let s be State of SCM+FSA; :: thesis:
let P be Instruction-Sequence of SCM+FSA; :: thesis:
assume that
A1: s . (intloc (1 + 1)) >= 1 and
A2: s . (intloc (1 + 1)) <= len (s . (fsloc 0)) ; :: thesis:
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 ; :: thesis:
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 ; :: thesis:
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 ; :: thesis:
take n ; :: thesis:
A15: intloc (4 + 1) <> intloc (1 + 1) by SCMFSA_2:101;
thus n = (s . (fsloc 0)) . (s . (intloc (1 + 1))) ; :: thesis:
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 :: thesis:

assume A17: n - (s . (intloc (5 + 1))) > 0 ; :: thesis:
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 ; :: thesis:
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 ; :: thesis:

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 ; :: thesis:
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 ; :: thesis:
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 ; :: thesis: