let s be State of SCM+FSA; for Q being Instruction-Sequence of SCM+FSA st s . (intloc (1 + 1)) >= 1 & s . (intloc (1 + 1)) <= len (s . (fsloc 0)) holds
( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),Q,s)) . (intloc (2 + 1)) = s . (intloc (2 + 1)) & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),Q,s)) . (intloc (5 + 1)) = s . (intloc (5 + 1)) & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),Q,s)) . (fsloc 0) = s . (fsloc 0) & ex x1 being Integer st
( x1 = (s . (fsloc 0)) . (s . (intloc (1 + 1))) & ( x1 - (s . (intloc (5 + 1))) > 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),Q,s)) . (intloc (1 + 1)) = 0 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),Q,s)) . (intloc (3 + 1)) = s . (intloc (3 + 1)) ) ) & ( x1 - (s . (intloc (5 + 1))) <= 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),Q,s)) . (intloc (1 + 1)) = (s . (intloc (1 + 1))) - 1 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),Q,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1 ) ) ) )
let P be Instruction-Sequence of SCM+FSA; ( s . (intloc (1 + 1)) >= 1 & s . (intloc (1 + 1)) <= len (s . (fsloc 0)) implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (2 + 1)) = s . (intloc (2 + 1)) & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (5 + 1)) = s . (intloc (5 + 1)) & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (fsloc 0) = s . (fsloc 0) & ex x1 being Integer st
( x1 = (s . (fsloc 0)) . (s . (intloc (1 + 1))) & ( x1 - (s . (intloc (5 + 1))) > 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = 0 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = s . (intloc (3 + 1)) ) ) & ( x1 - (s . (intloc (5 + 1))) <= 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = (s . (intloc (1 + 1))) - 1 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1 ) ) ) ) )
assume that
A1:
s . (intloc (1 + 1)) >= 1
and
A2:
s . (intloc (1 + 1)) <= len (s . (fsloc 0))
; ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (2 + 1)) = s . (intloc (2 + 1)) & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (5 + 1)) = s . (intloc (5 + 1)) & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (fsloc 0) = s . (fsloc 0) & ex x1 being Integer st
( x1 = (s . (fsloc 0)) . (s . (intloc (1 + 1))) & ( x1 - (s . (intloc (5 + 1))) > 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = 0 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = s . (intloc (3 + 1)) ) ) & ( x1 - (s . (intloc (5 + 1))) <= 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = (s . (intloc (1 + 1))) - 1 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1 ) ) ) )
A3:
|.(s . (intloc (1 + 1))).| = s . (intloc (1 + 1))
by A1, ABSVALUE:def 1;
set s0 = Initialized s;
set s1 = Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s));
set s2 = IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s);
A4:
intloc (3 + 1) <> intloc (2 + 1)
by SCMFSA_2:101;
reconsider k1 = s . (intloc (1 + 1)) as Element of NAT by A1, INT_1:3;
reconsider n = (s . (fsloc 0)) . k1 as Integer ;
A5:
(s . (fsloc 0)) /. k1 = n
by A1, A2, FINSEQ_4:15;
A6: (Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))) . (intloc (4 + 1)) =
((Initialized s) . (fsloc 0)) /. |.((Initialized s) . (intloc (1 + 1))).|
by SCMBSORT:2
.=
((Initialized s) . (fsloc 0)) /. |.(s . (intloc (1 + 1))).|
by SCMFSA_M:37
.=
n
by A3, A5, SCMFSA_M:37
;
A7:
intloc (1 + 1) <> intloc (2 + 1)
by SCMFSA_2:101;
A8:
intloc (4 + 1) <> intloc (5 + 1)
by SCMFSA_2:101;
then A9: (Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))) . (intloc (5 + 1)) =
(Initialized s) . (intloc (5 + 1))
by SCMFSA_2:72
.=
s . (intloc (5 + 1))
by SCMFSA_M:37
;
A10:
intloc (4 + 1) <> intloc (2 + 1)
by SCMFSA_2:101;
A11: (IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)) . (intloc (4 + 1)) =
(Exec ((SubFrom ((intloc (4 + 1)),(intloc (5 + 1)))),(Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))))) . (intloc (4 + 1))
by SCMFSA6C:8
.=
n - (s . (intloc (5 + 1)))
by A9, A6, SCMFSA_2:65
;
thus (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (2 + 1)) =
(IExec ((if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))),P,(IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)))) . (intloc (2 + 1))
by SCM_HALT:20
.=
(IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)) . (intloc (2 + 1))
by A4, A7, Lm15
.=
(Exec ((SubFrom ((intloc (4 + 1)),(intloc (5 + 1)))),(Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))))) . (intloc (2 + 1))
by SCMFSA6C:8
.=
(Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))) . (intloc (2 + 1))
by A10, SCMFSA_2:65
.=
(Initialized s) . (intloc (2 + 1))
by A10, SCMFSA_2:72
.=
s . (intloc (2 + 1))
by SCMFSA_M:37
; ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (5 + 1)) = s . (intloc (5 + 1)) & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (fsloc 0) = s . (fsloc 0) & ex x1 being Integer st
( x1 = (s . (fsloc 0)) . (s . (intloc (1 + 1))) & ( x1 - (s . (intloc (5 + 1))) > 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = 0 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = s . (intloc (3 + 1)) ) ) & ( x1 - (s . (intloc (5 + 1))) <= 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = (s . (intloc (1 + 1))) - 1 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1 ) ) ) )
A12:
intloc (3 + 1) <> intloc (5 + 1)
by SCMFSA_2:101;
A13:
intloc (1 + 1) <> intloc (5 + 1)
by SCMFSA_2:101;
thus (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (5 + 1)) =
(IExec ((if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))),P,(IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)))) . (intloc (5 + 1))
