let s be State of SCMPDS ; :: thesis: ( s . (intpos 4) >= 7 + (s . (DataLoc (s . GBP ),6)) & s . GBP = 0 implies ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 1) = s . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 2) = s . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 3) = s . (intpos 3) ) )
set b = DataLoc (s . GBP ),6;
set a = GBP ;
assume that
A1: s . (intpos 4) >= 7 + (s . (DataLoc (s . GBP ),6)) and
A2: s . GBP = 0 ; :: thesis: ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 1) = s . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 2) = s . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 3) = s . (intpos 3) )
A3: DataLoc (s . GBP ),6 = intpos (0 + 6) by A2, SCMP_GCD:5;
defpred S₁[ Nat] means for t being State of SCMPDS st t . (intpos 6) <= $1 & t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 holds
( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) );
A4: S₁[ 0 ]

proof

let t be State of SCMPDS ; :: thesis: ( t . (intpos 6) <= 0 & t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 implies ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) ) )
assume that
A5: t . (intpos 6) <= 0 and
t . (intpos 4) >= 7 + (t . (intpos 6)) and
A6: t . GBP = 0 ; :: thesis: ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) )
A7: DataLoc (t . GBP ),6 = intpos (0 + 6) by A6, SCMP_GCD:5;
hence (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . GBP = 0 by A5, A6, SCMPDS_8:23; :: thesis: ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) )
thus ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) ) by A5, A7, SCMPDS_8:23; :: thesis: verum

end;

A8: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A9: S₁[k] ; :: thesis: S₁[k + 1]
thus S₁[k + 1] :: thesis: verum

proof

let t be State of SCMPDS ; :: thesis: ( t . (intpos 6) <= k + 1 & t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 implies ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) ) )
set bt = DataLoc (t . GBP ),6;
assume that
A10: t . (intpos 6) <= k + 1 and
A11: t . (intpos 4) >= 7 + (t . (intpos 6)) and
A12: t . GBP = 0 ; :: thesis: ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) )
A13: DataLoc (t . GBP ),6 = intpos (0 + 6) by A12, SCMP_GCD:5;

per cases ( t . (DataLoc (t . GBP ),6) <= 0 or t . (DataLoc (t . GBP ),6) > 0 ) ;

suppose t . (DataLoc (t . GBP ),6) <= 0 ; :: thesis: ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) )

hence ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) ) by A12, SCMPDS_8:23; :: thesis: verum

end;

suppose A14: t . (DataLoc (t . GBP ),6) > 0 ; :: thesis: ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) )

A15: now

let v be State of SCMPDS ; :: thesis: ( ( for x being Int_position st x in {(intpos 4)} holds
v . x >= 7 + (v . (DataLoc (t . GBP ),6)) ) & v . GBP = t . GBP & v . (DataLoc (t . GBP ),6) > 0 implies ( (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . GBP = v . GBP & ((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))) is_closed_on v & ((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))) is_halting_on v & (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6) < v . (DataLoc (t . GBP ),6) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . x >= 7 + ((IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6)) ) ) )
assume that
A16: for x being Int_position st x in {(intpos 4)} holds
v . x >= 7 + (v . (DataLoc (t . GBP ),6)) and
A17: ( v . GBP = t . GBP & v . (DataLoc (t . GBP ),6) > 0 ) ; :: thesis: ( (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . GBP = v . GBP & ((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))) is_closed_on v & ((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))) is_halting_on v & (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6) < v . (DataLoc (t . GBP ),6) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . x >= 7 + ((IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6)) ) )
set Iv = IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v;
A18: intpos 4 in {(intpos 4)} by TARSKI:def 1;
then A19: v . (intpos 4) >= 7 + (v . (intpos 6)) by A13, A16;
hence (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . GBP = v . GBP by A12, A13, A17, Lm7; :: thesis: ( ((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))) is_closed_on v & ((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))) is_halting_on v & (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6) < v . (DataLoc (t . GBP ),6) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . x >= 7 + ((IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6)) ) )
thus ( ((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))) is_closed_on v & ((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))) is_halting_on v ) by SCMPDS_6:34, SCMPDS_6:35; :: thesis: ( (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6) < v . (DataLoc (t . GBP ),6) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . x >= 7 + ((IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6)) ) )
thus (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6) < v . (DataLoc (t . GBP ),6) by A12, A13, A17, A19, Lm8; :: thesis: for x being Int_position st x in {(intpos 4)} holds
(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . x >= 7 + ((IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6))
v . (intpos 4) >= 7 + (v . (DataLoc (t . GBP ),6)) by A16, A18;
then (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (intpos 4) >= 7 + ((IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (intpos 6)) by A12, A13, A17, Lm8;
hence for x being Int_position st x in {(intpos 4)} holds
(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . x >= 7 + ((IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),v) . (DataLoc (t . GBP ),6)) by A13, TARSKI:def 1; :: thesis: verum

end;

set It = IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t;
(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 6) < t . (intpos 6) by A11, A12, A13, A14, Lm8;
then ((IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 6)) + 1 <= t . (intpos 6) by INT_1:20;
then ((IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 6)) + 1 <= k + 1 by A10, XXREAL_0:2;
then A20: (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 6) <= k by XREAL_1:8;
A21: ( (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . GBP = 0 & (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 4) >= 7 + ((IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 6)) ) by A11, A12, A13, A14, Lm7, Lm8;
then A22: (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t)) . (intpos 1) = (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 1) by A9, A20;
A23: (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t)) . (intpos 3) = (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 3) by A9, A21, A20;
A24: (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t)) . (intpos 2) = (IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 2) by A9, A21, A20;
A25: for x being Int_position st x in {(intpos 4)} holds
t . x >= 7 + (t . (DataLoc (t . GBP ),6)) by A11, A13, TARSKI:def 1;
(IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t)) . GBP = 0 by A9, A21, A20;
hence (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . GBP = 0 by A14, A25, A15, Lm5, SCMPDS_8:29; :: thesis: ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) )
(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 1) = t . (intpos 1) by A11, A12, A13, A14, Lm7;
hence (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 1) = t . (intpos 1) by A14, A25, A15, A22, Lm5, SCMPDS_8:29; :: thesis: ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) )
(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 2) = t . (intpos 2) by A11, A12, A13, A14, Lm8;
hence (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 2) = t . (intpos 2) by A14, A25, A15, A24, Lm5, SCMPDS_8:29; :: thesis: (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3)
(IExec (((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))),t) . (intpos 3) = t . (intpos 3) by A11, A12, A13, A14, Lm8;
hence (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),t) . (intpos 3) = t . (intpos 3) by A14, A25, A15, A23, Lm5, SCMPDS_8:29; :: thesis: verum

end;

end;

end;

end;

per cases ( s . (intpos 6) <= 0 or s . (intpos 6) > 0 ) ;

suppose s . (intpos 6) <= 0 ; :: thesis: ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 1) = s . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 2) = s . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 3) = s . (intpos 3) )

hence ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 1) = s . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 2) = s . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 3) = s . (intpos 3) ) by A2, A3, SCMPDS_8:23; :: thesis: verum

end;

suppose s . (intpos 6) > 0 ; :: thesis: ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 1) = s . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 2) = s . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 3) = s . (intpos 3) )

then reconsider m = s . (intpos 6) as Element of NAT by INT_1:16;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A4, A8);
then S₁[m] ;
hence ( (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . GBP = 0 & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 1) = s . (intpos 1) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 2) = s . (intpos 2) & (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),s) . (intpos 3) = s . (intpos 3) ) by A1, A2, A3; :: thesis: verum

end;

end;