for p being autonomic FinPartState of SCM+FSA st DataPart p <> {} holds
IC in dom p
proof
let p be
autonomic FinPartState of
SCM+FSA;
( DataPart p <> {} implies IC in dom p )
assume
DataPart p <> {}
;
IC in dom p
then A1:
dom (DataPart p) <> {}
;
assume
not
IC in dom p
;
contradiction
then A2:
dom p misses {(IC )}
by ZFMISC_1:56;
not
p is
autonomic
proof
set il = the
Element of
NAT \ (dom p);
set d2 = the
Element of
Int-Locations \ (dom p);
set d1 = the
Element of
dom (DataPart p);
A3:
dom (DataPart p) c= Data-Locations SCM+FSA
by RELAT_1:87;
not
NAT c= dom p
;
then A4:
NAT \ (dom p) <> {}
by XBOOLE_1:37;
then
the
Element of
NAT \ (dom p) in NAT
by XBOOLE_0:def 5;
then reconsider il = the
Element of
NAT \ (dom p) as
Element of
NAT ;
not
Int-Locations c= dom p
;
then A5:
Int-Locations \ (dom p) <> {}
by XBOOLE_1:37;
then
the
Element of
Int-Locations \ (dom p) in Int-Locations
by XBOOLE_0:def 5;
then reconsider d2 = the
Element of
Int-Locations \ (dom p) as
Int-Location by SCMFSA_2:11;
A6:
the
Element of
dom (DataPart p) in dom (DataPart p)
by A1;
DataPart p c= NPP p
by COMPOS_1:169;
then
dom (DataPart p) c= dom (NPP p)
by RELAT_1:25;
then A7:
the
Element of
dom (DataPart p) in dom (NPP p)
by A6;
dom (DataPart p) c= the
carrier of
SCM+FSA
by RELAT_1:def 18;
then reconsider d1 = the
Element of
dom (DataPart p) as
Element of
SCM+FSA by A6;
per cases
( d1 in Int-Locations or d1 in FinSeq-Locations )
by A6, A3, XBOOLE_0:def 3, SCMFSA_2:127;
suppose
d1 in Int-Locations
;
not p is autonomic then reconsider d1 =
d1 as
Int-Location by SCMFSA_2:11;
set p1 =
p +* (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)));
set p2 =
p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)));
consider s1 being
State of
SCM+FSA such that A8:
p +* (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) c= s1
by PBOOLE:156;
not
il in dom p
by A4, XBOOLE_0:def 5;
then A9:
dom p misses {il}
by ZFMISC_1:56;
not
d2 in dom p
by A5, XBOOLE_0:def 5;
then A10:
dom p misses {d2}
by ZFMISC_1:56;
consider s2 being
State of
SCM+FSA such that A11:
p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) c= s2
by PBOOLE:156;
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)) +* (Start-At (il,SCM+FSA))) =
(dom ((il .--> (d1 := d2)) +* (d2 .--> 0))) \/ (dom (Start-At (il,SCM+FSA)))
by FUNCT_4:def 1
.=
(dom ((il .--> (d1 := d2)) +* (d2 .--> 0))) \/ {(IC )}
by FUNCOP_1:19
.=
((dom (il .--> (d1 := d2))) \/ (dom (d2 .--> 0))) \/ {(IC )}
by FUNCT_4:def 1
.=
({il} \/ (dom (d2 .--> 0))) \/ {(IC )}
by FUNCOP_1:19
.=
({il} \/ {d2}) \/ {(IC )}
by FUNCOP_1:19
;
then (dom p) /\ (dom (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))) =
((dom p) /\ ({il} \/ {d2})) \/ ((dom p) /\ {(IC )})
by XBOOLE_1:23
.=
((dom p) /\ ({il} \/ {d2})) \/ {}
by A2, XBOOLE_0:def 7
.=
((dom p) /\ {il}) \/ ((dom p) /\ {d2})
by XBOOLE_1:23
.=
((dom p) /\ {il}) \/ {}
by A10, XBOOLE_0:def 7
.=
{}
by A9, XBOOLE_0:def 7
;
then
dom p misses dom (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))
by XBOOLE_0:def 7;
then
p c= p +* (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))
by FUNCT_4:33;
then A12:
p c= s1
by A8, 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 .--> 1)) +* (Start-At (il,SCM+FSA))) =
(dom ((il .--> (d1 := d2)) +* (d2 .--> 1))) \/ (dom (Start-At (il,SCM+FSA)))
by FUNCT_4:def 1
.=
(dom ((il .--> (d1 := d2)) +* (d2 .--> 1))) \/ {(IC )}
by FUNCOP_1:19
.=
