let k be natural number ; for R being good Ring
for p being FinPartState of
for s being State of st not R is trivial & IC (SCM R) in dom p & p c= s & Relocated p,k is autonomic holds
for i being Element of NAT holds Computation s,i = (((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
let R be good Ring; for p being FinPartState of
for s being State of st not R is trivial & IC (SCM R) in dom p & p c= s & Relocated p,k is autonomic holds
for i being Element of NAT holds Computation s,i = (((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
let p be FinPartState of ; for s being State of st not R is trivial & IC (SCM R) in dom p & p c= s & Relocated p,k is autonomic holds
for i being Element of NAT holds Computation s,i = (((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
let s be State of ; ( not R is trivial & IC (SCM R) in dom p & p c= s & Relocated p,k is autonomic implies for i being Element of NAT holds Computation s,i = (((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) )
assume that
A1:
not R is trivial
and
A2:
IC (SCM R) in dom p
and
A3:
p c= s
and
A4:
Relocated p,k is autonomic
; for i being Element of NAT holds Computation s,i = (((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
A5:
IC (SCM R) in dom (Relocated p,k)
by AMISTD_2:72;
defpred S1[ Element of NAT ] means Computation s,$1 = (((Computation (s +* (Relocated p,k)),$1) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),$1)) -' k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p);
A6:
for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
reconsider pp =
[(ProgramPart p)] as
NAT -defined FinPartState of ;
reconsider sdom =
s | (dom (ProgramPart (Relocated p,k))) as
Element of
sproduct the
Object-Kind of
(SCM R) ;
let i be
Element of
NAT ;
( S1[i] implies S1[i + 1] )
assume A7:
S1[
i]
;
S1[i + 1]
reconsider ii =
IC (Computation (s +* (Relocated p,k)),i) as
Element of
NAT by ORDINAL1:def 13;
A8:
ProgramPart (Relocated p,k) c= Computation (s +* (Relocated p,k)),
i
by AMI_1:99, FUNCT_4:26;
dom (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k)) = {(IC (SCM R))}
by FUNCOP_1:19;
then A9:
IC (SCM R) in dom (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))
by TARSKI:def 1;
dom [(ProgramPart (Relocated p,k))] c= the
carrier of
(SCM R)
by AMI_1:80;
then
dom (ProgramPart (Relocated p,k)) c= dom s
by AMI_1:79;
then A10:
dom (ProgramPart (Relocated p,k)) = dom (s | (dom (ProgramPart (Relocated p,k))))
by RELAT_1:91;
then reconsider sdom =
sdom as
FinPartState of
by FINSET_1:29;
dom (s | (dom (ProgramPart (Relocated p,k)))) c= NAT
by A10, RELAT_1:87;
then reconsider sdom =
sdom as
NAT -defined FinPartState of
by RELAT_1:def 18;
consider jk being
natural number such that A11:
IC (Computation (s +* (Relocated p,k)),i) = il. (SCM R),
jk
by AMISTD_1:26;
IC (SCM R) in dom (Relocated p,k)
by AMISTD_2:72;
then
not
Relocated p,
k is
NAT -defined
by AMI_1:109;
then A12:
IC (Computation (s +* (Relocated p,k)),i) in dom (ProgramPart (Relocated p,k))
by A1, A4, Th27, FUNCT_4:26;
then A13:
il. (SCM R),
jk in { (il. (SCM R),(j + k)) where j is Element of NAT : il. (SCM R),j in dom [(ProgramPart p)] }
by A11, AMISTD_2:70;
(
dom sdom = (dom s) /\ (dom (ProgramPart (Relocated p,k))) & not
IC (SCM R) in dom (ProgramPart (Relocated p,k)) )
by AMI_1:101, RELAT_1:90;
then A14:
not
IC (SCM R) in dom sdom
by XBOOLE_0:def 4;
consider j being
Element of
NAT such that A15:
il. (SCM R),
jk = il. (SCM R),
(j + k)
and A16:
il. (SCM R),
j in dom [(ProgramPart p)]
by A13;
dom (Shift pp,k) = { (il. (SCM R),(m + k)) where m is Element of NAT : il. (SCM R),m in dom pp }
by AMISTD_2:def 16;
then A17:
il. (SCM R),
