set a = GBP ;
let s be 0 -started State of SCMPDS; for m3, md being Element of NAT st s . GBP = 0 & s . (intpos 7) > 0 & (s . (intpos 3)) + (s . (intpos 7)) = m3 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos mE3) ) ) )
let m3, md be Element of NAT ; ( s . GBP = 0 & s . (intpos 7) > 0 & (s . (intpos 3)) + (s . (intpos 7)) = m3 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 implies ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos mE3) ) ) ) )
assume that
A1:
s . GBP = 0
and
A2:
s . (intpos 7) > 0
and
A3:
(s . (intpos 3)) + (s . (intpos 7)) = m3
and
A4:
s . (intpos 3) >= 8
and
A5:
s . (intpos 2) = md
and
A6:
md >= 8
; ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos mE3) ) ) )
s . (intpos 7) >= 1 + 0
by A2, INT_1:20;
then reconsider m7 = (s . (intpos 7)) - 1 as Element of NAT by INT_1:16, XREAL_1:50;
A7:
s . (intpos 7) = m7 + 1
;
defpred S1[ Element of NAT ] means for t being State of SCMPDS st t . GBP = 0 & t . (intpos 7) = $1 + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) );
A8:
now let k be
Element of
NAT ;
( S1[k] implies S1[k + 1] )assume A9:
S1[
k]
;
S1[k + 1]now let t be
State of
SCMPDS;
( t . GBP = 0 & t . (intpos 7) = (k + 1) + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md implies ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b2 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),i)) . (intpos b2) = i . (intpos b2) ) & ex m5, mE3 being Element of NAT st
( mE3 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos 3) = b3 & b3 + mE3 = m3 & mE3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b4 & b4 < i holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos b4) ) & ( mE3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos b3) ) ) ) )set b =
DataLoc (
(t . GBP),7);
assume that A10:
t . GBP = 0
and A11:
t . (intpos 7) = (k + 1) + 1
and A12:
(t . (intpos 3)) + (t . (intpos 7)) = m3
and A13:
t . (intpos 3) >= 8
and A14:
t . (intpos 2) = md
;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b2 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),i)) . (intpos b2) = i . (intpos b2) ) & ex m5, mE3 being Element of NAT st
( mE3 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos 3) = b3 & b3 + mE3 = m3 & mE3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b4 & b4 < i holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos b4) ) & ( mE3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos b3) ) ) )reconsider me =
t . (intpos 3) as
Element of
NAT by A13, INT_1:16;
set It =
IExec (
((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),
t);
X1:
IExec (
((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),
t)
= IExec (
((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),
(Initialize t))
by SCMPDS_5:48;
X0:
t . GBP = (Initialize t) . GBP
by SCMPDS_5:40;
X2:
t . (intpos 1) = (Initialize t) . (intpos 1)
by SCMPDS_5:40;
X3:
t . (intpos 2) = (Initialize t) . (intpos 2)
by SCMPDS_5:40;
X8:
t . (intpos 3) = (Initialize t) . (intpos 3)
by SCMPDS_5:40;
X4:
t . (intpos 4) = (Initialize t) . (intpos 4)
by SCMPDS_5:40;
X9:
t . (intpos 7) = (Initialize t) . (intpos 7)
by SCMPDS_5:40;
X6:
t . (intpos md) = (Initialize t) . (intpos md)
by SCMPDS_5:40;
X7:
t . (intpos me) = (Initialize t) . (intpos me)
by SCMPDS_5:40;
A15:
(
t . (intpos md) > t . (intpos me) implies (
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 7) = (t . (intpos 7)) - 1 &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 3) = (t . (intpos 3)) + 1 &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 5) = (t . (intpos 7)) - 1 ) )
by A6, A10, A13, A14, Lm14, X0, X1, X3, X6, X7, X8, X9;
A16:
DataLoc (
(t . GBP),7)
= intpos (0 + 7)
by A10, SCMP_GCD:5;
A17:
now let v be
State of
SCMPDS;
( v . (intpos 3) >= 8 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),7)) > 0 implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2) ) )assume that A18:
v . (intpos 3) >= 8
and A19:
v . (intpos 2) = t . (intpos 2)
and A20:
v . GBP = t . GBP
and A21:
v . (DataLoc ((t . GBP),7)) > 0
;
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2) )reconsider ME =
v . (intpos 3) as
Element of
NAT by A18, INT_1:16;
Y1:
IExec (
((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),
v)
= IExec (
((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),
(Initialize v))
by SCMPDS_5:48;
Y2:
v . (intpos md) = (Initialize v) . (intpos md)
by SCMPDS_5:40;
Y3:
v . (intpos ME) = (Initialize v) . (intpos ME)
by SCMPDS_5:40;
Y8:
v . (intpos 2) = (Initialize v) . (intpos 2)
by SCMPDS_5:40;
Y9:
v . (intpos 3) = (Initialize v) . (intpos 3)
by SCMPDS_5:40;
Y10:
v . (intpos 7) = (Initialize v) . (intpos 7)
by SCMPDS_5:40;
Y6:
v . GBP = (Initialize v) . GBP
by SCMPDS_5:40;
A22:
ME = v . (intpos 3)
;
hence
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . GBP = v . GBP
by A6, A10, A14, A18, A19, A20, Lm14, Y1, Y6, Y8, Y9;
( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2) )thus
(
(((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v &
(((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v )