((dom (il .--> (d1 := d2))) \/ (dom (d2 .--> 1))) \/ {(IC )}
by FUNCT_4:def 1
.=
({il} \/ (dom (d2 .--> 1))) \/ {(IC )}
by FUNCOP_1:19
.=
({il} \/ {d2}) \/ {(IC )}
by FUNCOP_1:19
;
then (dom p) /\ (dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))) =
((dom p) /\ ({il} \/ {d2})) \/ ((dom p) /\ {(IC )})
by XBOOLE_1:23
.=
((dom p) /\ ({il} \/ {d2})) \/ {}
by A2, XBOOLE_0:def 7
.=
((dom p) /\ {il}) \/ ((dom p) /\ {d2})
by XBOOLE_1:23
.=
((dom p) /\ {il}) \/ {}
by A10, XBOOLE_0:def 7
.=
{}
by A9, XBOOLE_0:def 7
;
then
dom p misses dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))
by XBOOLE_0:def 7;
then
p c= p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))
by FUNCT_4:33;
then A13:
p c= s2
by A11, 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 A12;
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 A13;
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))A14:
dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) = (dom ((il .--> (d1 := d2)) +* (d2 .--> 1))) \/ (dom (Start-At (il,SCM+FSA)))
by FUNCT_4:def 1;
A15:
dom (NPP p) c= the
carrier of
SCM+FSA
by RELAT_1:def 18;
A16:
dom (Comput ((ProgramPart s2),s2,1)) = the
carrier of
SCM+FSA
by PARTFUN1:def 4;
A17:
dom ((Comput ((ProgramPart s2),s2,1)) | (dom (NPP p))) = dom (NPP p)
by RELAT_1:91, A15, A16;
A18:
dom ((il .--> (d1 := d2)) +* (d2 .--> 1)) = (dom (il .--> (d1 := d2))) \/ (dom (d2 .--> 1))
by FUNCT_4:def 1;
A19:
dom (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) = (dom ((il .--> (d1 := d2)) +* (d2 .--> 0))) \/ (dom (Start-At (il,SCM+FSA)))
by FUNCT_4:def 1;
A20:
dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))) = (dom p) \/ (dom (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))))
by FUNCT_4:def 1;
A21:
dom (Start-At (il,SCM+FSA)) = {(IC )}
by FUNCOP_1:19;
then A22:
IC in dom (Start-At (il,SCM+FSA))
by TARSKI:def 1;
then A23:
IC in dom (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))
by A19, XBOOLE_0:def 3;
then
IC in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))))
by A20, XBOOLE_0:def 3;
then A24:
IC s1 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))) . (IC )
by A8, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) . (IC )
by A23, FUNCT_4:14
.=
(Start-At (il,SCM+FSA)) . (IC )
by A22, FUNCT_4:14
.=
il
by FUNCOP_1:87
;
il <> IC
by COMPOS_1:3;
then A25:
not
il in dom (Start-At (il,SCM+FSA))
by A21, TARSKI:def 1;
A26:
dom ((il .--> (d1 := d2)) +* (d2 .--> 0)) = (dom (il .--> (d1 := d2))) \/ (dom (d2 .--> 0))
by FUNCT_4:def 1;
d2 <> IC
by SCMFSA_2:81;
then A27:
not
d2 in dom (Start-At (il,SCM+FSA))
by A21, TARSKI:def 1;
A28:
dom (d2 .--> 0) = {d2}
by FUNCOP_1:19;
then A29:
d2 in dom (d2 .--> 0)
by TARSKI:def 1;
then
d2 in dom ((il .--> (d1 := d2)) +* (d2 .--> 0))
by A26, XBOOLE_0:def 3;
then A30:
d2 in dom (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))
by A19, XBOOLE_0:def 3;
then
d2 in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))))
by A20, XBOOLE_0:def 3;
then A31:
s1 . d2 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))) . d2
by A8, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) . d2
by A30, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> 0)) . d2
by A27, FUNCT_4:12
.=
(d2 .--> 0) . d2
by A29, FUNCT_4:14
.=
0
by FUNCOP_1:87
;
il <> d2
by SCMFSA_2:84;
then A32:
not
il in dom (d2 .--> 0)
by A28, 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))
by A26, XBOOLE_0:def 3;
then A33:
il in dom (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))
by A19, XBOOLE_0:def 3;
then
il in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))))
by A20, XBOOLE_0:def 3;
then A34:
s1 . il =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))) . il
by A8, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) . il
by A33, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> 0)) . il
by A25, FUNCT_4:12
.=
(il .--> (d1 := d2)) . il
by A32, FUNCT_4:12
.=
d1 := d2
by FUNCOP_1:87
;
A35:
dom (NPP p) c= the
carrier of
SCM+FSA
by RELAT_1:def 18;
A36:
dom (Comput ((ProgramPart s1),s1,1)) = the
carrier of
SCM+FSA
by PARTFUN1:def 4;
A37:
dom ((Comput ((ProgramPart s1),s1,1)) | (dom (NPP p))) = dom (NPP p)
by RELAT_1:91, A35, A36;
A38:
dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))) = (dom p) \/ (dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))))
by FUNCT_4:def 1;
A39:
dom (Start-At (il,SCM+FSA)) = {(IC )}
by FUNCOP_1:19;
then A40:
IC in dom (Start-At (il,SCM+FSA))
by TARSKI:def 1;
then A41:
IC in dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))
by A14, XBOOLE_0:def 3;
then
IC in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))))
by A38, XBOOLE_0:def 3;
then A42:
IC s2 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))) . (IC )
by A11, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) . (IC )
by A41, FUNCT_4:14
.=
(Start-At (il,SCM+FSA)) . (IC )
by A40, FUNCT_4:14
.=
il
by FUNCOP_1:87
;
d2 <> IC
by SCMFSA_2:81;
then A43:
not
d2 in dom (Start-At (il,SCM+FSA))
by A39, TARSKI:def 1;
A44:
dom (d2 .--> 1) = {d2}
by FUNCOP_1:19;
then A45:
d2 in dom (d2 .--> 1)
by TARSKI:def 1;
then
d2 in dom ((il .--> (d1 := d2)) +* (d2 .--> 1))
by A18, XBOOLE_0:def 3;
then A46:
d2 in dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))
by A14, XBOOLE_0:def 3;
then
d2 in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))))
by A38, XBOOLE_0:def 3;
then A47:
s2 . d2 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))) . d2
by A11, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) . d2
by A46, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> 1)) . d2
by A43, FUNCT_4:12
.=
(d2 .--> 1) . d2
by A45, FUNCT_4:14
.=
1
by FUNCOP_1:87
;
il <> IC
by COMPOS_1:3;
then A48:
not
il in dom (Start-At (il,SCM+FSA))
by A39, TARSKI:def 1;
il <> d2
by SCMFSA_2:84;
then A49:
not
il in dom (d2 .--> 1)
by A44, 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 .--> 1))
by A18, XBOOLE_0:def 3;
then A50:
il in dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))
by A14, XBOOLE_0:def 3;
then
il in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))))
by A38, XBOOLE_0:def 3;
then A51:
s2 . il =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))) . il
by A11, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) . il
by A50, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> 1)) . il
by A48, FUNCT_4:12
.=
(il .--> (d1 := d2)) . il
by A49, FUNCT_4:12
.=
d1 := d2
by FUNCOP_1:87
;
A52:
(ProgramPart s2) /. (IC s2) = s2 . (IC s2)
by COMPOS_1:38;
A53:
(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
.=
1
by A42, A51, A47, A52, SCMFSA_2:89
;
A54:
(ProgramPart s1) /. (IC s1) = s1 . (IC s1)
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
by A24, A34, A31, A54, SCMFSA_2:89
;
then
((Comput ((ProgramPart s1),s1,1)) | (dom (NPP p))) . d1 = 0
by FUNCT_1:70, A7, A37;
hence
(Comput (P,s1,1)) | (dom (NPP p)) <> (Comput (Q,s2,1)) | (dom (NPP p))
