)) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = s . (intpos 2) ) ) & ( s . (intpos 3) <= 0 implies ( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = (s . (intpos 2)) - (s . (intpos 1)) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = s . (intpos 1) ) ) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . GBP = 0 & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) = ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2)) )
then A21:
(Initialize s) . GBP = 0
by SCMPDS_5:40;
A22:
DataLoc ((s . GBP),3) = intpos (0 + 3)
by A20, SCMP_GCD:5;
A23:
(Initialize s) . (intpos 2) = s . (intpos 2)
by SCMPDS_5:40;
A24:
now
2
<> abs (((Initialize s) . GBP) + 1)
by A21, ABSVALUE:def 1;
then A25:
intpos 2
<> DataLoc (
((Initialize s) . GBP),1)
by ZFMISC_1:33;
0 <> abs (((Initialize s) . GBP) + 1)
by A21, ABSVALUE:def 1;
then A26:
GBP <> DataLoc (
((Initialize s) . GBP),1)
by ZFMISC_1:33;
assume A27:
s . (intpos 3) > 0
;
( (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP = 0 & (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1) = (s . (intpos 1)) - (s . (intpos 2)) & (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2) = s . (intpos 2) )hence (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP =
(IExec ((Load (SubFrom (GBP,1,GBP,2))),P,s)) . GBP
by A22, SCMPDS_6:87
.=
(Exec ((SubFrom (GBP,1,GBP,2)),(Initialize s))) . GBP
by SCMPDS_5:45
.=
0
by A21, A26, SCMPDS_2:62
;
( (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1) = (s . (intpos 1)) - (s . (intpos 2)) & (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2) = s . (intpos 2) )A28:
DataLoc (
((Initialize s) . GBP),1)
= intpos (0 + 1)
by A21, SCMP_GCD:5;
thus (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1) =
(IExec ((Load (SubFrom (GBP,1,GBP,2))),P,s)) . (intpos 1)
by A22, A27, SCMPDS_6:87
.=
(Exec ((SubFrom (GBP,1,GBP,2)),(Initialize s))) . (intpos 1)
by SCMPDS_5:45
.=
((Initialize s) . (intpos 1)) - ((Initialize s) . (DataLoc (((Initialize s) . GBP),2)))
by A28, SCMPDS_2:62
.=
((Initialize s) . (intpos 1)) - ((Initialize s) . (intpos (0 + 2)))
by A21, SCMP_GCD:5
.=
(s . (intpos 1)) - (s . (intpos 2))
by A23, SCMPDS_5:40
;
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2) = s . (intpos 2)thus (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2) =
(IExec ((Load (SubFrom (GBP,1,GBP,2))),P,s)) . (intpos 2)
by A22, A27, SCMPDS_6:87
.=
(Exec ((SubFrom (GBP,1,GBP,2)),(Initialize s))) . (intpos 2)
by SCMPDS_5:45
.=
s . (intpos 2)
by A23, A25, SCMPDS_2:62
;
verum
hence
( s . (intpos 3) > 0 implies ( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = (s . (intpos 1)) - (s . (intpos 2)) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = s . (intpos 2) ) )
by A4; ( ( s . (intpos 3) <= 0 implies ( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = (s . (intpos 2)) - (s . (intpos 1)) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = s . (intpos 1) ) ) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . GBP = 0 & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) = ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2)) )
A29:
(Initialize s) . (intpos 1) = s . (intpos 1)
by SCMPDS_5:40;
A30:
now
1
<> abs (((Initialize s) . GBP) + 2)
by A21, ABSVALUE:def 1;
then A31:
intpos 1
<> DataLoc (
((Initialize s) . GBP),2)
by ZFMISC_1:33;
0 <> abs (((Initialize s) . GBP) + 2)
by A21, ABSVALUE:def 1;
then A32:
GBP <> DataLoc (
((Initialize s) . GBP),2)
by ZFMISC_1:33;
assume A33:
s . (intpos 3) <= 0
;
( (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP = 0 & (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2) = (s . (intpos 2)) - (s . (intpos 1)) & (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1) = s . (intpos 1) )hence (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP =
(IExec ((Load (SubFrom (GBP,2,GBP,1))),P,s)) . GBP
by A22, SCMPDS_6:88
.=
(Exec ((SubFrom (GBP,2,GBP,1)),(Initialize s))) . GBP
by SCMPDS_5:45
.=
0
by A21, A32, SCMPDS_2:62
;
( (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2) = (s . (intpos 2)) - (s . (intpos 1)) & (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1) = s . (intpos 1) )A34:
DataLoc (
((Initialize s) . GBP),2)
= intpos (0 + 2)
by A21, SCMP_GCD:5;
thus (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 2) =
(IExec ((Load (SubFrom (GBP,2,GBP,1))),P,s)) . (intpos 2)
by A22, A33, SCMPDS_6:88
.=
(Exec ((SubFrom (GBP,2,GBP,1)),(Initialize s))) . (intpos 2)
by SCMPDS_5:45
.=
((Initialize s) . (intpos 2)) - ((Initialize s) . (DataLoc (((Initialize s) . GBP),1)))
by A34, SCMPDS_2:62
.=
((Initialize s) . (intpos 2)) - ((Initialize s) . (intpos (0 + 1)))
by A21, SCMP_GCD:5
.=
(s . (intpos 2)) - (s . (intpos 1))
by A23, SCMPDS_5:40
;
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1) = s . (intpos 1)thus (IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . (intpos 1) =
(IExec ((Load (SubFrom (GBP,2,GBP,1))),P,s)) . (intpos 1)
by A22, A33, SCMPDS_6:88
.=
(Exec ((SubFrom (GBP,2,GBP,1)),(Initialize s))) . (intpos 1)
by SCMPDS_5:45
.=
s . (intpos 1)
by A29, A31, SCMPDS_2:62
;
verum
hence
( s . (intpos 3) <= 0 implies ( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2) = (s . (intpos 2)) - (s . (intpos 1)) & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1) = s . (intpos 1) ) )
by A4; ( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . GBP = 0 & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) = ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2)) )
now per cases
( s . (intpos 3) > 0 or s . (intpos 3) <= 0 )
;
suppose
s . (intpos 3) > 0
;
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP = 0 hence
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP = 0
by A24;
verum end; suppose
s . (intpos 3) <= 0
;
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP = 0 hence
(IExec ((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))),P,s)) . GBP = 0
by A30;
verum end;
hence
( (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . GBP = 0 & (IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 3) = ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 1)) - ((IExec ((((if>0 (GBP,3,(Load (SubFrom (GBP,1,GBP,2))),(Load (SubFrom (GBP,2,GBP,1))))) ';' ((GBP,3) := (GBP,1))) ';' (SubFrom (GBP,3,GBP,2))),P,s)) . (intpos 2)) )
by A4; verum