let P be Instruction-Sequence of SCMPDS; :: thesis: 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; :: thesis: ( 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 :: thesis: 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; :: thesis: 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; :: thesis: ( 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) ; :: thesis: ( (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; :: thesis: ( ((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; :: thesis: ( ((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; :: thesis: ( ( 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 :: thesis: ( ( 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) ; :: thesis: ( (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; :: thesis: verum

end;

hereby :: thesis: (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) ; :: thesis: ( (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; :: thesis: 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; :: thesis: verum

end;

assume A5: ( s . GBP = 0 & s . (intpos 1) > 0 & s . (intpos 2) > 0 & s . (intpos 3) = (s . (intpos 1)) - (s . (intpos 2)) ) ; :: thesis: ( (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; :: thesis: ( 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; :: thesis: verum