by SCM_HALT:20
.=
(IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)) . (intloc (5 + 1))
by A12, A13, Lm15
.=
(Exec ((SubFrom ((intloc (4 + 1)),(intloc (5 + 1)))),(Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))))) . (intloc (5 + 1))
by SCMFSA6C:8
.=
s . (intloc (5 + 1))
by A8, A9, SCMFSA_2:65
; ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (fsloc 0) = s . (fsloc 0) & ex x1 being Integer st
( x1 = (s . (fsloc 0)) . (s . (intloc (1 + 1))) & ( x1 - (s . (intloc (5 + 1))) > 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = 0 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = s . (intloc (3 + 1)) ) ) & ( x1 - (s . (intloc (5 + 1))) <= 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = (s . (intloc (1 + 1))) - 1 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1 ) ) ) )
A14:
intloc (4 + 1) <> intloc (3 + 1)
by SCMFSA_2:101;
thus (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (fsloc 0) =
(IExec ((if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))),P,(IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)))) . (fsloc 0)
by SCM_HALT:21
.=
(IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)) . (fsloc 0)
by Lm13
.=
(Exec ((SubFrom ((intloc (4 + 1)),(intloc (5 + 1)))),(Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))))) . (fsloc 0)
by SCMFSA6C:9
.=
(Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))) . (fsloc 0)
by SCMFSA_2:65
.=
(Initialized s) . (fsloc 0)
by SCMFSA_2:72
.=
s . (fsloc 0)
by SCMFSA_M:37
; ex x1 being Integer st
( x1 = (s . (fsloc 0)) . (s . (intloc (1 + 1))) & ( x1 - (s . (intloc (5 + 1))) > 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = 0 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = s . (intloc (3 + 1)) ) ) & ( x1 - (s . (intloc (5 + 1))) <= 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = (s . (intloc (1 + 1))) - 1 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1 ) ) )
take
n
; ( n = (s . (fsloc 0)) . (s . (intloc (1 + 1))) & ( n - (s . (intloc (5 + 1))) > 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = 0 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = s . (intloc (3 + 1)) ) ) & ( n - (s . (intloc (5 + 1))) <= 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = (s . (intloc (1 + 1))) - 1 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1 ) ) )
A15:
intloc (4 + 1) <> intloc (1 + 1)
by SCMFSA_2:101;
thus
n = (s . (fsloc 0)) . (s . (intloc (1 + 1)))
; ( ( n - (s . (intloc (5 + 1))) > 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = 0 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = s . (intloc (3 + 1)) ) ) & ( n - (s . (intloc (5 + 1))) <= 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = (s . (intloc (1 + 1))) - 1 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1 ) ) )
A16: (IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)) . (intloc (3 + 1)) =
(Exec ((SubFrom ((intloc (4 + 1)),(intloc (5 + 1)))),(Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))))) . (intloc (3 + 1))
by SCMFSA6C:8
.=
(Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))) . (intloc (3 + 1))
by A14, SCMFSA_2:65
.=
(Initialized s) . (intloc (3 + 1))
by A14, SCMFSA_2:72
.=
s . (intloc (3 + 1))
by SCMFSA_M:37
;
hereby ( n - (s . (intloc (5 + 1))) <= 0 implies ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = (s . (intloc (1 + 1))) - 1 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1 ) )
assume A17:
n - (s . (intloc (5 + 1))) > 0
;
( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = 0 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = s . (intloc (3 + 1)) )thus (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) =
(IExec ((if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))),P,(IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)))) . (intloc (1 + 1))
by SCM_HALT:20
.=
0
by A11, A17, Lm3
;
(IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = s . (intloc (3 + 1))thus (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) =
(IExec ((if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))),P,(IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)))) . (intloc (3 + 1))
by SCM_HALT:20
.=
s . (intloc (3 + 1))
by A16, A11, A17, Lm14
;
verum
end;
A18: (IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)) . (intloc (1 + 1)) =
(Exec ((SubFrom ((intloc (4 + 1)),(intloc (5 + 1)))),(Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))))) . (intloc (1 + 1))
by SCMFSA6C:8
.=
(Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))),(Initialized s))) . (intloc (1 + 1))
by A15, SCMFSA_2:65
.=
(Initialized s) . (intloc (1 + 1))
by A15, SCMFSA_2:72
.=
s . (intloc (1 + 1))
by SCMFSA_M:37
;
assume A19:
n - (s . (intloc (5 + 1))) <= 0
; ( (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) = (s . (intloc (1 + 1))) - 1 & (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1 )
thus (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (1 + 1)) =
(IExec ((if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))),P,(IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)))) . (intloc (1 + 1))
by SCM_HALT:20
.=
(s . (intloc (1 + 1))) - 1
by A18, A11, A19, Lm3
; (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) = (s . (intloc (3 + 1))) + 1
thus (IExec (((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))),P,s)) . (intloc (3 + 1)) =
(IExec ((if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))),P,(IExec ((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))),P,s)))) . (intloc (3 + 1))
by SCM_HALT:20
.=
(s . (intloc (3 + 1))) + 1
by A16, A11, A19, Lm14
; verum