let k be Element of NAT ; for p being FinPartState of SCM+FSA st IC SCM+FSA in dom p holds
for s being State of SCM+FSA 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+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
let p be FinPartState of SCM+FSA ; ( IC SCM+FSA in dom p implies for s being State of SCM+FSA 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+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) )
assume
IC SCM+FSA in dom p
; for s being State of SCM+FSA 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+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
then A1:
Start-At (IC p),SCM+FSA c= p
by AMI_1:110;
let s be State of SCM+FSA ; ( 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+FSA )) +* (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+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
A4:
Start-At (IC p),SCM+FSA c= s
by A2, A1, XBOOLE_1:1;
defpred S1[ 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+FSA )) +* (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+FSA ;
set sdom =
s | (dom (ProgramPart (Relocated p,k)));
let i be
Element of
NAT ;
( S1[i] implies S1[i + 1] )
assume A6:
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+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
;
S1[i + 1]
dom (ProgramPart (Relocated p,k)) c= the
carrier of
SCM+FSA
by RELAT_1:def 18;
then
dom (ProgramPart (Relocated p,k)) c= dom s
by PARTFUN1:def 4;
then A7:
dom (ProgramPart (Relocated p,k)) = dom (s | (dom (ProgramPart (Relocated p,k))))
by RELAT_1:91;
reconsider sdom =
s | (dom (ProgramPart (Relocated p,k))) as
FinPartState of
SCM+FSA ;
dom (s | (dom (ProgramPart (Relocated p,k)))) c= NAT
by A7, RELAT_1:def 18;
then reconsider sdom =
sdom as
preProgram of
SCM+FSA by RELAT_1:def 18;
reconsider ii =
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+FSA in dom (Relocated p,k)
by Th5;
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, FUNCT_4:26, SCMFSA_3:17;
then A10:
jk in { (j + k) where j is Element of NAT : j in dom (ProgramPart p) }
by Th3;
A11:
not
IC SCM+FSA in dom (ProgramPart (Relocated p,k))
by AMI_1:101;
dom sdom = (dom s) /\ (dom (ProgramPart (Relocated p,k)))
by RELAT_1:90;
then A12:
not
IC SCM+FSA in dom sdom
by A11, XBOOLE_0:def 4;
dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM+FSA ) = {(IC SCM+FSA )}
by FUNCOP_1:19;
then A13:
IC SCM+FSA in dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM+FSA )
by TARSKI:def 1;
not
IC SCM+FSA in dom (ProgramPart p)
by AMI_1:101;
then A14:
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+FSA )) +* 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+FSA )) +* sdom) . (IC SCM+FSA )
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+FSA )) . (IC SCM+FSA )
by A12, FUNCT_4:12
.=
(Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM+FSA ) . (IC SCM+FSA )
by A13, FUNCT_4:14
.=
(IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k
by FUNCOP_1:87
;
consider j being
Element of
NAT such that A15:
jk = j + k
and A16:
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 A17:
j + k in dom (Shift (ProgramPart p),k)
by A16;
then A18:
IncAddr (pi (Shift (ProgramPart p),k),ii),
k =
(IncAddr (Shift (ProgramPart p),k),k) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))
by A15, SCMFSA_4:24
.=
(ProgramPart (Relocated p,k)) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))
by Th2
.=
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
;
Y:
(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;
A19:
((j + k) -' k) + k = j + k
by NAT_D:34;
(j + k) -' k = j
by NAT_D:34;
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 A6, A14, A15, A16, FUNCT_4:14, Y
.=
(Shift (ProgramPart p),k) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))
by A15, A16, A19, VALUED_1:def 12
.=
pi (Shift (ProgramPart p),k),
ii
by A15, A17, AMI_1:def 47
;
then A20:
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+FSA )) = (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+FSA )
by A15, A18, SCMFSA_4:29;
T:
ProgramPart s = ProgramPart (Comput (ProgramPart s),s,i)
by AMI_1:144;
S:
ProgramPart (s +* (Relocated p,k)) = ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)
by AMI_1:144;
A21:
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)
by AMI_1:14;
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+FSA )) +* sdom)) +* (ProgramPart p)
