let P be Instruction-Sequence of SCMPDS; for s being 0 -started State of SCMPDS st s . GBP = 0 & s . (intpos 1) > 0 & s . (intpos 2) > 0 & s . (intpos 3) = (s . (intpos 1)) - (s . (intpos 2)) holds
( (IExec ((while<>0 (GBP,3,(((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)) gcd (s . (intpos 2)) & (IExec ((while<>0 (GBP,3,(((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 1)) gcd (s . (intpos 2)) & while<>0 (GBP,3,(((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)))) is_closed_on s,P & while<>0 (GBP,3,(((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)))) is_halting_on s,P )
set a = GBP ;
let s be 0 -started State of SCMPDS; ( s . GBP = 0 & s . (intpos 1) > 0 & s . (intpos 2) > 0 & s . (intpos 3) = (s . (intpos 1)) - (s . (intpos 2)) implies ( (IExec ((while<>0 (GBP,3,(((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)) gcd (s . (intpos 2)) & (IExec ((while<>0 (GBP,3,(((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 1)) gcd (s . (intpos 2)) & while<>0 (GBP,3,(((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)))) is_closed_on s,P & while<>0 (GBP,3,(((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)))) is_halting_on s,P ) )
A1:
DataLoc (0,3) = intpos (0 + 3)
by SCMP_GCD:1;
A2:
now for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . (intpos 1) > 0 & t . (intpos 2) > 0 & t . GBP = 0 & t . (DataLoc (0,3)) = (t . (intpos 1)) - (t . (intpos 2)) & t . (intpos 1) <> t . (intpos 2) holds
( (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))),Q,t)) . GBP = 0 & ((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)) is_closed_on t,Q & ((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)) is_halting_on t,Q & ( t . (intpos 1) > t . (intpos 2) 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))),Q,t)) . (intpos 1) = (t . (intpos 1)) - (t . (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))),Q,t)) . (intpos 2) = t . (intpos 2) ) ) & ( t . (intpos 1) <= t . (intpos 2) 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))),Q,t)) . (intpos 2) = (t . (intpos 2)) - (t . (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))),Q,t)) . (intpos 1) = t . (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))),Q,t)) . (DataLoc (0,3)) = ((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))),Q,t)) . (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))),Q,t)) . (intpos 2)) )let t be
0 -started State of
SCMPDS;
for Q being Instruction-Sequence of SCMPDS st t . (intpos 1) > 0 & t . (intpos 2) > 0 & t . GBP = 0 & t . (DataLoc (0,3)) = (t . (intpos 1)) - (t . (intpos 2)) & t . (intpos 1) <> t . (intpos 2) holds
( (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))),Q,t)) . GBP = 0 & ((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)) is_closed_on t,Q & ((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)) is_halting_on t,Q & ( t . (intpos 1) > t . (intpos 2) 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))),Q,t)) . (intpos 1) = (t . (intpos 1)) - (t . (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))),Q,t)) . (intpos 2) = t . (intpos 2) ) ) & ( t . (intpos 1) <= t . (intpos 2) 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))),Q,t)) . (intpos 2) = (t . (intpos 2)) - (t . (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))),Q,t)) . (intpos 1) = t . (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))),Q,t)) . (DataLoc (0,3)) = ((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))),Q,t)) . (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))),Q,t)) . (intpos 2)) )let Q be
Instruction-Sequence of
SCMPDS;
( t . (intpos 1) > 0 & t . (intpos 2) > 0 & t . GBP = 0 & t . (DataLoc (0,3)) = (t . (intpos 1)) - (t . (intpos 2)) & t . (intpos 1) <> t . (intpos 2) 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))),Q,t)) . GBP = 0 & ((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)) is_closed_on t,Q & ((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)) is_halting_on t,Q & ( t . (intpos 1) > t . (intpos 2) 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))),Q,t)) . (intpos 1) = (t . (intpos 1)) - (t . (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))),Q,t)) . (intpos 2) = t . (intpos 2) ) ) & ( t . (intpos 1) <= t . (intpos 2) 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))),Q,t)) . (intpos 2) = (t . (intpos 2)) - (t . (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))),Q,t)) . (intpos 1) = t . (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))),Q,t)) . (DataLoc (0,3)) = ((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))),Q,t)) . (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))),Q,t)) . (intpos 2)) ) )assume that
t . (intpos 1) > 0
and
t . (intpos 2) > 0
and A3:
t . GBP = 0
and A4:
t . (DataLoc (0,3)) = (t . (intpos 1)) - (t . (intpos 2))
and
t . (intpos 1) <> t . (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))),Q,t)) . GBP = 0 & ((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)) is_closed_on t,Q & ((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)) is_halting_on t,Q & ( t . (intpos 1) > t . (intpos 2) 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))),Q,t)) . (intpos 1) = (t . (intpos 1)) - (t . (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))),Q,t)) . (intpos 2) = t . (intpos 2) ) ) & ( t . (intpos 1) <= t . (intpos 2) 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))),Q,t)) . (intpos 2) = (t . (intpos 2)) - (t . (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))),Q,t)) . (intpos 1) = t . (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))),Q,t)) . (DataLoc (0,3)) = ((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))),Q,t)) . (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))),Q,t)) . (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))),Q,t)) . GBP = 0
