let k be Element of NAT ; for R being good Ring
for p being FinPartState of (SCM R)
for s being State of (SCM R) 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 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 R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
let R be good Ring; for p being FinPartState of (SCM R)
for s being State of (SCM R) 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 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 R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
let p be FinPartState of (SCM R); for s being State of (SCM R) 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 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 R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
let s be State of (SCM R); ( 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 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 R))) +* (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 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 R))) +* (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 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 R))) +* (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 ;
set sdom =
s | (dom (ProgramPart (Relocated p,k)));
let i be
Element of
NAT ;
( S1[i] implies S1[i + 1] )
assume A7:
S1[
i]
;
S1[i + 1]
reconsider ii =
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;
dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),(SCM R)) = {(IC (SCM R))}
by FUNCOP_1:19;
then A9:
IC (SCM R) in dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),(SCM R))
by TARSKI:def 1;
dom (ProgramPart (Relocated p,k)) c= the
carrier of
(SCM R)
by RELAT_1:def 18;
then
dom (ProgramPart (Relocated p,k)) c= dom s
by PARTFUN1:def 4;
then A10:
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 R) ;
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 (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) = jk
;
IC (SCM R) in dom (Relocated p,k)
by AMISTD_2:72;
then
not
Relocated p,
k is
NAT -defined
by COMPOS_1:19;
then A12:
IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) in dom (ProgramPart (Relocated p,k))
by A1, A4, Th27, FUNCT_4:26;
then
IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) in dom (Reloc (ProgramPart p),k)
by AMISTD_2:69;
then A13:
jk in { (j + k) where j is Element of NAT : 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 COMPOS_1:12, 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:
jk = j + k
and A16:
j in dom (ProgramPart p)
by A13;
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 COMPOS_1:38;
reconsider sh =
Shift (ProgramPart p),
k as
NAT -defined the
Instructions of
(SCM R) -valued finite Function ;
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 ((Shift (ProgramPart p),k) /. ii),
k =
(IncAddr sh,k) . ii
by A11, A15, AMISTD_2:def 15
.=
(ProgramPart (Relocated p,k)) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))
by AMISTD_2:69
.=
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 A12, A8, Y, GRFUNC_1:8
;
A19:
((j + k) -' k) + k = j + k
by NAT_D:34;
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 COMPOS_1:38;
A20:
(j + k) -' k = j
by NAT_D:34;
T:
ProgramPart (s +* (Relocated p,k)) = ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)
by AMI_1:123;
S:
ProgramPart s = ProgramPart (Comput (ProgramPart s),s,i)
by AMI_1:123;
not
IC (SCM R) in dom (ProgramPart p)
by COMPOS_1:12;
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 R))) +* 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 R))) +* sdom) . (IC (SCM R))
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 R))) . (IC (SCM R))
by A14, FUNCT_4:12
.=
(Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),(SCM R)) . (IC (SCM R))
by A9, 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, A11, A15, A16, A20, Y, FUNCT_4:14
.=
(Shift (ProgramPart p),k) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))
by A11, A15, A16, A19, VALUED_1:def 12
.=
(Shift (ProgramPart p),k) /. ii
by A11, A15, A17, PARTFUN1:def 8
;
then A21:
(
Comput (ProgramPart (s +* (Relocated p,k))),
(s +* (Relocated p,k)),
(i + 1) = Following (ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)),
(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 R))) = (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 (s +* (Relocated p,k))),(Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))) -' k),(SCM R)) )
by A11, A15, A18, Th23, T, AMI_1:14;
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 R))) +* sdom)) +* (ProgramPart p)
by A7, S, AMISTD_2:67
.=
(((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 R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A21, T, AMISTD_2:67
;
verum
end;
set PR = ProgramPart (Relocated p,k);
set IP = Start-At ((IC p) + k),(SCM R);
set IS = Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R);
set SD = s | (dom (ProgramPart (Relocated p,k)));
A23: IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 ) =
(s +* (Relocated p,k)) . (IC (SCM R))
by AMI_1:13
.=
IC (Relocated p,k)
by A5, FUNCT_4:14
;
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),(SCM R) c= p
by A2, FUNCOP_1:99;
then A25:
Start-At (IC p),(SCM R) 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;
A28:
{(IC (SCM R))} misses dom (ProgramPart (Relocated p,k))
by COMPOS_1:14;
(dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) /\ (dom (s | (dom (ProgramPart (Relocated p,k))))) =
(dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) /\ ((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 (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R)) 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 RELAT_1:def 18;
then
dom (ProgramPart (Relocated p,k)) c= dom s
by PARTFUN1:def 4;
then A30:
dom (ProgramPart (Relocated p,k)) = dom (s | (dom (ProgramPart (Relocated p,k))))
by RELAT_1:91;
dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R)) = {(IC (SCM R))}
by FUNCOP_1:19;
then A31:
dom (Start-At ((IC p) + k),(SCM R)) = dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))
by FUNCOP_1:19;
U2:
dom (Start-At ((IC p) + k),(SCM R)) = {(IC (SCM R))}
by FUNCOP_1:19;
Comput (ProgramPart s),s,0 =
s
by AMI_1:13
.=
s +* (ProgramPart p)
by A24, FUNCT_4:79
.=
(s +* (Start-At (IC p),(SCM R))) +* (ProgramPart p)
by A25, FUNCT_4:79
.=
(s +* (Start-At (((IC p) + k) -' k),(SCM R))) +* (ProgramPart p)
by NAT_D:34
.=
(s +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) +* (ProgramPart p)
by A23, A2, AMISTD_2:73
.=
((s +* (s | (dom (ProgramPart (Relocated p,k))))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) +* (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 R))) +* (ProgramPart p)
by A30, 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 R)))) +* (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 R)) +* (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 R))) +* (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 R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A26, FUNCT_4:79
.=
(((((s +* (DataPart p)) +* (ProgramPart (Relocated p,k))) +* (Start-At ((IC p) + k),(SCM R))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A31, FUNCT_4:78
.=
((((s +* ((DataPart p) +* (ProgramPart (Relocated p,k)))) +* (Start-At ((IC p) + k),(SCM R))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* (((DataPart p) +* (ProgramPart (Relocated p,k))) +* (Start-At ((IC p) + k),(SCM R)))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* ((DataPart p) +* ((ProgramPart (Relocated p,k)) +* (Start-At ((IC p) + k),(SCM R))))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* ((DataPart p) +* ((Start-At ((IC p) + k),(SCM R)) +* (ProgramPart (Relocated p,k))))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by U2, COMPOS_1:14, FUNCT_4:36
.=
(((s +* (((DataPart p) +* (Start-At ((IC p) + k),(SCM R))) +* (ProgramPart (Relocated p,k)))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by FUNCT_4:15
.=
(((s +* ((IncrIC (NPP p),k) +* (ProgramPart (Relocated p,k)))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by A2, COMPOS_1:75
.=
(((s +* (Relocated p,k)) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),(SCM R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
by AMISTD_2:69
.=
(((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 R))) +* (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 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 R))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p)
; verum