let p be autonomic FinPartState of SCM ; :: thesis: ( DataPart p <> {} implies IC SCM in dom p )
assume
DataPart p <> {}
; :: thesis: IC SCM in dom p
then A1:
dom (DataPart p) <> {}
;
assume A2:
not IC SCM in dom p
; :: thesis: contradiction
not p is autonomic
proof
consider d1 being
Element of
dom (DataPart p);
A3:
d1 in dom (DataPart p)
by A1;
dom (DataPart p) c= the
carrier of
SCM
by AMI_1:80;
then reconsider d1 =
d1 as
Element of
SCM by A3;
dom (DataPart p) c= SCM-Data-Loc
by AMI_3:72, RELAT_1:87;
then reconsider d1 =
d1 as
Data-Location by A3, AMI_3:def 2;
consider d2 being
Element of
SCM-Data-Loc \ (dom p);
not
SCM-Data-Loc c= dom p
;
then A4:
SCM-Data-Loc \ (dom p) <> {}
by XBOOLE_1:37;
then
d2 in SCM-Data-Loc
by XBOOLE_0:def 5;
then reconsider d2 =
d2 as
Data-Location by AMI_3:def 2;
consider il being
Element of
NAT \ (dom p);
not
NAT c= dom p
;
then A5:
NAT \ (dom p) <> {}
by XBOOLE_1:37;
then
il is
Element of
NAT
by XBOOLE_0:def 5;
then reconsider il =
il as
Instruction-Location of
SCM by AMI_1:def 4;
set p1 =
p +* (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il));
set p2 =
p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il));
consider s1 being
State of
SCM such that A6:
p +* (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il)) c= s1
by CARD_3:97;
consider s2 being
State of
SCM such that A7:
p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il)) c= s2
by CARD_3:97;
take
s1
;
:: according to AMI_1:def 25 :: thesis: ex b1 being Element of K242(the Object-Kind of SCM ) st
( p c= s1 & p c= b1 & not for b2 being Element of K97() holds (Computation s1,b2) | (dom p) = (Computation b1,b2) | (dom p) )
take
s2
;
:: thesis: ( p c= s1 & p c= s2 & not for b1 being Element of K97() holds (Computation s1,b1) | (dom p) = (Computation s2,b1) | (dom p) )
A8:
not
d2 in dom p
by A4, XBOOLE_0:def 5;
A9:
not
il in dom p
by A5, XBOOLE_0:def 5;
dom p misses {(IC SCM )}
by A2, ZFMISC_1:56;
then A10:
(dom p) /\ {(IC SCM )} = {}
by XBOOLE_0:def 7;
dom p misses {d2}
by A8, ZFMISC_1:56;
then A11:
(dom p) /\ {d2} = {}
by XBOOLE_0:def 7;
A12:
dom p misses {il}
by A9, ZFMISC_1:56;
dom (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il)) =
(dom ((il .--> (d1 := d2)) +* (d2 .--> 0 ))) \/ (dom (Start-At il))
by FUNCT_4:def 1
.=
(dom ((il .--> (d1 := d2)) +* (d2 .--> 0 ))) \/ {(IC SCM )}
by FUNCOP_1:19
.=
((dom (il .--> (d1 := d2))) \/ (dom (d2 .--> 0 ))) \/ {(IC SCM )}
by FUNCT_4:def 1
.=
({il} \/ (dom (d2 .--> 0 ))) \/ {(IC SCM )}
by FUNCOP_1:19
.=
({il} \/ {d2}) \/ {(IC SCM )}
by FUNCOP_1:19
;
then (dom p) /\ (dom (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il))) =
((dom p) /\ ({il} \/ {d2})) \/ {}
by A10, XBOOLE_1:23
.=
((dom p) /\ {il}) \/ {}
by A11, XBOOLE_1:23
.=
{}
by A12, XBOOLE_0:def 7
;
then
dom p misses dom (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il))
by XBOOLE_0:def 7;
then
p c= p +* (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il))
by FUNCT_4:33;
hence
p c= s1
by A6, XBOOLE_1:1;
:: thesis: ( p c= s2 & not for b1 being Element of K97() holds (Computation s1,b1) | (dom p) = (Computation s2,b1) | (dom p) )
dom p misses {(IC SCM )}
by A2, ZFMISC_1:56;
then A13:
(dom p) /\ {(IC SCM )} = {}
by XBOOLE_0:def 7;
dom p misses {d2}
by A8, ZFMISC_1:56;
then A14:
(dom p) /\ {d2} = {}
by XBOOLE_0:def 7;
A15:
dom p misses {il}
by A9, ZFMISC_1:56;
dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il)) =
(dom ((il .--> (d1 := d2)) +* (d2 .--> 1))) \/ (dom (Start-At il))
by FUNCT_4:def 1
.=
