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 COMPOS_1:119;
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 COMPOS_1:119;
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 COMPOS_1:116;
then A13:
jk in { (j + k) where j is Element of NAT : j in dom (ProgramPart p) }
by A11, COMPOS_1:117;
(
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, COMPOS_1:def 40
.=
(ProgramPart (Relocated (p,k))) . (IC (Comput ((ProgramPart (s +* (Relocated (p,k)))),(s +* (Relocated (p,k))),i)))
by COMPOS_1:116
.=
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, EXTPRO_1:4;
thus Comput (
(ProgramPart s),
s,
(i + 1)) =
Following (
(ProgramPart s),
(Comput ((ProgramPart s),s,i)))
by EXTPRO_1:4
.=
(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, AMI_1:127
.=
(((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, AMI_1:127
;
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 EXTPRO_1:3
.=
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 EXTPRO_1:3
.=
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, COMPOS_1:120
.=
((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 COMPOS_1:116
.=
(((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 EXTPRO_1:3
;
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