let k be Element of NAT ; for p being FinPartState of SCM st IC SCM in dom p holds
for s being State of SCM st p c= s & Relocated p,k is autonomic holds
for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
let p be FinPartState of SCM ; ( IC SCM in dom p implies for s being State of SCM st p c= s & Relocated p,k is autonomic holds
for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) )
assume
IC SCM in dom p
; for s being State of SCM st p c= s & Relocated p,k is autonomic holds
for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
then A1:
Start-At (IC p),SCM c= p
by AMI_1:110;
let s be State of SCM ; ( p c= s & Relocated p,k is autonomic implies for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) )
assume that
A2:
p c= s
and
A3:
Relocated p,k is autonomic
; for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
A4:
Start-At (IC p),SCM c= s
by A2, A1, XBOOLE_1:1;
defpred S1[ Element of NAT ] means Comput (ProgramPart s),s,$1 = (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),$1) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),$1)) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p);
A5:
for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
reconsider pp =
ProgramPart p as
preProgram of
SCM ;
set sdom =
s | (dom (ProgramPart (Relocated p,k)));
dom (ProgramPart (Relocated p,k)) c= the
carrier of
SCM
by RELAT_1:def 18;
then
dom (ProgramPart (Relocated p,k)) c= dom s
by PARTFUN1:def 4;
then
dom (ProgramPart (Relocated p,k)) = dom (s | (dom (ProgramPart (Relocated p,k))))
by RELAT_1:91;
then
(
rng (s | (dom (ProgramPart (Relocated p,k)))) c= the
Instructions of
SCM &
dom (s | (dom (ProgramPart (Relocated p,k)))) c= NAT )
by AMI_1:140, RELAT_1:87;
then reconsider sdom =
s | (dom (ProgramPart (Relocated p,k))) as
NAT -defined finite FinPartState of
by RELSET_1:11;
let i be
Element of
NAT ;
( S1[i] implies S1[i + 1] )
assume A7:
Comput (ProgramPart s),
s,
i = (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
;
S1[i + 1]
reconsider kk =
IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) as
Element of
NAT ;
reconsider jk =
IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) as
Element of
NAT ;
A8:
ProgramPart (Relocated p,k) c= Comput (ProgramPart (s +* (Relocated p,k))),
(s +* (Relocated p,k)),
i
by AMI_1:99, FUNCT_4:26;
IC SCM in dom (Relocated p,k)
by Th25;
then
not
Relocated p,
k is
NAT -defined
by AMI_1:109;
then A9:
IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) in dom (ProgramPart (Relocated p,k))
by A3, AMI_5:86, FUNCT_4:26;
then A10:
jk in { (j + k) where j is Element of NAT : j in dom (ProgramPart p) }
by Th23;
(
dom sdom = (dom s) /\ (dom (ProgramPart (Relocated p,k))) & not
IC SCM in dom (ProgramPart (Relocated p,k)) )
by AMI_1:101, RELAT_1:90;
then A11:
not
IC SCM in dom sdom
by XBOOLE_0:def 4;
dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM ) = {(IC SCM )}
by FUNCOP_1:19;
then A12:
IC SCM in dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )
by TARSKI:def 1;
consider j being
Element of
NAT such that A13:
jk = j + k
and A14:
j in dom (ProgramPart p)
by A10;
Y:
(ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) /. (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) = (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))
by AMI_1:150;
dom (Shift pp,k) = { (m + k) where m is Element of NAT : m in dom pp }
by VALUED_1:def 12;
then A15:
j + k in dom (Shift (ProgramPart p),k)
by A14;
then A16:
IncAddr ((Shift (ProgramPart p),k) /. kk),
k =
(IncAddr (Shift (ProgramPart p),k),k) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))
by A13, Def5
.=
(ProgramPart (Relocated p,k)) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))
by Th22
.=
CurInstr (ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)),
(Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)
by A9, A8, GRFUNC_1:8, Y
;
A17:
(j + k) -' k = j
by NAT_D:34;
Z:
(ProgramPart (Comput (ProgramPart s),s,i)) /. (IC (Comput (ProgramPart s),s,i)) = (Comput (ProgramPart s),s,i) . (IC (Comput (ProgramPart s),s,i))
by AMI_1:150;
T:
ProgramPart (s +* (Relocated p,k)) = ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)
by AMI_1:144;
not
IC SCM in dom (ProgramPart p)
by AMI_1:101;
then IC ((((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* sdom) +* (ProgramPart p)) =
(((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* sdom) . (IC SCM )
by FUNCT_4:12
.=
((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) . (IC SCM )
by A11, FUNCT_4:12
.=
(Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM ) . (IC SCM )
by A12, FUNCT_4:14
.=
(IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k
by FUNCOP_1:87
;
then CurInstr (ProgramPart (Comput (ProgramPart s),s,i)),
(Comput (ProgramPart s),s,i) =
(ProgramPart p) . ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k)
by A7, A13, A14, A17, FUNCT_4:14, Z
.=
(Shift (ProgramPart p),k) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))
by A13, A14, A17, VALUED_1:def 12
.=
(Shift (ProgramPart p),k) /. kk
by A13, A15, PARTFUN1:def 8
;
then A18:
(
Comput (ProgramPart (s +* (Relocated p,k))),
(s +* (Relocated p,k)),
(i + 1) = Following (ProgramPart (s +* (Relocated p,k))),
(Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) &
Exec (CurInstr (ProgramPart (Comput (ProgramPart s),s,i)),(Comput (ProgramPart s),s,i)),