(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 (Computation (s +* (Relocated p,k)),i))
by A11, A15, AMISTD_2:74
.=
(ProgramPart (Relocated p,k)) . (IC (Computation (s +* (Relocated p,k)),i))
by AMISTD_2:69
.=
CurInstr (Computation (s +* (Relocated p,k)),i)
by A12, A8, GRFUNC_1:8
;
A19:
((il. (SCM R),(j + k)) -' k) + k =
(((il. (SCM R),j) + k) -' k) + k
by AMISTD_1:def 13
.=
(il. (SCM R),j) + k
by AMISTD_1:61
.=
il. (SCM R),
(j + k)
by AMISTD_1:def 13
;
A20:
(il. (SCM R),(j + k)) -' k =
((il. (SCM R),j) + k) -' k
by AMISTD_1:def 13
.=
il. (SCM R),
j
by AMISTD_1:61
;
not
IC (SCM R) in dom (ProgramPart p)
by AMI_1:101;
then IC ((((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) +* sdom) +* [(ProgramPart p)]) =
(((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) +* sdom) . (IC (SCM R))
by FUNCT_4:12
.=
((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) . (IC (SCM R))
by A14, FUNCT_4:12
.=
(Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k)) . (IC (SCM R))
by A9, FUNCT_4:14
.=
(IC (Computation (s +* (Relocated p,k)),i)) -' k
by FUNCOP_1:87
;
then CurInstr (Computation s,i) =
(ProgramPart p) . ((IC (Computation (s +* (Relocated p,k)),i)) -' k)
by A7, A11, A15, A16, A20, FUNCT_4:14
.=
(Shift [(ProgramPart p)],k) . (IC (Computation (s +* (Relocated p,k)),i))
by A11, A15, A16, A19, A20, AMISTD_2:65
.=
pi (Shift [(ProgramPart p)],k),
ii
by A11, A15, A17, AMI_1:def 47
;
then A21:
(
Computation (s +* (Relocated p,k)),
(i + 1) = Following (Computation (s +* (Relocated p,k)),i) &
Exec (CurInstr (Computation s,i)),
((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) = (Following (Computation (s +* (Relocated p,k)),i)) +* (Start-At ((IC (Following (Computation (s +* (Relocated p,k)),i))) -' k)) )
by A11, A15, A18, Th23, AMI_1:14;
thus Computation s,
(i + 1) =
Following (Computation s,i)
by AMI_1:14
.=
(Exec (CurInstr (Computation s,i)),(((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) +* sdom)) +* (ProgramPart p)
by A7, AMISTD_2:67
.=
(((Computation (s +* (Relocated p,k)),(i + 1)) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),(i + 1))) -' k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A21, AMISTD_2:67
;
verum
end;
set PR = [(ProgramPart (Relocated p,k))];
set IP = Start-At ((IC p) + k);
set IS = Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k);
set SD = s | (dom [(ProgramPart (Relocated p,k))]);
A22:
dom (Start-At ((IC p) + k)) = {(IC (SCM R))}
by FUNCOP_1:19;
A23: IC (Computation (s +* (Relocated p,k)),0 ) =
(s +* (Relocated p,k)) . (IC (SCM R))
by AMI_1:13
.=
(Relocated p,k) . (IC (SCM R))
by A5, FUNCT_4:14
.=
IC (Relocated p,k)
by A5, AMI_1:def 43
;
set PP = ProgramPart p;
ProgramPart p c= p
by RELAT_1:88;
then A24:
ProgramPart p c= s
by A3, XBOOLE_1:1;
Start-At (IC p) c= p
by A2, AMI_1:110;
then A25:
Start-At (IC p) c= s
by A3, XBOOLE_1:1;
set DP = DataPart p;
DataPart p c= p
by RELAT_1:88;
then A26:
DataPart p c= s
by A3, XBOOLE_1:1;
A27:
{(IC (SCM R))} misses dom (DataPart p)
by AMI_1:102;
A28:
{(IC (SCM R))} misses dom [(ProgramPart (Relocated p,k))]
by AMI_1:103;
(dom (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) /\ (dom (s | (dom [(ProgramPart (Relocated p,k))]))) =
(dom (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) /\ ((dom s) /\ (dom [(ProgramPart (Relocated p,k))]))
by RELAT_1:90
.=
{(IC (SCM R))} /\ ((dom s) /\ (dom [(ProgramPart (Relocated p,k))]))
by FUNCOP_1:19
.=
({(IC (SCM R))} /\ (dom [(ProgramPart (Relocated p,k))])) /\ (dom s)
by XBOOLE_1:16
.=
{} /\ (dom s)
by A28, XBOOLE_0:def 7
.=
{}
;
then A29:
dom (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k)) misses dom (s | (dom [(ProgramPart (Relocated p,k))]))
by XBOOLE_0:def 7;