by A53, FUNCT_1:70, A7, A17;
verum end; suppose
d1 in FinSeq-Locations
;
not p is autonomic then reconsider d1 =
d1 as
FinSeq-Location by SCMFSA_2:12;
set p1 =
p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)));
set p2 =
p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)));
consider s1 being
State of
SCM+FSA such that A55:
p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) c= s1
by PBOOLE:156;
not
il in dom p
by A4, XBOOLE_0:def 5;
then A56:
dom p misses {il}
by ZFMISC_1:56;
A57:
dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) = (dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0))) \/ (dom (Start-At (il,SCM+FSA)))
by FUNCT_4:def 1;
consider k being
Element of
NAT such that A58:
k = abs (s1 . d2)
and A59:
(Exec ((d1 :=<0,...,0> d2),s1)) . d1 = k |-> 0
by SCMFSA_2:101;
A60:
dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) = (dom (il .--> (d1 :=<0,...,0> d2))) \/ (dom (d2 .--> 0))
by FUNCT_4:def 1;
A61:
dom (p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))) = (dom p) \/ (dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))))
by FUNCT_4:def 1;
A62:
dom (Start-At (il,SCM+FSA)) = {(IC )}
by FUNCOP_1:19;
then A63:
IC in dom (Start-At (il,SCM+FSA))
by TARSKI:def 1;
then A64:
IC in dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))
by A57, XBOOLE_0:def 3;
then
IC in dom (p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))))
by A61, XBOOLE_0:def 3;
then A65:
IC s1 =
(p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))) . (IC )
by A55, GRFUNC_1:8
.=
(((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) . (IC )
by A64, FUNCT_4:14
.=
(Start-At (il,SCM+FSA)) . (IC )
by A63, FUNCT_4:14
.=
il
by FUNCOP_1:87
;
il <> IC
by COMPOS_1:3;
then A66:
not
il in dom (Start-At (il,SCM+FSA))
by A62, TARSKI:def 1;
consider s2 being
State of
SCM+FSA such that A67:
p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) c= s2
by PBOOLE:156;
d2 <> IC
by SCMFSA_2:81;
then A68:
not
d2 in dom (Start-At (il,SCM+FSA))
by A62, TARSKI:def 1;
A69:
dom (d2 .--> 0) = {d2}
by FUNCOP_1:19;
then A70:
d2 in dom (d2 .--> 0)
by TARSKI:def 1;
then
d2 in dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0))
by A60, XBOOLE_0:def 3;
then A71:
d2 in dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))
by A57, XBOOLE_0:def 3;
then
d2 in dom (p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))))
by A61, XBOOLE_0:def 3;
then s1 . d2 =
(p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))) . d2
by A55, GRFUNC_1:8
.=
(((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) . d2
by A71, FUNCT_4:14
.=
((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) . d2
by A68, FUNCT_4:12
.=
(d2 .--> 0) . d2
by A70, FUNCT_4:14
.=
0
by FUNCOP_1:87
;
then A72:
k |-> 0 =
0 |-> 0
by A58, ABSVALUE:7
.=
{}
by FINSEQ_2:72
;
not
d2 in dom p
by A5, XBOOLE_0:def 5;
then A73:
dom p misses {d2}
by ZFMISC_1:56;
il <> d2
by SCMFSA_2:84;
then A74:
not
il in dom (d2 .--> 0)
by A69, TARSKI:def 1;
dom (il .--> (d1 :=<0,...,0> d2)) = {il}
by FUNCOP_1:19;
then
il in dom (il .--> (d1 :=<0,...,0> d2))
by TARSKI:def 1;
then
il in dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0))
by A60, XBOOLE_0:def 3;
then A75:
il in dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))
by A57, XBOOLE_0:def 3;
then
il in dom (p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))))
by A61, XBOOLE_0:def 3;
then A76:
s1 . il =