((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) = (Following (ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)),(Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) +* (Start-At ((IC (Following (ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)),(Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))) -' k),SCM ) )
by A13, A16, Th32, AMI_1:14;
S:
ProgramPart s = ProgramPart (Comput (ProgramPart s),s,i)
by AMI_1:144;
thus Comput (ProgramPart s),
s,
(i + 1) =
Following (ProgramPart s),
(Comput (ProgramPart s),s,i)
by AMI_1:14
.=
(Exec (CurInstr (ProgramPart (Comput (ProgramPart s),s,i)),(Comput (ProgramPart s),s,i)),(((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* sdom)) +* (ProgramPart p)
by A7, AMI_5:77, S
.=
(((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),(i + 1)) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),(i + 1))) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A18, AMI_5:77, T
;
verum
end;
A19:
IC SCM in dom (Relocated p,k)
by Th25;
A20: IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 ) =
(s +* (Relocated p,k)) . (IC SCM )
by AMI_1:13
.=
(Relocated p,k) . (IC SCM )
by A19, FUNCT_4:14
.=
IC (Relocated p,k)
;
DataPart p c= p
by RELAT_1:88;
then A21:
DataPart p c= s
by A2, XBOOLE_1:1;
ProgramPart p c= p
by RELAT_1:88;
then A22:
ProgramPart p c= s
by A2, XBOOLE_1:1;
set DP = DataPart p;
set IP = Start-At ((IC p) + k),SCM ;
A23:
dom (DataPart p) misses dom (ProgramPart (Relocated p,k))
by AMI_1:104;
set PP = ProgramPart p;
set IS = Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM ;
A24:
dom (Start-At ((IC p) + k),SCM ) = {(IC SCM )}
by FUNCOP_1:19;
set PR = ProgramPart (Relocated p,k);
set SD = s | (dom (ProgramPart (Relocated p,k)));
A25:
{(IC SCM )} misses dom (DataPart p)
by AMI_1:102;
dom (ProgramPart (Relocated p,k)) c= the carrier of SCM
by RELAT_1:def 18;
then
dom (ProgramPart (Relocated p,k)) c= dom s
by PARTFUN1:def 4;
then A26:
dom (ProgramPart (Relocated p,k)) = dom (s | (dom (ProgramPart (Relocated p,k))))
by RELAT_1:91;
{(IC SCM )} misses dom (ProgramPart (Relocated p,k))
by AMI_1:103;
then A27:
{(IC SCM )} /\ (dom (ProgramPart (Relocated p,k))) = {}
by XBOOLE_0:def 7;
A28:
dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM ) = {(IC SCM )}
by FUNCOP_1:19;
then (dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) /\ (dom (s | (dom (ProgramPart (Relocated p,k))))) =
{(IC SCM )} /\ ((dom s) /\ (dom (ProgramPart (Relocated p,k))))
by RELAT_1:90
.=
({(IC SCM )} /\ (dom (ProgramPart (Relocated p,k)))) /\ (dom s)
by XBOOLE_1:16
.=
{}
by A27
;
then A29:
dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM ) misses dom (s | (dom (ProgramPart (Relocated p,k))))
by XBOOLE_0:def 7;
Comput (ProgramPart s),s,0 =
s
by AMI_1:13
.=
s +* (ProgramPart p)
by A22, FUNCT_4:79
.=
(s +* (Start-At (IC p),SCM )) +* (ProgramPart p)
by A4, FUNCT_4:79
.=
(s +* (Start-At (((IC p) + k) -' k),SCM )) +* (ProgramPart p)
by NAT_D:34
.=
(s +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (ProgramPart p)
by A20, Th26
.=
((s +* (s | (dom (ProgramPart (Relocated p,k))))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (ProgramPart p)
by FUNCT_4:80
.=
(((s +* (ProgramPart (Relocated p,k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (ProgramPart p)
by A26, FUNCT_4:78
.=
((s +* (ProgramPart (Relocated p,k))) +* ((s | (dom (ProgramPart (Relocated p,k)))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM ))) +* (ProgramPart p)
by FUNCT_4:15
.=
((s +* (ProgramPart (Relocated p,k))) +* ((Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM ) +* (s | (dom (ProgramPart (Relocated p,k)))))) +* (ProgramPart p)
by A29, FUNCT_4:36
.=
(((s +* (ProgramPart (Relocated p,k))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
((((s +* (DataPart p)) +* (ProgramPart (Relocated p,k))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A21, FUNCT_4:79
.=
(((s +* ((DataPart p) +* (ProgramPart (Relocated p,k)))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* ((ProgramPart (Relocated p,k)) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A23, FUNCT_4:36
.=
((((s +* (ProgramPart (Relocated p,k))) +* (DataPart p)) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((((s +* (ProgramPart (Relocated p,k))) +* (DataPart p)) +* (Start-At ((IC p) + k),SCM )) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A28, A24, FUNCT_4:78
.=
((((s +* ((ProgramPart (Relocated p,k)) +* (DataPart p))) +* (Start-At ((IC p) + k),SCM )) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* (((ProgramPart (Relocated p,k)) +* (DataPart p)) +* (Start-At ((IC p) + k),SCM ))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* ((ProgramPart (Relocated p,k)) +* ((DataPart p) +* (Start-At ((IC p) + k),SCM )))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* ((ProgramPart (Relocated p,k)) +* ((Start-At ((IC p) + k),SCM ) +* (DataPart p)))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A24, A25, FUNCT_4:36
.=
(((s +* (((ProgramPart (Relocated p,k)) +* (Start-At ((IC p) + k),SCM )) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* (((Start-At ((IC p) + k),SCM ) +* (ProgramPart (Relocated p,k))) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A24, AMI_1:103, FUNCT_4:36
.=
(((s +* (Relocated p,k)) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by Th22
.=
(((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 ) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by AMI_1:13
;
then A30:
S1[ 0 ]
;
thus
for i being Element of NAT holds S1[i]
from NAT_1:sch 1(A30, A5); verum