by A3, Lm12;
( ((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)) is_closed_on t,Q & ((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)) is_halting_on t,Q & ( t . (intpos 1) > t . (intpos 2) 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))),Q,t)) . (intpos 1) = (t . (intpos 1)) - (t . (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))),Q,t)) . (intpos 2) = t . (intpos 2) ) ) & ( t . (intpos 1) <= t . (intpos 2) 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))),Q,t)) . (intpos 2) = (t . (intpos 2)) - (t . (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))),Q,t)) . (intpos 1) = t . (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))),Q,t)) . (DataLoc (0,3)) = ((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))),Q,t)) . (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))),Q,t)) . (intpos 2)) )thus
((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)) is_closed_on t,
Q
by SCMPDS_6:20;
( ((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)) is_halting_on t,Q & ( t . (intpos 1) > t . (intpos 2) 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))),Q,t)) . (intpos 1) = (t . (intpos 1)) - (t . (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))),Q,t)) . (intpos 2) = t . (intpos 2) ) ) & ( t . (intpos 1) <= t . (intpos 2) 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))),Q,t)) . (intpos 2) = (t . (intpos 2)) - (t . (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))),Q,t)) . (intpos 1) = t . (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))),Q,t)) . (DataLoc (0,3)) = ((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))),Q,t)) . (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))),Q,t)) . (intpos 2)) )thus
((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)) is_halting_on t,
Q
by SCMPDS_6:21;
( ( t . (intpos 1) > t . (intpos 2) 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))),Q,t)) . (intpos 1) = (t . (intpos 1)) - (t . (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))),Q,t)) . (intpos 2) = t . (intpos 2) ) ) & ( t . (intpos 1) <= t . (intpos 2) 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))),Q,t)) . (intpos 2) = (t . (intpos 2)) - (t . (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))),Q,t)) . (intpos 1) = t . (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))),Q,t)) . (DataLoc (0,3)) = ((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))),Q,t)) . (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))),Q,t)) . (intpos 2)) )hereby ( ( t . (intpos 1) <= t . (intpos 2) 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))),Q,t)) . (intpos 2) = (t . (intpos 2)) - (t . (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))),Q,t)) . (intpos 1) = t . (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))),Q,t)) . (DataLoc (0,3)) = ((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))),Q,t)) . (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))),Q,t)) . (intpos 2)) )
assume
t . (intpos 1) > t . (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))),Q,t)) . (intpos 1) = (t . (intpos 1)) - (t . (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))),Q,t)) . (intpos 2) = t . (intpos 2) )then
t . (intpos 3) > 0
by A1, A4, XREAL_1:50;
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))),Q,t)) . (intpos 1) = (t . (intpos 1)) - (t . (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))),Q,t)) . (intpos 2) = t . (intpos 2) )
by A3, Lm12;
verum
end; hereby (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))),Q,t)) . (DataLoc (0,3)) = ((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))),Q,t)) . (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))),Q,t)) . (intpos 2))
assume
t . (intpos 1) <= t . (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))),Q,t)) . (intpos 2) = (t . (intpos 2)) - (t . (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))),Q,t)) . (intpos 1) = t . (intpos 1) )then
t . (intpos 3) <= 0
by A1, A4, XREAL_1:47;
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))),Q,t)) . (intpos 2) = (t . (intpos 2)) - (t . (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))),Q,t)) . (intpos 1) = t . (intpos 1) )
by A3, Lm12;
verum
end; 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))),Q,t)) . (DataLoc (0,3)) = ((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))),Q,t)) . (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))),Q,t)) . (intpos 2))
by A1, A3, Lm12;
verum end;
assume A5:
( s . GBP = 0 & s . (intpos 1) > 0 & s . (intpos 2) > 0 & s . (intpos 3) = (s . (intpos 1)) - (s . (intpos 2)) )
; ( (IExec ((while<>0 (GBP,3,(((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)) gcd (s . (intpos 2)) & (IExec ((while<>0 (GBP,3,(((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 1)) gcd (s . (intpos 2)) & while<>0 (GBP,3,(((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)))) is_closed_on s,P & while<>0 (GBP,3,(((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)))) is_halting_on s,P )
hence
( (IExec ((while<>0 (GBP,3,(((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)) gcd (s . (intpos 2)) & (IExec ((while<>0 (GBP,3,(((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 1)) gcd (s . (intpos 2)) )
by A1, A2, Th17; ( while<>0 (GBP,3,(((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)))) is_closed_on s,P & while<>0 (GBP,3,(((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)))) is_halting_on s,P )
thus
( while<>0 (GBP,3,(((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)))) is_closed_on s,P & while<>0 (GBP,3,(((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)))) is_halting_on s,P )
by A5, A1, A2, Lm10, Th16; verum