(p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))) . il
by A55, GRFUNC_1:8
.=
(((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) . il
by A75, FUNCT_4:14
.=
((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) . il
by A66, FUNCT_4:12
.=
(il .--> (d1 :=<0,...,0> d2)) . il
by A74, FUNCT_4:12
.=
d1 :=<0,...,0> d2
by FUNCOP_1:87
;
A77:
dom (NPP p) c= the
carrier of
SCM+FSA
by RELAT_1:def 18;
A78:
dom (Comput ((ProgramPart s1),s1,1)) = the
carrier of
SCM+FSA
by PARTFUN1:def 4;
A79:
dom ((Comput ((ProgramPart s1),s1,1)) | (dom (NPP p))) = dom (NPP p)
by RELAT_1:91, A77, A78;
A80:
dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) = (dom (il .--> (d1 :=<0,...,0> d2))) \/ (dom (d2 .--> 1))
by FUNCT_4:def 1;
consider k being
Element of
NAT such that A81:
k = abs (s2 . d2)
and A82:
(Exec ((d1 :=<0,...,0> d2),s2)) . d1 = k |-> 0
by SCMFSA_2:101;
A83:
dom (p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))) = (dom p) \/ (dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))))
by FUNCT_4:def 1;
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 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA))) =
(dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0))) \/ (dom (Start-At (il,SCM+FSA)))
by FUNCT_4:def 1
.=
(dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0))) \/ {(IC )}
by FUNCOP_1:19
.=
((dom (il .--> (d1 :=<0,...,0> d2))) \/ (dom (d2 .--> 0))) \/ {(IC )}
by FUNCT_4:def 1
.=
({il} \/ (dom (d2 .--> 0))) \/ {(IC )}
by FUNCOP_1:19
.=
({il} \/ {d2}) \/ {(IC )}
by FUNCOP_1:19
;
then (dom p) /\ (dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))) =
((dom p) /\ ({il} \/ {d2})) \/ ((dom p) /\ {(IC )})
by XBOOLE_1:23
.=
((dom p) /\ ({il} \/ {d2})) \/ {}
by A2, XBOOLE_0:def 7
.=
((dom p) /\ {il}) \/ ((dom p) /\ {d2})
by XBOOLE_1:23
.=
((dom p) /\ {il}) \/ {}
by A73, XBOOLE_0:def 7
.=
{}
by A56, XBOOLE_0:def 7
;
then
dom p misses dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))
by XBOOLE_0:def 7;
then
p c= p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 0)) +* (Start-At (il,SCM+FSA)))
by FUNCT_4:33;
then A84:
p c= s1
by A55, 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 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) =
(dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1))) \/ (dom (Start-At (il,SCM+FSA)))
by FUNCT_4:def 1
.=
(dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1))) \/ {(IC )}
by FUNCOP_1:19
.=
((dom (il .--> (d1 :=<0,...,0> d2))) \/ (dom (d2 .--> 1))) \/ {(IC )}
by FUNCT_4:def 1
.=
({il} \/ (dom (d2 .--> 1))) \/ {(IC )}
by FUNCOP_1:19
.=
({il} \/ {d2}) \/ {(IC )}
by FUNCOP_1:19
;
then (dom p) /\ (dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))) =
((dom p) /\ ({il} \/ {d2})) \/ ((dom p) /\ {(IC )})
by XBOOLE_1:23
.=
((dom p) /\ ({il} \/ {d2})) \/ {}
by A2, XBOOLE_0:def 7
.=
((dom p) /\ {il}) \/ ((dom p) /\ {d2})
by XBOOLE_1:23
.=
((dom p) /\ {il}) \/ {}
by A73, XBOOLE_0:def 7
.=
{}
by A56, XBOOLE_0:def 7
;
then
dom p misses dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))
by XBOOLE_0:def 7;
then
p c= p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))
by FUNCT_4:33;
then A85:
p c= s2
by A67, 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 A84;
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 A85;
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))A86:
dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) = (dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1))) \/ (dom (Start-At (il,SCM+FSA)))
by FUNCT_4:def 1;
A87:
dom (Start-At (il,SCM+FSA)) = {(IC )}
by FUNCOP_1:19;
then A88:
IC in dom (Start-At (il,SCM+FSA))
by TARSKI:def 1;
then A89:
IC in dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))
by A86, XBOOLE_0:def 3;
then
IC in dom (p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))))
by A83, XBOOLE_0:def 3;
then A90:
IC s2 =
(p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))) . (IC )
by A67, GRFUNC_1:8
.=
(((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) . (IC )
by A89, FUNCT_4:14
.=
(Start-At (il,SCM+FSA)) . (IC )
by A88, FUNCT_4:14
.=
il
by FUNCOP_1:87
;
d2 <> IC
by SCMFSA_2:81;
then A91:
not
d2 in dom (Start-At (il,SCM+FSA))
by A87, TARSKI:def 1;
A92:
dom (d2 .--> 1) = {d2}
by FUNCOP_1:19;
then A93:
d2 in dom (d2 .--> 1)
by TARSKI:def 1;
then
d2 in dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1))
by A80, XBOOLE_0:def 3;
then A94:
d2 in dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))
by A86, XBOOLE_0:def 3;
then
d2 in dom (p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))))
by A83, XBOOLE_0:def 3;
then s2 . d2 =
(p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))) . d2
by A67, GRFUNC_1:8
.=
(((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) . d2
by A94, FUNCT_4:14
.=
((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) . d2
by A91, FUNCT_4:12
.=
(d2 .--> 1) . d2
by A93, FUNCT_4:14
.=
1
by FUNCOP_1:87
;
then A95:
k |-> 0 =
1
|-> 0
by A81, ABSVALUE:def 1
.=
<*0*>
by FINSEQ_2:73
;
il <> IC
by COMPOS_1:3;
then A96:
not
il in dom (Start-At (il,SCM+FSA))
by A87, TARSKI:def 1;
il <> d2
by SCMFSA_2:84;
then A97:
not
il in dom (d2 .--> 1)
by A92, TARSKI:def 1;
dom (il .--> (d1 :=<0,...,0> d2)) = {il}
by FUNCOP_1:19;
then
il in dom (il .--> (d1 :=<0,...,0> d2))
by TARSKI:def 1;
then
il in dom ((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1))
by A80, XBOOLE_0:def 3;
then A98:
il in dom (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))
by A86, XBOOLE_0:def 3;
then
il in dom (p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))))
by A83, XBOOLE_0:def 3;
then A99:
s2 . il =
(p +* (((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA)))) . il
by A67, GRFUNC_1:8
.=
(((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) +* (Start-At (il,SCM+FSA))) . il
by A98, FUNCT_4:14
.=
((il .--> (d1 :=<0,...,0> d2)) +* (d2 .--> 1)) . il
by A96, FUNCT_4:12
.=
(il .--> (d1 :=<0,...,0> d2)) . il
by A97, FUNCT_4:12
.=
d1 :=<0,...,0> d2
by FUNCOP_1:87
;
A100:
(ProgramPart s2) /. (IC s2) = s2 . (IC s2)
by COMPOS_1:38;
A101:
(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
.=
<*0*>
by A90, A99, A82, A95, A100
;
A102:
dom (NPP p) c= the
carrier of
SCM+FSA
by RELAT_1:def 18;
A103:
dom (Comput ((ProgramPart s2),s2,1)) = the
carrier of
SCM+FSA
by PARTFUN1:def 4;
A104:
dom ((Comput ((ProgramPart s2),s2,1)) | (dom (NPP p))) = dom (NPP p)
by RELAT_1:91, A102, A103;
A105:
(ProgramPart s1) /. (IC s1) = s1 . (IC s1)
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
.=
{}
by A65, A76, A59, A72, A105
;
then
((Comput ((ProgramPart s1),s1,1)) | (dom (NPP p))) . d1 = {}
by FUNCT_1:70, A7, A79;
hence
(Comput (P,s1,1)) | (dom (NPP p)) <> (Comput (Q,s2,1)) | (dom (NPP p))
by A101, FUNCT_1:70, A7, A104;
verum end;
end;
hence
contradiction
;
verum
end;
hence
SCM+FSA is IC-recognized
by AMISTD_5:3; verum