by SCMPDS_6:34, SCMPDS_6:35;
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2) )A23:
(
v . (intpos md) > v . (intpos ME) implies (
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 7) = (v . (intpos 7)) - 1 &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) = (v . (intpos 3)) + 1 &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 5) = (v . (intpos 7)) - 1 ) )
by A6, A10, A14, A18, A19, A20, Lm14, Y1, Y2, Y3, Y6, Y8, Y9, Y10;
hereby (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2)
per cases
( v . (intpos md) > v . (intpos ME) or v . (intpos md) <= v . (intpos ME) )
;
suppose A24:
v . (intpos md) > v . (intpos ME)
;
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 )hence
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7))
by A16, A23, XREAL_1:148;
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8A25:
(v . (intpos 3)) + 1
> v . (intpos 3)
by XREAL_1:31;
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) = (v . (intpos 3)) + 1
by A6, A10, A14, A18, A19, A20, A24, Lm14, Y1, Y2, Y3, Y6, Y8, Y9;
hence
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8
by A18, A25, XXREAL_0:2;
verum end; suppose A26:
v . (intpos md) <= v . (intpos ME)
;
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 )hence
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7))
by A6, A10, A14, A16, A18, A19, A20, A21, Lm14, Y1, Y2, Y3, Y6, Y8, Y9;
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8thus
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8
by A6, A10, A14, A18, A19, A20, A26, Lm14, Y1, Y2, Y3, Y6, Y8, Y9;
verum end; thus
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2)
by A6, A10, A14, A18, A19, A20, A22, Lm14, Y1, Y6, Y8, Y9;
verum end; A27:
(
t . (intpos md) <= t . (intpos me) implies (
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 7) = 0 &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 3) = t . (intpos 3) &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 5) = t . (intpos 7) ) )
by A6, A10, A13, A14, Lm14, X1, X3, X6, X7, X8, X9, X0;
A28:
me = t . (intpos 3)
;
then A29:
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . GBP = 0
by A6, A10, A13, A14, Lm14, X1, X3, X8, X0;
then A30:
DataLoc (
((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . GBP),7)
= intpos (0 + 7)
by SCMP_GCD:5;
A31:
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 1) = t . (intpos 1)
by A6, A10, A13, A14, A28, Lm14, X1, X2, X3, X8, X0;
A32:
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 4) = t . (intpos 4)
by A6, A10, A13, A14, A28, Lm14, X1, X3, X4, X8, X0;
A33:
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 2) = t . (intpos 2)
by A6, A10, A13, A14, A28, Lm14, X1, X3, X8, X0;
per cases
( t . (intpos md) > t . (intpos me) or t . (intpos md) <= t . (intpos me) )
;
suppose A34:
t . (intpos md) > t . (intpos me)
;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b2 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),i)) . (intpos b2) = i . (intpos b2) ) & ex m5, mE3 being Element of NAT st
( mE3 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos 3) = b3 & b3 + mE3 = m3 & mE3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b4 & b4 < i holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos b4) ) & ( mE3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos b3) ) ) )then
t . (intpos 3) < (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 3)
by A15, XREAL_1:31;
then A35:
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 3) >= 8
by A13, XXREAL_0:2;
A36:
((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 3)) + ((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 7)) = m3
by A12, A15, A34;
then consider m5,
mE3 being
Element of
NAT such that A37:
m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)))) . (intpos 5)
and A38:
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)))) . (intpos 3) = mE3
and A39:
mE3 + m5 = m3
and A40:
m5 <= (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 7)
and A41:
for
i being
Element of
NAT st
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 3) <= i &
i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)))) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)))) . (intpos i)
and A42:
(
m5 = 0 or
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)))) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)))) . (intpos mE3) )
by A9, A11, A12, A14, A29, A33, A15, A27, A35;
A43:
IExec (
(while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),
t)
= IExec (
(while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)))
by A11, A13, A16, A17, Th6;
hence
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . GBP = 0
by A9, A11, A12, A14, A29, A33, A15, A27, A35, A36;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 1) = t . (intpos 1)
by A9, A11, A12, A14, A29, A31, A33, A15, A27, A35, A36, A43;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0
by A9, A11, A12, A14, A29, A33, A15, A27, A35, A36, A43;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2)
