( not R is trivial implies for p being autonomic FinPartState of (SCM R) st DataPart p <> {} holds
IC in dom p )
proof
assume
not
R is
trivial
;
for p being autonomic FinPartState of (SCM R) st DataPart p <> {} holds
IC in dom p
let p be
autonomic FinPartState of
(SCM R);
( DataPart p <> {} implies IC in dom p )
assume
DataPart p <> {}
;
IC in dom p
then A1:
dom (DataPart p) <> {}
;
assume A2:
not
IC in dom p
;
contradiction
not
p is
autonomic
proof
set il = the
Element of
NAT \ (dom p);
set d2 = the
Element of
(Data-Locations SCM) \ (dom p);
set d1 = the
Element of
dom (DataPart p);
A3:
the
Element of
dom (DataPart p) in dom (DataPart p)
by A1;
DataPart p c= NPP p
by COMPOS_1:169;
then A4:
dom (DataPart p) c= dom (NPP p)
by RELAT_1:25;
dom (DataPart p) c= the
carrier of
(SCM R)
by RELAT_1:def 18;
then reconsider d1 = the
Element of
dom (DataPart p) as
Element of
(SCM R) by A3;
not
Data-Locations SCM c= dom p
;
then A5:
(Data-Locations SCM) \ (dom p) <> {}
by XBOOLE_1:37;
then
the
Element of
(Data-Locations SCM) \ (dom p) in Data-Locations SCM
by XBOOLE_0:def 5;
then reconsider d2 = the
Element of
(Data-Locations SCM) \ (dom p) as
Data-Location of
R by SCMRING2:1;
A6:
not
d2 in dom p
by A5, XBOOLE_0:def 5;
not
NAT c= dom p
;
then A7:
NAT \ (dom p) <> {}
by XBOOLE_1:37;
then reconsider il = the
Element of
NAT \ (dom p) as
Element of
NAT by XBOOLE_0:def 5;
A8:
not
il in dom p
by A7, XBOOLE_0:def 5;
Data-Locations (SCM R) = Data-Locations SCM
by SCMRING2:31;
then
dom (DataPart p) c= Data-Locations SCM
by RELAT_1:87;
then reconsider d1 =
d1 as
Data-Location of
R by A3, SCMRING2:1;
A9:
dom (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R)))) = (dom ((il .--> (d1 := d2)) +* (d2 .--> (0. R)))) \/ (dom (Start-At (il,(SCM R))))
by FUNCT_4:def 1;
set p1 =
p +* (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))));
consider s1 being
State of
(SCM R) such that A10:
p +* (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R)))) c= s1
by PBOOLE:156;
A11:
dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))))) = (dom p) \/ (dom (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R)))))
by FUNCT_4:def 1;
A12:
dom (Start-At (il,(SCM R))) = {(IC )}
by FUNCOP_1:19;
then A13:
IC in dom (Start-At (il,(SCM R)))
by TARSKI:def 1;
then A14:
IC in dom (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))))
by A9, XBOOLE_0:def 3;
then
IC in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R)))))
by A11, XBOOLE_0:def 3;
then A15:
IC s1 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))))) . (IC )
by A10, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R)))) . (IC )
by A14, FUNCT_4:14
.=
(Start-At (il,(SCM R))) . (IC )
by A13, FUNCT_4:14
.=
il
by FUNCOP_1:87
;
A16:
il <> IC
by COMPOS_1:3;
then A17:
not
il in dom (Start-At (il,(SCM R)))
by A12, TARSKI:def 1;
A18:
dom (Start-At (il,(SCM R))) = {(IC )}
by FUNCOP_1:19;
then A19:
not
il in dom (Start-At (il,(SCM R)))
by A16, TARSKI:def 1;
A20:
d2 <> IC
by SCMRING3:3;
then A21:
not
d2 in dom (Start-At (il,(SCM R)))
by A12, TARSKI:def 1;
A22:
not
d2 in dom (Start-At (il,(SCM R)))