dom [(ProgramPart (Relocated p,k))] c= the carrier of (SCM R)
by AMI_1:80;
then
dom [(ProgramPart (Relocated p,k))] c= dom s
by AMI_1:79;
then A30:
dom [(ProgramPart (Relocated p,k))] = dom (s | (dom [(ProgramPart (Relocated p,k))]))
by RELAT_1:91;
dom (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k)) = {(IC (SCM R))}
by FUNCOP_1:19;
then A31:
dom (Start-At ((IC p) + k)) = dom (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))
by FUNCOP_1:19;
A32:
dom (DataPart p) misses dom [(ProgramPart (Relocated p,k))]
by AMI_1:104;
Computation s,0 =
s
by AMI_1:13
.=
s +* (ProgramPart p)
by A24, FUNCT_4:79
.=
(s +* (Start-At (IC p))) +* (ProgramPart p)
by A25, FUNCT_4:79
.=
(s +* (Start-At (((IC p) + k) -' k))) +* (ProgramPart p)
by AMISTD_1:61
.=
(s +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (ProgramPart p)
by A23, AMISTD_2:73
.=
((s +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (ProgramPart p)
by FUNCT_4:80
.=
(((s +* [(ProgramPart (Relocated p,k))]) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (ProgramPart p)
by A30, FUNCT_4:78
.=
((s +* [(ProgramPart (Relocated p,k))]) +* ((s | (dom [(ProgramPart (Relocated p,k))])) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k)))) +* (ProgramPart p)
by FUNCT_4:15
.=
((s +* [(ProgramPart (Relocated p,k))]) +* ((Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k)) +* (s | (dom [(ProgramPart (Relocated p,k))])))) +* (ProgramPart p)
by A29, FUNCT_4:36
.=
(((s +* [(ProgramPart (Relocated p,k))]) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by FUNCT_4:15
.=
((((s +* (DataPart p)) +* [(ProgramPart (Relocated p,k))]) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by A26, FUNCT_4:79
.=
(((s +* ((DataPart p) +* [(ProgramPart (Relocated p,k))])) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* ([(ProgramPart (Relocated p,k))] +* (DataPart p))) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by A32, FUNCT_4:36
.=
((((s +* [(ProgramPart (Relocated p,k))]) +* (DataPart p)) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((((s +* [(ProgramPart (Relocated p,k))]) +* (DataPart p)) +* (Start-At ((IC p) + k))) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by A31, FUNCT_4:78
.=
((((s +* ([(ProgramPart (Relocated p,k))] +* (DataPart p))) +* (Start-At ((IC p) + k))) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* (([(ProgramPart (Relocated p,k))] +* (DataPart p)) +* (Start-At ((IC p) + k)))) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* ([(ProgramPart (Relocated p,k))] +* ((DataPart p) +* (Start-At ((IC p) + k))))) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* ([(ProgramPart (Relocated p,k))] +* ((Start-At ((IC p) + k)) +* (DataPart p)))) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by A22, A27, FUNCT_4:36
.=
(((s +* (([(ProgramPart (Relocated p,k))] +* (Start-At ((IC p) + k))) +* (DataPart p))) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* (((Start-At ((IC p) + k)) +* [(ProgramPart (Relocated p,k))]) +* (DataPart p))) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by A22, AMI_1:103, FUNCT_4:36
.=
(((s +* (Relocated p,k)) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom [(ProgramPart (Relocated p,k))]))) +* (ProgramPart p)
by AMISTD_2:69
.=
(((Computation (s +* (Relocated p,k)),0 ) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),0 )) -' k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by AMI_1:13
;
then A33:
S1[ 0 ]
;
for n being Element of NAT holds S1[n]
from NAT_1:sch 1(A33, A6);
hence
for i being Element of NAT holds Computation s,i = (((Computation (s +* (Relocated p,k)),i) +* (Start-At ((IC (Computation (s +* (Relocated p,k)),i)) -' k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
; verum