(dom ((il .--> (d1 := d2)) +* (d2 .--> 1))) \/ {(IC SCM )}
by FUNCOP_1:19
.=
((dom (il .--> (d1 := d2))) \/ (dom (d2 .--> 1))) \/ {(IC SCM )}
by FUNCT_4:def 1
.=
({il} \/ (dom (d2 .--> 1))) \/ {(IC SCM )}
by FUNCOP_1:19
.=
({il} \/ {d2}) \/ {(IC SCM )}
by FUNCOP_1:19
;
then (dom p) /\ (dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il))) =
((dom p) /\ ({il} \/ {d2})) \/ {}
by A13, XBOOLE_1:23
.=
((dom p) /\ {il}) \/ {}
by A14, XBOOLE_1:23
.=
{}
by A15, XBOOLE_0:def 7
;
then
dom p misses dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il))
by XBOOLE_0:def 7;
then
p c= p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il))
by FUNCT_4:33;
hence
p c= s2
by A7, XBOOLE_1:1;
:: thesis: not for b1 being Element of K97() holds (Computation s1,b1) | (dom p) = (Computation s2,b1) | (dom p)
take
1
;
:: thesis: not (Computation s1,1) | (dom p) = (Computation s2,1) | (dom p)
DataPart p c= p
by RELAT_1:88;
then A16:
dom (DataPart p) c= dom p
by RELAT_1:25;
dom (Computation s1,1) = the
carrier of
SCM
by AMI_1:79;
then A17:
dom ((Computation s1,1) | (dom p)) = dom p
by AMI_1:80, RELAT_1:91;
A18:
dom (Start-At il) = {(IC SCM )}
by FUNCOP_1:19;
then A19:
IC SCM in dom (Start-At il)
by TARSKI:def 1;
A20:
dom (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il)) = (dom ((il .--> (d1 := d2)) +* (d2 .--> 0 ))) \/ (dom (Start-At il))
by FUNCT_4:def 1;
then A21:
IC SCM in dom (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il))
by A19, XBOOLE_0:def 3;
A22:
dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il))) = (dom p) \/ (dom (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il)))
by FUNCT_4:def 1;
then
IC SCM in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il)))
by A21, XBOOLE_0:def 3;
then A23:
IC s1 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il))) . (IC SCM )
by A6, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il)) . (IC SCM )
by A21, FUNCT_4:14
.=
(Start-At il) . (IC SCM )
by A19, FUNCT_4:14
.=
il
by FUNCOP_1:87
;
dom (il .--> (d1 := d2)) = {il}
by FUNCOP_1:19;
then A24:
il in dom (il .--> (d1 := d2))
by TARSKI:def 1;
A25:
dom (d2 .--> 0 ) = {d2}
by FUNCOP_1:19;
il <> d2
by Th22;
then A26:
not
il in dom (d2 .--> 0 )
by A25, TARSKI:def 1;
A27:
dom ((il .--> (d1 := d2)) +* (d2 .--> 0 )) = (dom (il .--> (d1 := d2))) \/ (dom (d2 .--> 0 ))
by FUNCT_4:def 1;
then A28:
il in dom ((il .--> (d1 := d2)) +* (d2 .--> 0 ))
by A24, XBOOLE_0:def 3;
il <> IC SCM
by AMI_1:48;
then A29:
not
il in dom (Start-At il)
by A18, TARSKI:def 1;
A30:
il in dom (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il))
by A20, A28, XBOOLE_0:def 3;
then
il in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il)))
by A22, XBOOLE_0:def 3;
then A31:
s1 . il =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il))) . il
by A6, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il)) . il
by A30, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> 0 )) . il
by A29, FUNCT_4:12
.=
(il .--> (d1 := d2)) . il
by A26, FUNCT_4:12
.=
d1 := d2
by FUNCOP_1:87
;
A32:
d2 in dom (d2 .--> 0 )
by A25, TARSKI:def 1;
then A33:
d2 in dom ((il .--> (d1 := d2)) +* (d2 .--> 0 ))
by A27, XBOOLE_0:def 3;
d2 <> IC SCM
by Th20;
then A34:
not
d2 in dom (Start-At il)
by A18, TARSKI:def 1;
A35:
d2 in dom (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il))
by A20, A33, XBOOLE_0:def 3;
then
d2 in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il)))
by A22, XBOOLE_0:def 3;
then A36:
s1 . d2 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il))) . d2