by A20, A18, TARSKI:def 1;
A23:
dom ((il .--> (d1 := d2)) +* (d2 .--> (0. R))) = (dom (il .--> (d1 := d2))) \/ (dom (d2 .--> (0. R)))
by FUNCT_4:def 1;
A24:
dom (d2 .--> (0. R)) = {d2}
by FUNCOP_1:19;
then A25:
d2 in dom (d2 .--> (0. R))
by TARSKI:def 1;
then
d2 in dom ((il .--> (d1 := d2)) +* (d2 .--> (0. R)))
by A23, XBOOLE_0:def 3;
then A26:
d2 in dom (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))))
by A9, XBOOLE_0:def 3;
then
d2 in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R)))))
by A11, XBOOLE_0:def 3;
then A27:
s1 . d2 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))))) . d2
by A10, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R)))) . d2
by A26, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> (0. R))) . d2
by A21, FUNCT_4:12
.=
(d2 .--> (0. R)) . d2
by A25, FUNCT_4:14
.=
0. R
by FUNCOP_1:87
;
A28:
il <> d2
by Th5;
then A29:
not
il in dom (d2 .--> (0. R))
by A24, TARSKI:def 1;
dom (il .--> (d1 := d2)) = {il}
by FUNCOP_1:19;
then
il in dom (il .--> (d1 := d2))
by TARSKI:def 1;
then
il in dom ((il .--> (d1 := d2)) +* (d2 .--> (0. R)))
by A23, XBOOLE_0:def 3;
then A30:
il in dom (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))))
by A9, XBOOLE_0:def 3;
then
il in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R)))))
by A11, XBOOLE_0:def 3;
then A31:
s1 . il =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))))) . il
by A10, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R)))) . il
by A30, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> (0. R))) . il
by A17, FUNCT_4:12
.=
(il .--> (d1 := d2)) . il
by A29, FUNCT_4:12
.=
d1 := d2
by FUNCOP_1:87
;
A32:
dom (NPP p) c= the
carrier of
(SCM R)
by RELAT_1:def 18;
dom (Comput ((ProgramPart s1),s1,1)) = the
carrier of
(SCM R)
by PARTFUN1:def 4;
then A33:
dom ((Comput ((ProgramPart s1),s1,1)) | (dom (NPP p))) = dom (NPP p)
by A32, RELAT_1:91;
consider e being
Element of
R such that A34:
e <> 0. R
by STRUCT_0:def 19;
set p2 =
p +* (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))));
consider s2 being
State of
(SCM R) such that A35:
p +* (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R)))) c= s2
by PBOOLE:156;
A36:
dom (Comput ((ProgramPart s2),s2,1)) = the
carrier of
(SCM R)
by PARTFUN1:def 4;
A37:
dom ((Comput ((ProgramPart s2),s2,1)) | (dom (NPP p))) = dom (NPP p)
by RELAT_1:91, A32, A36;
take P =
ProgramPart s1;
EXTPRO_1:def 9 ex b1 being set st
( ProgramPart p c= P & ProgramPart p c= b1 & ex b2, b3 being set st
( NPP p c= b2 & NPP p c= b3 & not for b4 being Element of NAT holds (Comput (P,b2,b4)) | (proj1 (NPP p)) = (Comput (b1,b3,b4)) | (proj1 (NPP p)) ) )
take Q =
ProgramPart s2;
( ProgramPart p c= P & ProgramPart p c= Q & ex b1, b2 being set st
( NPP p c= b1 & NPP p c= b2 & not for b3 being Element of NAT holds (Comput (P,b1,b3)) | (proj1 (NPP p)) = (Comput (Q,b2,b3)) | (proj1 (NPP p)) ) )
dom (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R)))) =
(dom ((il .--> (d1 := d2)) +* (d2 .--> (0. R)))) \/ (dom (Start-At (il,(SCM R))))
by FUNCT_4:def 1
.=
(dom ((il .--> (d1 := d2)) +* (d2 .--> (0. R)))) \/ {(IC )}