by A9, A11, A12, A14, A29, A33, A15, A27, A35, A36, A43;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4)
by A9, A11, A12, A14, A29, A33, A32, A15, A27, A35, A36, A43;
( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )A44:
now let i be
Element of
NAT ;
( i >= 8 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) )U1:
t . (intpos i) = (Initialize t) . (intpos i)
by SCMPDS_5:40;
assume A45:
i >= 8
;
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i)hence (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) =
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos i)
by A9, A11, A12, A14, A29, A33, A15, A27, A35, A36, A43
.=
t . (intpos i)
by A6, A10, A13, A14, A28, A45, Lm14, X1, X3, X8, U1, X0
;
verum end; hence
for
i being
Element of
NAT st
i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i)
;
ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )take m5 =
m5;
ex mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )take mE3 =
mE3;
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5)
by A11, A13, A16, A17, A37, Th6;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3
by A11, A13, A16, A17, A38, Th6;
( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
mE3 + m5 = m3
by A39;
( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
m5 <= t . (intpos 7)
by A15, A34, A40, XREAL_1:148, XXREAL_0:2;
( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )A46:
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 3) = (t . (intpos 3)) + 1
by A6, A10, A13, A14, A34, Lm14, X1, X3, X6, X7, X8, X0;
hereby ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) )
let i be
Element of
NAT ;
( t . (intpos 3) <= i & i < mE3 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos b1) )assume that A47:
t . (intpos 3) <= i
and A48:
i < mE3
;
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos b1)per cases
( i = t . (intpos 3) or i <> t . (intpos 3) )
;
suppose A49:
i = t . (intpos 3)
;
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos b1)
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos me) = t . (intpos me)
by A13, A44;
hence
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i)
by A6, A34, A44, A49;
verum end; suppose
i <> t . (intpos 3)
;
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos b1)then
t . (intpos 3) < i
by A47, XXREAL_0:1;
then
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 3) <= i
by A46, INT_1:20;
hence
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i)
by A43, A41, A48;
verum end;
end; thus
(
m5 = 0 or
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) )
by A43, A42;
verum end; suppose A50:
t . (intpos md) <= t . (intpos me)
;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b2 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),i)) . (intpos b2) = i . (intpos b2) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos 5) = mE3 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos 3) = b3 & b3 + mE3 = m3 & mE3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b4 & b4 < i holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos b4) ) & ( mE3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),m5)) . (intpos b3) ) ) )A51:
now let x be
Int_position ;
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . x = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . xthus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . x =
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)))) . x
by A11, A13, A16, A17, Th6
.=
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . x
by A27, A30, A50, SCMPDS_8:23
;
verum end; hence
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . GBP = 0
by A29;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 1) = t . (intpos 1)
by A31, A51;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0
by A27, A50, A51;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2)
by A33, A51;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4)
by A32, A51;
( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )A52:
now let i be
Element of
NAT ;
( i >= 8 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) )U1:
t . (intpos i) = (Initialize t) . (intpos i)
by SCMPDS_5:40;
assume A53:
i >= 8
;
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i)thus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) =
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos i)
by A51
.=
t . (intpos i)
by A6, A10, A13, A14, A28, A53, Lm14, U1, X1, X3, X8, X0
;
verum end; hence
for
i being
Element of
NAT st
i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i)
;
ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )take m5 =
(k + 1) + 1;
ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )take mE3 =
me;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5
by A11, A27, A50, A51;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3
by A27, A50, A51;
( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
mE3 + m5 = m3
by A11, A12;
( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
m5 <= t . (intpos 7)
by A11;
( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
