let k be Element of NAT ; for p being autonomic FinPartState of st IC SCM+FSA in dom p holds
for s being State of st Relocated p,k c= s holds
for i being Element of NAT holds Computation s,i = (((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
let p be autonomic FinPartState of ; ( IC SCM+FSA in dom p implies for s being State of st Relocated p,k c= s holds
for i being Element of NAT holds Computation s,i = (((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k)) )
assume A1:
IC SCM+FSA in dom p
; for s being State of st Relocated p,k c= s holds
for i being Element of NAT holds Computation s,i = (((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
let s be State of ; ( Relocated p,k c= s implies for i being Element of NAT holds Computation s,i = (((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k)) )
assume A2:
Relocated p,k c= s
; for i being Element of NAT holds Computation s,i = (((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
defpred S1[ Element of NAT ] means Computation s,$1 = (((Computation (s +* p),$1) +* (Start-At ((IC (Computation (s +* p),$1)) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k));
A3:
for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
A4:
not
IC SCM+FSA in dom (ProgramPart p)
by AMI_1:101;
reconsider sdom =
s | (dom (ProgramPart p)) as
Element of
sproduct the
Object-Kind of
SCM+FSA ;
let i be
Element of
NAT ;
( S1[i] implies S1[i + 1] )
assume A5:
Computation s,
i = (((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
;
S1[i + 1]
dom [(ProgramPart p)] c= the
carrier of
SCM+FSA
by AMI_1:80;
then
dom (ProgramPart p) c= dom s
by AMI_1:79;
then A6:
dom (ProgramPart p) = dom (s | (dom (ProgramPart p)))
by RELAT_1:91;
then reconsider sdom =
sdom as
FinPartState of
by FINSET_1:29;
dom (s | (dom [(ProgramPart p)])) c= NAT
by A6, RELAT_1:def 18;
then reconsider sdom =
sdom as
preProgram of
by RELAT_1:def 18;
reconsider ii =
IC (Computation (s +* p),i) as
Element of
NAT by ORDINAL1:def 13;
A7:
ProgramPart p c= Computation (s +* p),
i
by AMI_1:99, FUNCT_4:26;
dom (Start-At ((IC (Computation (s +* p),i)) + k)) = {(IC SCM+FSA )}
by FUNCOP_1:19;
then A8:
IC SCM+FSA in dom (Start-At ((IC (Computation (s +* p),i)) + k))
by TARSKI:def 1;
dom sdom = (dom s) /\ (dom (ProgramPart p))
by RELAT_1:90;
then A9:
not
IC SCM+FSA in dom sdom
by A4, XBOOLE_0:def 4;
not
p is
NAT -defined
by A1, AMI_1:109;
then A10:
IC (Computation (s +* p),i) in dom (ProgramPart p)
by FUNCT_4:26, SCMFSA_3:17;
then A11:
IC (Computation (s +* p),i) in dom (IncAddr [(ProgramPart p)],k)
by SCMFSA_4:def 6;
ProgramPart p c= p
by RELAT_1:88;
then
dom (ProgramPart p) c= dom p
by GRFUNC_1:8;
then
(IC (Computation (s +* p),i)) + k in dom (Relocated p,k)
by A10, Th4;
then A12:
(IC (Computation (s +* p),i)) + k in dom (ProgramPart (Relocated p,k))
by AMI_1:106;
A13:
pi [(ProgramPart p)],
ii =
[(ProgramPart p)] . (IC (Computation (s +* p),i))
by A10, AMI_1:def 47
.=
(Computation (s +* p),i) . (IC (Computation (s +* p),i))
by A10, A7, GRFUNC_1:8
;
not
IC SCM+FSA in dom (ProgramPart (Relocated p,k))
by AMI_1:101;
then IC ((((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k))) +* sdom) +* [(ProgramPart (Relocated p,k))]) =
(((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k))) +* sdom) . (IC SCM+FSA )
by FUNCT_4:12
.=
((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k))) . (IC SCM+FSA )
by A9, FUNCT_4:12
.=
(Start-At ((IC (Computation (s +* p),i)) + k)) . (IC SCM+FSA )
by A8, FUNCT_4:14
.=
(IC (Computation (s +* p),i)) + k
by FUNCOP_1:87
;
then A14:
CurInstr (Computation s,i) =
(ProgramPart (Relocated p,k)) . ((IC (Computation (s +* p),i)) + k)
by A5, A12, FUNCT_4:14
.=
(IncAddr [(Shift (ProgramPart p),k)],k) . ((IC (Computation (s +* p),i)) + k)
by Th2
.=
(Shift (IncAddr [(ProgramPart p)],k),k) . ((IC (Computation (s +* p),i)) + k)
by SCMFSA_4:35
.=
(IncAddr [(ProgramPart p)],k) . ii
by A11, VALUED_1:def 12
.=
IncAddr (CurInstr (Computation (s +* p),i)),
k
by A10, A13, SCMFSA_4:24
;
A15:
Computation (s +* p),
(i + 1) = Following (Computation (s +* p),i)
by AMI_1:14;
thus Computation s,
(i + 1) =
Following (Computation s,i)
by AMI_1:14
