let k be Element of NAT ; for R being good Ring
for s being State of (SCM R) st not R is trivial holds
for p being autonomic FinPartState of (SCM R) st IC (SCM R) in dom p & Relocated p,k c= s holds
for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
let R be good Ring; for s being State of (SCM R) st not R is trivial holds
for p being autonomic FinPartState of (SCM R) st IC (SCM R) in dom p & Relocated p,k c= s holds
for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
let s be State of (SCM R); ( not R is trivial implies for p being autonomic FinPartState of (SCM R) st IC (SCM R) in dom p & Relocated p,k c= s holds
for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k)) )
assume A1:
not R is trivial
; for p being autonomic FinPartState of (SCM R) st IC (SCM R) in dom p & Relocated p,k c= s holds
for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
let p be autonomic FinPartState of (SCM R); ( IC (SCM R) in dom p & Relocated p,k c= s implies for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k)) )
assume that
A2:
IC (SCM R) in dom p
and
A3:
Relocated p,k c= s
; for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
A4:
Start-At ((IC p) + k),(SCM R) c= Relocated p,k
by A2, AMISTD_2:77;
defpred S1[ Element of NAT ] means Comput (ProgramPart s),s,$1 = (((Comput (ProgramPart (s +* p)),(s +* p),$1) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),$1)) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k));
A5:
for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
set sdom =
s | (dom (ProgramPart p));
let i be
Element of
NAT ;
( S1[i] implies S1[i + 1] )
assume A6:
Comput (ProgramPart s),
s,
i = (((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
;
S1[i + 1]
dom (ProgramPart p) c= the
carrier of
(SCM R)
by RELAT_1:def 18;
then
dom (ProgramPart p) c= dom s
by PARTFUN1:def 4;
then A7:
dom (ProgramPart p) = dom (s | (dom (ProgramPart p)))
by RELAT_1:91;
reconsider sdom =
s | (dom (ProgramPart p)) as
FinPartState of
(SCM R) ;
dom (s | (dom (ProgramPart p))) c= NAT
by A7, RELAT_1:87;
then reconsider sdom =
sdom as
NAT -defined FinPartState of
by RELAT_1:def 18;
reconsider ii =
IC (Comput (ProgramPart (s +* p)),(s +* p),i) as
Element of
NAT ;
A8:
ProgramPart p c= Comput (ProgramPart (s +* p)),
(s +* p),
i
by AMI_1:99, FUNCT_4:26;
dom (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R)) = {(IC (SCM R))}
by FUNCOP_1:19;
then A9:
IC (SCM R) in dom (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))
by TARSKI:def 1;
(
dom sdom = (dom s) /\ (dom (ProgramPart p)) & not
IC (SCM R) in dom (ProgramPart p) )
by COMPOS_1:12, RELAT_1:90;
then A10:
not
IC (SCM R) in dom sdom
by XBOOLE_0:def 4;
not
p is
NAT -defined
by A2, COMPOS_1:19;
then A11:
IC (Comput (ProgramPart (s +* p)),(s +* p),i) in dom (ProgramPart p)
by A1, Th27, FUNCT_4:26;
then A12:
IC (Comput (ProgramPart (s +* p)),(s +* p),i) in dom (IncAddr (ProgramPart p),k)
by AMISTD_2:def 15;
ProgramPart p c= p
by RELAT_1:88;
then
dom (ProgramPart p) c= dom p
by GRFUNC_1:8;
then
(IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k in dom (Relocated p,k)
by A11, AMISTD_2:71;
then A13:
(IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k in dom (ProgramPart (Relocated p,k))
by COMPOS_1:16;
A14:
(ProgramPart p) /. ii =
(ProgramPart p) . (IC (Comput (ProgramPart (s +* p)),(s +* p),i))
by A11, PARTFUN1:def 8
.=
(Comput (ProgramPart (s +* p)),(s +* p),i) . (IC (Comput (ProgramPart (s +* p)),(s +* p),i))
by A11, A8, GRFUNC_1:8
;
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;
Z:
(ProgramPart (Comput (ProgramPart (s +* p)),(s +* p),i)) /. (IC (Comput (ProgramPart (s +* p)),(s +* p),i)) = (Comput (ProgramPart (s +* p)),(s +* p),i) . (IC (Comput (ProgramPart (s +* p)),(s +* p),i))
by COMPOS_1:38;
not
IC (SCM R) in dom (ProgramPart (Relocated p,k))
by COMPOS_1:12;
then IC ((((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* sdom) +* (ProgramPart (Relocated p,k))) =
(((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* sdom) . (IC (SCM R))
by FUNCT_4:12
.=
((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) . (IC (SCM R))
by A10, FUNCT_4:12
.=
(Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R)) . (IC (SCM R))
by A9, FUNCT_4:14
.=
(IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k
by FUNCOP_1:87
;
then A15:
CurInstr (ProgramPart (Comput (ProgramPart s),s,i)),
(Comput (ProgramPart s),s,i) =
(ProgramPart (Relocated p,k)) . ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k)
by A6, A13, Y, FUNCT_4:14
.=
(IncAddr (Shift (ProgramPart p),k),k) . ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k)
by AMISTD_2:69
.=
(Shift (IncAddr (ProgramPart p),k),k) . ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k)
by AMISTD_2:75
.=
(IncAddr (ProgramPart p),k) . (IC (Comput (ProgramPart (s +* p)),(s +* p),i))
by A12, VALUED_1:def 12
.=
IncAddr (CurInstr (ProgramPart (Comput (ProgramPart (s +* p)),(s +* p),i)),(Comput (ProgramPart (s +* p)),(s +* p),i)),
k
by A11, A14, Z, AMISTD_2:def 15
;
T:
ProgramPart (s +* p) = ProgramPart (Comput (ProgramPart (s +* p)),(s +* p),i)
by AMI_1:123;
A16:
Comput (ProgramPart (s +* p)),
(s +* p),
(i + 1) = Following (ProgramPart (s +* p)),
(Comput (ProgramPart (s +* p)),(s +* p),i)
by AMI_1:14;
S:
ProgramPart s = ProgramPart (Comput (ProgramPart s),s,i)
by AMI_1:123;
thus Comput (ProgramPart s),
s,
(i + 1) =
Following (ProgramPart s),
(Comput (ProgramPart s),s,i)
by AMI_1:14
.=
(Exec (IncAddr (CurInstr (ProgramPart (Comput (ProgramPart (s +* p)),(s +* p),i)),(Comput (ProgramPart (s +* p)),(s +* p),i)),k),(((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* sdom)) +* (ProgramPart (Relocated p,k))
by A6, A15, S, AMISTD_2:67
.=
((Exec (IncAddr (CurInstr (ProgramPart (Comput (ProgramPart (s +* p)),(s +* p),i)),(Comput (ProgramPart (s +* p)),(s +* p),i)),k),((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R)))) +* sdom) +* (ProgramPart (Relocated p,k))
by AMISTD_2:67
.=
(((Comput (ProgramPart (s +* p)),(s +* p),(i + 1)) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),(i + 1))) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A16, Th22, T
;
verum
end;
set SD = s | (dom (ProgramPart p));
A17:
dom (DataPart p) misses dom (ProgramPart p)
by COMPOS_1:15;
set PR = ProgramPart (Relocated p,k);
set IP = Start-At (IC p),(SCM R);
A18:
s | (dom (ProgramPart p)) c= s
by RELAT_1:88;
set PP = ProgramPart p;
A19:
{(IC (SCM R))} misses dom (DataPart p)
by COMPOS_1:13;
ProgramPart (Relocated p,k) c= Relocated p,k
by RELAT_1:88;
then A20:
ProgramPart (Relocated p,k) c= s
by A3, XBOOLE_1:1;
dom (ProgramPart p) c= the carrier of (SCM R)
by RELAT_1:def 18;
then
dom (ProgramPart p) c= dom s
by PARTFUN1:def 4;
then A21:
dom (ProgramPart p) = dom (s | (dom (ProgramPart p)))
by RELAT_1:91;
DataPart (Relocated p,k) c= Relocated p,k
by RELAT_1:88;
then
DataPart (Relocated p,k) c= s
by A3, XBOOLE_1:1;
then A22:
DataPart p c= s
by AMISTD_2:68;
set DP = DataPart p;
set IS = Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R);
A23:
dom (Start-At (IC p),(SCM R)) = {(IC (SCM R))}
by FUNCOP_1:19;
IC (Comput (ProgramPart (s +* p)),(s +* p),0 ) =
(s +* p) . (IC (SCM R))
by AMI_1:13
.=
IC p
by A2, FUNCT_4:14
;
then A24:
Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R) c= s
by A3, A4, XBOOLE_1:1;
A25:
dom (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R)) = {(IC (SCM R))}
by FUNCOP_1:19;
then A26:
dom (Start-At (IC p),(SCM R)) = dom (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))
by FUNCOP_1:19;
Comput (ProgramPart s),s,0 =
s
by AMI_1:13
.=
s +* (ProgramPart (Relocated p,k))
by A20, FUNCT_4:79
.=
(s +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A18, FUNCT_4:79
.=
((s +* (ProgramPart p)) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A21, FUNCT_4:78
.=
(((s +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (ProgramPart p)) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A24, FUNCT_4:79
.=
((s +* ((Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R)) +* (ProgramPart p))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
((s +* ((ProgramPart p) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R)))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A25, COMPOS_1:14, FUNCT_4:36
.=
(((s +* (ProgramPart p)) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
((((s +* (DataPart p)) +* (ProgramPart p)) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A22, FUNCT_4:79
.=
(((s +* ((DataPart p) +* (ProgramPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((s +* ((ProgramPart p) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A17, FUNCT_4:36
.=
((((s +* (ProgramPart p)) +* (DataPart p)) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((((s +* (ProgramPart p)) +* (DataPart p)) +* (Start-At (IC p),(SCM R))) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A26, FUNCT_4:78
.=
((((s +* ((ProgramPart p) +* (DataPart p))) +* (Start-At (IC p),(SCM R))) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((s +* (((ProgramPart p) +* (DataPart p)) +* (Start-At (IC p),(SCM R)))) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((s +* ((ProgramPart p) +* ((DataPart p) +* (Start-At (IC p),(SCM R))))) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((s +* ((ProgramPart p) +* ((Start-At (IC p),(SCM R)) +* (DataPart p)))) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A23, A19, FUNCT_4:36
.=
(((s +* (((ProgramPart p) +* (Start-At (IC p),(SCM R))) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((s +* (((Start-At (IC p),(SCM R)) +* (ProgramPart p)) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A23, COMPOS_1:14, FUNCT_4:36
.=
(((s +* p) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A2, COMPOS_1:18
.=
(((Comput (ProgramPart (s +* p)),(s +* p),0 ) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),0 )) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by AMI_1:13
;
then A27:
S1[ 0 ]
;
for n being Element of NAT holds S1[n]
from NAT_1:sch 1(A27, A5);
hence
for i being Element of NAT holds Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* p)),(s +* p),i) +* (Start-At ((IC (Comput (ProgramPart (s +* p)),(s +* p),i)) + k),(SCM R))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
; verum