by A6, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 0 )) +* (Start-At il)) . d2
by A35, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> 0 )) . d2
by A34, FUNCT_4:12
.=
(d2 .--> 0 ) . d2
by A32, FUNCT_4:14
.=
0
by FUNCOP_1:87
;
(Computation s1,(0 + 1)) . d1 =
(Following (Computation s1,0 )) . d1
by AMI_1:14
.=
(Following s1) . d1
by AMI_1:13
.=
0
by A23, A31, A36, AMI_3:8
;
then A37:
((Computation s1,1) | (dom p)) . d1 = 0
by A3, A16, A17, FUNCT_1:70;
dom (Computation s2,1) = the
carrier of
SCM
by AMI_1:79;
then A38:
dom ((Computation s2,1) | (dom p)) = dom p
by AMI_1:80, RELAT_1:91;
A39:
dom (Start-At il) = {(IC SCM )}
by FUNCOP_1:19;
then A40:
IC SCM in dom (Start-At il)
by TARSKI:def 1;
A41:
dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il)) = (dom ((il .--> (d1 := d2)) +* (d2 .--> 1))) \/ (dom (Start-At il))
by FUNCT_4:def 1;
then A42:
IC SCM in dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il))
by A40, XBOOLE_0:def 3;
A43:
dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il))) = (dom p) \/ (dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il)))
by FUNCT_4:def 1;
then
IC SCM in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il)))
by A42, XBOOLE_0:def 3;
then A44:
IC s2 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il))) . (IC SCM )
by A7, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il)) . (IC SCM )
by A42, FUNCT_4:14
.=
(Start-At il) . (IC SCM )
by A40, FUNCT_4:14
.=
il
by FUNCOP_1:87
;
dom (il .--> (d1 := d2)) = {il}
by FUNCOP_1:19;
then A45:
il in dom (il .--> (d1 := d2))
by TARSKI:def 1;
A46:
dom (d2 .--> 1) = {d2}
by FUNCOP_1:19;
il <> d2
by Th22;
then A47:
not
il in dom (d2 .--> 1)
by A46, TARSKI:def 1;
A48:
dom ((il .--> (d1 := d2)) +* (d2 .--> 1)) = (dom (il .--> (d1 := d2))) \/ (dom (d2 .--> 1))
by FUNCT_4:def 1;
then A49:
il in dom ((il .--> (d1 := d2)) +* (d2 .--> 1))
by A45, XBOOLE_0:def 3;
il <> IC SCM
by AMI_1:48;
then A50:
not
il in dom (Start-At il)
by A39, TARSKI:def 1;
A51:
il in dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il))
by A41, A49, XBOOLE_0:def 3;
then
il in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il)))
by A43, XBOOLE_0:def 3;
then A52:
s2 . il =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il))) . il
by A7, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il)) . il
by A51, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> 1)) . il
by A50, FUNCT_4:12
.=
(il .--> (d1 := d2)) . il
by A47, FUNCT_4:12
.=
d1 := d2
by FUNCOP_1:87
;
A53:
d2 in dom (d2 .--> 1)
by A46, TARSKI:def 1;
then A54:
d2 in dom ((il .--> (d1 := d2)) +* (d2 .--> 1))
by A48, XBOOLE_0:def 3;
d2 <> IC SCM
by Th20;
then A55:
not
d2 in dom (Start-At il)
by A39, TARSKI:def 1;
A56:
d2 in dom (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il))
by A41, A54, XBOOLE_0:def 3;
then
d2 in dom (p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il)))
by A43, XBOOLE_0:def 3;
then A57:
s2 . d2 =
(p +* (((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il))) . d2
by A7, GRFUNC_1:8
.=
(((il .--> (d1 := d2)) +* (d2 .--> 1)) +* (Start-At il)) . d2
by A56, FUNCT_4:14
.=
((il .--> (d1 := d2)) +* (d2 .--> 1)) . d2
by A55, FUNCT_4:12
.=
(d2 .--> 1) . d2
by A53, FUNCT_4:14
.=
1
by FUNCOP_1:87
;
(Computation s2,(0 + 1)) . d1 =
(Following (Computation s2,0 )) . d1
by AMI_1:14
.=
(Following s2) . d1
by AMI_1:13
.=
1
by A44, A52, A57, AMI_3:8
;
hence
(Computation s1,1) | (dom p) <> (Computation s2,1) | (dom p)
by A3, A16, A37, A38, FUNCT_1:70;
:: thesis: verum
end;
hence
contradiction
; :: thesis: verum