for
i being
Element of
NAT st
t . (intpos 3) <= i &
i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i)
;
( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) )
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos me) = t . (intpos me)
by A13, A52;
hence
(
m5 = 0 or
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) )
by A6, A50, A52;
verum end; hence
S1[
k + 1]
;
verum end;
now let t be
State of
SCMPDS;
( t . GBP = 0 & t . (intpos 7) = 0 + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md implies ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) ) )set b =
DataLoc (
(t . GBP),7);
assume that A54:
t . GBP = 0
and A55:
t . (intpos 7) = 0 + 1
and A56:
(t . (intpos 3)) + (t . (intpos 7)) = m3
and A57:
t . (intpos 3) >= 8
and A58:
t . (intpos 2) = md
;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )reconsider me =
t . (intpos 3) as
Element of
NAT by A57, INT_1:16;
set It =
IExec (
((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),
t);
X1:
IExec (
((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),
t)
= IExec (
((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),
(Initialize t))
by SCMPDS_5:48;
X0:
t . GBP = (Initialize t) . GBP
by SCMPDS_5:40;
X2:
t . (intpos 1) = (Initialize t) . (intpos 1)
by SCMPDS_5:40;
X3:
t . (intpos 2) = (Initialize t) . (intpos 2)
by SCMPDS_5:40;
X8:
t . (intpos 3) = (Initialize t) . (intpos 3)
by SCMPDS_5:40;
X4:
t . (intpos 4) = (Initialize t) . (intpos 4)
by SCMPDS_5:40;
X9:
t . (intpos 7) = (Initialize t) . (intpos 7)
by SCMPDS_5:40;
X6:
t . (intpos md) = (Initialize t) . (intpos md)
by SCMPDS_5:40;
X7:
t . (intpos me) = (Initialize t) . (intpos me)
by SCMPDS_5:40;
A59:
(
t . (intpos md) > t . (intpos me) implies (
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 7) = (t . (intpos 7)) - 1 &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 3) = (t . (intpos 3)) + 1 &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 5) = (t . (intpos 7)) - 1 ) )
by A6, A54, A57, A58, Lm14, X1, X3, X6, X7, X8, X9, X0;
A60:
DataLoc (
(t . GBP),7)
= intpos (0 + 7)
by A54, SCMP_GCD:5;
A61:
now let v be
State of
SCMPDS;
( v . (intpos 3) >= 8 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),7)) > 0 implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2) ) )assume that A62:
v . (intpos 3) >= 8
and A63:
v . (intpos 2) = t . (intpos 2)
and A64:
v . GBP = t . GBP
and A65:
v . (DataLoc ((t . GBP),7)) > 0
;
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2) )reconsider ME =
v . (intpos 3) as
Element of
NAT by A62, INT_1:16;
A66:
ME = v . (intpos 3)
;
Y1:
IExec (
((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),
v)
= IExec (
((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),
(Initialize v))
by SCMPDS_5:48;
Y2:
v . (intpos md) = (Initialize v) . (intpos md)
by SCMPDS_5:40;
Y3:
v . (intpos ME) = (Initialize v) . (intpos ME)
by SCMPDS_5:40;
Y8:
v . (intpos 2) = (Initialize v) . (intpos 2)
by SCMPDS_5:40;
Y9:
v . (intpos 3) = (Initialize v) . (intpos 3)
by SCMPDS_5:40;
Y10:
v . (intpos 7) = (Initialize v) . (intpos 7)
by SCMPDS_5:40;
Y6:
v . GBP = (Initialize v) . GBP
by SCMPDS_5:40;
thus
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . GBP = v . GBP
by A6, A54, A58, A62, A63, A64, Lm14, Y1, Y3, Y6, Y8, Y9;
( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2) )thus
(
(((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v &
(((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v )
by SCMPDS_6:34, SCMPDS_6:35;
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2) )A67:
(
v . (intpos md) > v . (intpos ME) implies (
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 7) = (v . (intpos 7)) - 1 &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) = (v . (intpos 3)) + 1 &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 5) = (v . (intpos 7)) - 1 ) )
by A6, A54, A58, A62, A63, A64, Lm14, Y1, Y2, Y3, Y6, Y8, Y9, Y10;
hereby (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2)
per cases
( v . (intpos md) > v . (intpos ME) or v . (intpos md) <= v . (intpos ME) )
;
suppose A68:
v . (intpos md) > v . (intpos ME)
;
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 )hence
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7))
by A60, A67, XREAL_1:148;
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8A69:
(v . (intpos 3)) + 1
> v . (intpos 3)
by XREAL_1:31;
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) = (v . (intpos 3)) + 1
by A6, A54, A58, A62, A63, A64, A68, Lm14, Y1, Y2, Y3, Y6, Y8, Y9;
hence
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8
by A62, A69, XXREAL_0:2;
verum end; suppose A70:
v . (intpos md) <= v . (intpos ME)
;
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8 )hence
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7))
by A6, A54, A58, A60, A62, A63, A64, A65, Lm14, Y1, Y2, Y3, Y6, Y8, Y9;
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8thus
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 3) >= 8
by A6, A54, A58, A62, A63, A64, A70, Lm14, Y1, Y2, Y3, Y6, Y8, Y9;