.=
(Exec (IncAddr (CurInstr (Computation (s +* p),i)),k),(((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k))) +* sdom)) +* [(ProgramPart (Relocated p,k))]
by A5, A14, SCMFSA_3:10
.=
((Exec (IncAddr (CurInstr (Computation (s +* p),i)),k),((Computation (s +* p),i) +* (Start-At ((IC (Computation (s +* p),i)) + k)))) +* sdom) +* (ProgramPart (Relocated p,k))
by SCMFSA_3:10
.=
(((Computation (s +* p),(i + 1)) +* (Start-At ((IC (Computation (s +* p),(i + 1))) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A15, SCMFSA_4:28
;
verum
end;
set IP = Start-At (IC p);
A16:
dom (Start-At (IC p)) = {(IC SCM+FSA )}
by FUNCOP_1:19;
A17:
Start-At ((IC p) + k) c= Relocated p,k
by Th8;
IC (Computation (s +* p),0 ) =
(s +* p) . (IC SCM+FSA )
by AMI_1:13
.=
p . (IC SCM+FSA )
by A1, FUNCT_4:14
.=
IC p
by A1, AMI_1:def 43
;
then A18:
Start-At ((IC (Computation (s +* p),0 )) + k) c= s
by A2, A17, XBOOLE_1:1;
set DP = DataPart p;
A19:
{(IC SCM+FSA )} misses dom (DataPart p)
by AMI_1:102;
set PP = ProgramPart p;
A20:
dom (DataPart p) misses dom (ProgramPart p)
by AMI_1:104;
set SD = s | (dom (ProgramPart p));
A21:
s | (dom (ProgramPart p)) c= s
by RELAT_1:88;
ProgramPart (Relocated p,k) c= Relocated p,k
by RELAT_1:88;
then A22:
ProgramPart (Relocated p,k) c= s
by A2, XBOOLE_1:1;
dom [(ProgramPart p)] c= the carrier of SCM+FSA
by AMI_1:80;
then
dom (ProgramPart p) c= dom s
by AMI_1:79;
then A23:
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 A2, XBOOLE_1:1;
then A24:
DataPart p c= s
by Th1;
set PR = ProgramPart (Relocated p,k);
set IS = Start-At ((IC (Computation (s +* p),0 )) + k);
A25:
dom (Start-At ((IC (Computation (s +* p),0 )) + k)) = {(IC SCM+FSA )}
by FUNCOP_1:19;
Computation s,0 =
s
by AMI_1:13
.=
s +* (ProgramPart (Relocated p,k))
by A22, FUNCT_4:79
.=
(s +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A21, FUNCT_4:79
.=
((s +* (ProgramPart p)) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A23, FUNCT_4:78
.=
(((s +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (ProgramPart p)) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A18, FUNCT_4:79
.=
((s +* ((Start-At ((IC (Computation (s +* p),0 )) + k)) +* (ProgramPart p))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
((s +* ((ProgramPart p) +* (Start-At ((IC (Computation (s +* p),0 )) + k)))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A25, AMI_1:103, FUNCT_4:36
.=
(((s +* (ProgramPart p)) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
((((s +* (DataPart p)) +* (ProgramPart p)) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A24, FUNCT_4:79
.=
(((s +* ((DataPart p) +* (ProgramPart p))) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((s +* ((ProgramPart p) +* (DataPart p))) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A20, FUNCT_4:36
.=
((((s +* (ProgramPart p)) +* (DataPart p)) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((((s +* (ProgramPart p)) +* (DataPart p)) +* (Start-At (IC p))) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A25, A16, FUNCT_4:78
.=
((((s +* ((ProgramPart p) +* (DataPart p))) +* (Start-At (IC p))) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((s +* (((ProgramPart p) +* (DataPart p)) +* (Start-At (IC p)))) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((s +* ((ProgramPart p) +* ((DataPart p) +* (Start-At (IC p))))) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((s +* ((ProgramPart p) +* ((Start-At (IC p)) +* (DataPart p)))) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A16, A19, FUNCT_4:36
.=
(((s +* (((ProgramPart p) +* (Start-At (IC p))) +* (DataPart p))) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by FUNCT_4:15
.=
(((s +* (((Start-At (IC p)) +* (ProgramPart p)) +* (DataPart p))) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A16, AMI_1:103, FUNCT_4:36
.=
(((s +* p) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by A1, AMI_1:108
.=
(((Computation (s +* p),0 ) +* (Start-At ((IC (Computation (s +* p),0 )) + k))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated p,k))
by AMI_1:13
;
then A26:
S1[ 0 ]
;
thus
for i being Element of NAT holds S1[i]
from NAT_1:sch 1(A26, A3); verum