by FUNCOP_1:19
.=
((dom (il .--> (d1 := d2))) \/ (dom (d2 .--> (0. R)))) \/ {(IC )}
by FUNCT_4:def 1
.=
({il} \/ (dom (d2 .--> (0. R)))) \/ {(IC )}
by FUNCOP_1:19
.=
({il} \/ {d2}) \/ {(IC )}
by FUNCOP_1:19
;
then (dom p) /\ (dom (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))))) =
((dom p) /\ ({il} \/ {d2})) \/ ((dom p) /\ {(IC )})
by XBOOLE_1:23
.=
((dom p) /\ ({il} \/ {d2})) \/ {}
by A2, Lm1
.=
((dom p) /\ {il}) \/ ((dom p) /\ {d2})
by XBOOLE_1:23
.=
((dom p) /\ {il}) \/ {}
by A6, Lm1
.=
{}
by A8, Lm1
;
then
dom p misses dom (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))))
by XBOOLE_0:def 7;
then
p c= p +* (((il .--> (d1 := d2)) +* (d2 .--> (0. R))) +* (Start-At (il,(SCM R))))
by FUNCT_4:33;
then A38:
p c= s1
by A10, XBOOLE_1:1;
hence
ProgramPart p c= P
by RELAT_1:105;
( ProgramPart p c= Q & ex b1, b2 being set st
( NPP p c= b1 & NPP p c= b2 & not for b3 being Element of NAT holds (Comput (P,b1,b3)) | (proj1 (NPP p)) = (Comput (Q,b2,b3)) | (proj1 (NPP p)) ) )
dom (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R)))) =
(dom ((il .--> (d1 := d2)) +* (d2 .--> e))) \/ (dom (Start-At (il,(SCM R))))
by FUNCT_4:def 1
.=
(dom ((il .--> (d1 := d2)) +* (d2 .--> e))) \/ {(IC )}
by FUNCOP_1:19
.=
((dom (il .--> (d1 := d2))) \/ (dom (d2 .--> e))) \/ {(IC )}
by FUNCT_4:def 1
.=
({il} \/ (dom (d2 .--> e))) \/ {(IC )}
by FUNCOP_1:19
.=
({il} \/ {d2}) \/ {(IC )}
by FUNCOP_1:19
;
then (dom p) /\ (dom (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))))) =
((dom p) /\ ({il} \/ {d2})) \/ ((dom p) /\ {(IC )})
by XBOOLE_1:23
.=
((dom p) /\ ({il} \/ {d2})) \/ {}
by A2, Lm1
.=
((dom p) /\ {il}) \/ ((dom p) /\ {d2})
by XBOOLE_1:23
.=
((dom p) /\ {il}) \/ {}
by A6, Lm1
.=
{}
by A8, Lm1
;
then
dom p misses dom (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))))
by XBOOLE_0:def 7;
then
p c= p +* (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))))
by FUNCT_4:33;
then A39:
p c= s2
by A35, XBOOLE_1:1;
hence
ProgramPart p c= Q
by RELAT_1:105;
ex b1, b2 being set st
( NPP p c= b1 & NPP p c= b2 & not for b3 being Element of NAT holds (Comput (P,b1,b3)) | (proj1 (NPP p)) = (Comput (Q,b2,b3)) | (proj1 (NPP p)) )
take
s1
;
ex b1 being set st
( NPP p c= s1 & NPP p c= b1 & not for b2 being Element of NAT holds (Comput (P,s1,b2)) | (proj1 (NPP p)) = (Comput (Q,b1,b2)) | (proj1 (NPP p)) )
take
s2
;
( NPP p c= s1 & NPP p c= s2 & not for b1 being Element of NAT holds (Comput (P,s1,b1)) | (proj1 (NPP p)) = (Comput (Q,s2,b1)) | (proj1 (NPP p)) )
p c= s1
by A38;
hence
NPP p c= s1
by XBOOLE_1:1;
( NPP p c= s2 & not for b1 being Element of NAT holds (Comput (P,s1,b1)) | (proj1 (NPP p)) = (Comput (Q,s2,b1)) | (proj1 (NPP p)) )
p c= s2
by A39;
hence
NPP p c= s2
by XBOOLE_1:1;
not for b1 being Element of NAT holds (Comput (P,s1,b1)) | (proj1 (NPP p)) = (Comput (Q,s2,b1)) | (proj1 (NPP p))
take
1
;
not (Comput (P,s1,1)) | (proj1 (NPP p)) = (Comput (Q,s2,1)) | (proj1 (NPP p))
A40:
dom (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R)))) = (dom ((il .--> (d1 := d2)) +* (d2 .--> e))) \/ (dom (Start-At (il,(SCM R))))
by FUNCT_4:def 1;
A41:
dom (d2 .--> e) = {d2}
by FUNCOP_1:19;
then A42:
not
il in dom (d2 .--> e)
by A28, TARSKI:def 1;
A43:
dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))))) = (dom p) \/ (dom (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R)))))
by FUNCT_4:def 1;
A44:
IC in dom (Start-At (il,(SCM R)))
by A18, TARSKI:def 1;
then A45:
IC in dom (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))))
by A40, XBOOLE_0:def 3;
then
IC in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R)))))
by A43, XBOOLE_0:def 3;
then A46:
IC s2 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))))) . (IC )
by A35, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R)))) . (IC )
by A45, FUNCT_4:14
.=
(Start-At (il,(SCM R))) . (IC )
by A44, FUNCT_4:14
.=
il
by FUNCOP_1:87
;
A47:
dom ((il .--> (d1 := d2)) +* (d2 .--> e)) = (dom (il .--> (d1 := d2))) \/ (dom (d2 .--> e))
by FUNCT_4:def 1;
dom (il .--> (d1 := d2)) = {il}
by FUNCOP_1:19;
then
il in dom (il .--> (d1 := d2))
by TARSKI:def 1;
then
il in dom ((il .--> (d1 := d2)) +* (d2 .--> e))
by A47, XBOOLE_0:def 3;
then A48:
il in dom (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))))
by A40, XBOOLE_0:def 3;
then
il in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R)))))
by A43, XBOOLE_0:def 3;
then A49:
s2 . il =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))))) . il
by A35, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R)))) . il
by A48, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> e)) . il
by A19, FUNCT_4:12
.=
(il .--> (d1 := d2)) . il
by A42, FUNCT_4:12
.=
d1 := d2
by FUNCOP_1:87
;
A50:
d2 in dom (d2 .--> e)
by A41, TARSKI:def 1;
then
d2 in dom ((il .--> (d1 := d2)) +* (d2 .--> e))
by A47, XBOOLE_0:def 3;
then A51:
d2 in dom (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))))
by A40, XBOOLE_0:def 3;
then
d2 in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R)))))
by A43, XBOOLE_0:def 3;
then A52:
s2 . d2 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R))))) . d2
by A35, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> e)) +* (Start-At (il,(SCM R)))) . d2
by A51, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> e)) . d2
by A22, FUNCT_4:12
.=
(d2 .--> e) . d2
by A50, FUNCT_4:14
.=
e
by FUNCOP_1:87
;
A53:
(ProgramPart s2) /. il = s2 . il
by COMPOS_1:38;
A54:
(Comput ((ProgramPart s2),s2,(0 + 1))) . d1 =
(Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,0)))) . d1
by EXTPRO_1:4
.=
(Following ((ProgramPart s2),s2)) . d1
by EXTPRO_1:3
.=
e
by A46, A49, A52, A53, SCMRING2:13
;
A55:
(ProgramPart s1) /. il = s1 . il
by COMPOS_1:38;
(Comput ((ProgramPart s1),s1,(0 + 1))) . d1 =
(Following ((ProgramPart s1),(Comput ((ProgramPart s1),s1,0)))) . d1
by EXTPRO_1:4
.=
(Following ((ProgramPart s1),s1)) . d1
by EXTPRO_1:3
.=
0. R
by A15, A31, A27, A55, SCMRING2:13
;
then
((Comput (P,s1,1)) | (dom (NPP p))) . d1 = 0. R
by A33, FUNCT_1:70, A3, A4;
hence
(Comput (P,s1,1)) | (dom (NPP p)) <> (Comput (Q,s2,1)) | (dom (NPP p))
by A34, A3, A37, A4, A54, FUNCT_1:70;
verum
end;
hence
contradiction
;
verum
end;
hence
SCM R is IC-recognized
by AMISTD_5:3; verum