verum end; thus
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),v)) . (intpos 2) = v . (intpos 2)
by A6, A54, A58, A62, A63, A64, A66, Lm14, Y1, Y6, Y8, Y9;
verum end; A71:
(
t . (intpos md) <= t . (intpos me) implies (
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 7) = 0 &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 3) = t . (intpos 3) &
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 5) = t . (intpos 7) ) )
by A6, A54, A57, A58, Lm14, X1, X3, X6, X7, X8, X9, X0;
A72:
me = t . (intpos 3)
;
then A73:
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . GBP = 0
by A6, A54, A57, A58, Lm14, X1, X3, X8, X0;
then A74:
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . GBP),7)) = 0
by A55, A59, A71, SCMP_GCD:5;
A75:
now let x be
Int_position ;
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . x = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . xthus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . x =
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)))) . x
by A55, A57, A60, A61, Th6
.=
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . x
by A74, SCMPDS_8:23
;
verum end; hence
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . GBP = 0
by A73;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 1) = t . (intpos 1)
by A6, A54, A57, A58, A72, Lm14, X1, X2, X3, X8, X0;
hence
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 1) = t . (intpos 1)
by A75;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 7) = 0
by A55, A59, A71, A75;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 2) = t . (intpos 2)
by A6, A54, A57, A58, A72, Lm14, X1, X3, X8, X0;
hence
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 2) = t . (intpos 2)
by A75;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos 4) = t . (intpos 4)
by A6, A54, A57, A58, A72, Lm14, X1, X3, X4, X8, X0;
hence
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 4) = t . (intpos 4)
by A75;
( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) ) )A76:
now let i be
Element of
NAT ;
( i >= 8 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i) )assume A77:
i >= 8
;
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i)U1:
t . (intpos i) = (Initialize t) . (intpos i)
by SCMPDS_5:40;
thus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) =
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),t)) . (intpos i)
by A75
.=
t . (intpos i)
by A6, A54, A57, A58, A72, A77, Lm14, X1, X3, X8, X0, U1
;
verum end; hence
for
i being
Element of
NAT st
i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) = t . (intpos i)
;
ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )A78:
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos me) = t . (intpos me)
by A57, A76;
hereby verum
per cases
( t . (intpos md) > t . (intpos me) or t . (intpos md) <= t . (intpos me) )
;
suppose A79:
t . (intpos md) > t . (intpos me)
;
ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )take m5 =
0 ;
ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )take mE3 =
m3;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5
by A55, A59, A75, A79;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3
by A55, A56, A59, A75, A79;
( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
mE3 + m5 = m3
;
( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
m5 <= t . (intpos 7)
by A55;
( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )hereby ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) )
let i be
Element of
NAT ;
( t . (intpos 3) <= i & i < mE3 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) )assume that A80:
t . (intpos 3) <= i
and A81:
i < mE3
;
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i)
i <= me
by A55, A56, A81, NAT_1:13;
then
i = t . (intpos 3)
by A80, XXREAL_0:1;
hence
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i)
by A6, A76, A78, A79;
verum
end; thus
(
m5 = 0 or
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) )
;
verum end; suppose A82:
t . (intpos md) <= t . (intpos me)
;
ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )take m5 = 1;
ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )take mE3 =
me;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 5) = m5
by A55, A71, A75, A82;
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos 3) = mE3
by A71, A75, A82;
( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
mE3 + m5 = m3
by A55, A56;
( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
m5 <= t . (intpos 7)
by A55;
( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) ) )thus
for
i being
Element of
NAT st
t . (intpos 3) <= i &
i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos i)
;
( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) )thus
(
m5 = 0 or
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),t)) . (intpos mE3) )
by A6, A76, A78, A82;
verum end; end;
then A83:
S1[ 0 ]
;
for k being Element of NAT holds S1[k]
from NAT_1:sch 1(A83, A8);
hence
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),s)) . (intpos mE3) ) ) )
by A1, A3, A4, A5, A7; verum