by A6, SCMFSA_3:10, T
.=
(((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+FSA )) +* sdom) +* (ProgramPart p)
by A20, SCMFSA_3:10
;
hence
S1[
i + 1]
by A21, S;
verum
end;
A22:
IC SCM+FSA in dom (Relocated p,k)
by Th5;
A23: IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 ) =
(s +* (Relocated p,k)) . (IC SCM+FSA )
by AMI_1:13
.=
IC (Relocated p,k)
by A22, FUNCT_4:14
;
DataPart p c= p
by RELAT_1:88;
then A24:
DataPart p c= s
by A2, XBOOLE_1:1;
ProgramPart p c= p
by RELAT_1:88;
then A25:
ProgramPart p c= s
by A2, XBOOLE_1:1;
set DP = DataPart p;
set IP = Start-At ((IC p) + k),SCM+FSA ;
A26:
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+FSA ;
A27:
dom (Start-At ((IC p) + k),SCM+FSA ) = {(IC SCM+FSA )}
by FUNCOP_1:19;
set PR = ProgramPart (Relocated p,k);
set SD = s | (dom (ProgramPart (Relocated p,k)));
A28:
{(IC SCM+FSA )} misses dom (DataPart p)
by AMI_1:102;
dom (ProgramPart (Relocated p,k)) c= the carrier of SCM+FSA
by RELAT_1:def 18;
then
dom (ProgramPart (Relocated p,k)) c= dom s
by PARTFUN1:def 4;
then A29:
dom (ProgramPart (Relocated p,k)) = dom (s | (dom (ProgramPart (Relocated p,k))))
by RELAT_1:91;
{(IC SCM+FSA )} misses dom (ProgramPart (Relocated p,k))
by AMI_1:103;
then A30:
{(IC SCM+FSA )} /\ (dom (ProgramPart (Relocated p,k))) = {}
by XBOOLE_0:def 7;
A31:
dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA ) = {(IC SCM+FSA )}
by FUNCOP_1:19;
then (dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) /\ (dom (s | (dom (ProgramPart (Relocated p,k))))) =
{(IC SCM+FSA )} /\ ((dom s) /\ (dom (ProgramPart (Relocated p,k))))
by RELAT_1:90
.=
({(IC SCM+FSA )} /\ (dom (ProgramPart (Relocated p,k)))) /\ (dom s)
by XBOOLE_1:16
.=
{}
by A30
;
then A32:
dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA ) 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 A25, FUNCT_4:79
.=
(s +* (Start-At (IC p),SCM+FSA )) +* (ProgramPart p)
by A4, FUNCT_4:79
.=
(s +* (Start-At (((IC p) + k) -' k),SCM+FSA )) +* (ProgramPart p)
by NAT_D:34
.=
(s +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) +* (ProgramPart p)
by A23, Th6
.=
((s +* (s | (dom (ProgramPart (Relocated p,k))))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) +* (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+FSA )) +* (ProgramPart p)
by A29, 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+FSA ))) +* (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+FSA ) +* (s | (dom (ProgramPart (Relocated p,k)))))) +* (ProgramPart p)
by A32, FUNCT_4:36
.=
(((s +* (ProgramPart (Relocated p,k))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) +* (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+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A24, 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+FSA )) +* (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+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A26, 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+FSA )) +* (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+FSA )) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A31, A27, FUNCT_4:78
.=
((((s +* ((ProgramPart (Relocated p,k)) +* (DataPart p))) +* (Start-At ((IC p) + k),SCM+FSA )) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) +* (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+FSA ))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) +* (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+FSA )))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* ((ProgramPart (Relocated p,k)) +* ((Start-At ((IC p) + k),SCM+FSA ) +* (DataPart p)))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A27, A28, FUNCT_4:36
.=
(((s +* (((ProgramPart (Relocated p,k)) +* (Start-At ((IC p) + k),SCM+FSA )) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
;
then Comput (ProgramPart s),s,0 =
(((s +* (((Start-At ((IC p) + k),SCM+FSA ) +* (ProgramPart (Relocated p,k))) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A27, 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+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by Th2
.=
(((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+FSA )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by AMI_1:13
;
then A33:
S1[ 0 ]
;
thus
for i being Element of NAT holds S1[i]
from NAT_1:sch 1(A33, A5); verum