let k be Element of NAT ; for p being autonomic FinPartState of SCM+FSA st IC SCM+FSA in dom p holds
for s being State of SCM+FSA st 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+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
let p be autonomic FinPartState of SCM+FSA; ( IC SCM+FSA in dom p implies for s being State of SCM+FSA st 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+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k))) )
assume A1:
IC SCM+FSA in dom p
; for s being State of SCM+FSA st 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+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
let s be State of SCM+FSA; ( 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+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k))) )
assume A2:
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+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
defpred S1[ 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+FSA))) +* (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 COMPOS_1:12;
set sdom =
s | (dom (ProgramPart p));
let i be
Element of
NAT ;
( S1[i] implies S1[i + 1] )
assume A5:
Comput (
(ProgramPart s),
s,
i)
= (((Comput ((ProgramPart (s +* p)),(s +* p),i)) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),i))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
;
S1[i + 1]
dom (ProgramPart p) c= the
carrier of
SCM+FSA
by RELAT_1:def 18;
then
dom (ProgramPart p) c= dom s
by PARTFUN1:def 4;
then A6:
dom (ProgramPart p) = dom (s | (dom (ProgramPart p)))
by RELAT_1:91;
reconsider sdom =
s | (dom (ProgramPart p)) as
FinPartState of
SCM+FSA ;
X:
rng (ProgramPart s) c= the
Instructions of
SCM+FSA
by RELAT_1:def 19;
dom (ProgramPart p) c= NAT
by RELAT_1:def 18;
then
s | (dom (ProgramPart p)) c= ProgramPart s
by RELAT_1:104;
then
rng (s | (dom (ProgramPart p))) c= rng (ProgramPart s)
by RELAT_1:25;
then B:
rng (s | (dom (ProgramPart p))) c= the
Instructions of
SCM+FSA
by X, XBOOLE_1:1;
dom (s | (dom (ProgramPart p))) c= NAT
by A6, RELAT_1:def 18;
then reconsider sdom =
sdom as
preProgram of
SCM+FSA by B, RELAT_1:def 18, RELAT_1:def 19;
reconsider ii =
IC (Comput ((ProgramPart (s +* p)),(s +* p),i)) as
Element of
NAT ;
A7:
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+FSA)) = {(IC SCM+FSA)}
by FUNCOP_1:19;
then A8:
IC SCM+FSA in dom (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),i))) + k),SCM+FSA))
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, COMPOS_1:19;
then A10:
IC (Comput ((ProgramPart (s +* p)),(s +* p),i)) in dom (ProgramPart p)
by FUNCT_4:26, SCMFSA_3:17;
then A11:
IC (Comput ((ProgramPart (s +* p)),(s +* p),i)) in dom (IncAddr ((ProgramPart p),k))
by COMPOS_1:def 40;
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 A10, COMPOS_1:118;
then A12:
(IC (Comput ((ProgramPart (s +* p)),(s +* p),i))) + k in dom (ProgramPart (Relocated (p,k)))
by COMPOS_1:16;
A13:
(ProgramPart p) /. ii =
(ProgramPart p) . (IC (Comput ((ProgramPart (s +* p)),(s +* p),i)))
by A10, PARTFUN1:def 8
.=
(Comput ((ProgramPart (s +* p)),(s +* p),i)) . (IC (Comput ((ProgramPart (s +* p)),(s +* p),i)))
by A10, A7, GRFUNC_1:8
;
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;
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;
not
IC SCM+FSA 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+FSA))) +* sdom) +* (ProgramPart (Relocated (p,k)))) =
(((Comput ((ProgramPart (s +* p)),(s +* p),i)) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),i))) + k),SCM+FSA))) +* sdom) . (IC SCM+FSA)
by FUNCT_4:12
.=
((Comput ((ProgramPart (s +* p)),(s +* p),i)) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),i))) + k),SCM+FSA))) . (IC SCM+FSA)
by A9, FUNCT_4:12
.=
(Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),i))) + k),SCM+FSA)) . (IC SCM+FSA)
by A8, FUNCT_4:14
.=
(IC (Comput ((ProgramPart (s +* p)),(s +* p),i))) + k
by FUNCOP_1:87
;
then A14:
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 A5, A12, Y, FUNCT_4:14
.=
(Reloc ((ProgramPart p),k)) . ((IC (Comput ((ProgramPart (s +* p)),(s +* p),i))) + k)
by COMPOS_1:116
.=
(Shift ((IncAddr ((ProgramPart p),k)),k)) . ((IC (Comput ((ProgramPart (s +* p)),(s +* p),i))) + k)
by COMPOS_1:121
.=
(IncAddr ((ProgramPart p),k)) . ii
by A11, VALUED_1:def 12
.=
IncAddr (
(CurInstr ((ProgramPart (Comput ((ProgramPart (s +* p)),(s +* p),i))),(Comput ((ProgramPart (s +* p)),(s +* p),i)))),
k)
by A10, A13, Z, SCMFSA_4:24
;
T:
ProgramPart (s +* p) = ProgramPart (Comput ((ProgramPart (s +* p)),(s +* p),i))
by AMI_1:123;
A15:
Comput (
(ProgramPart (s +* p)),
(s +* p),
(i + 1))
= Following (
(ProgramPart (s +* p)),
(Comput ((ProgramPart (s +* p)),(s +* p),i)))
by EXTPRO_1:4;
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 EXTPRO_1:4
.=
(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+FSA))) +* sdom))) +* (ProgramPart (Relocated (p,k)))
by A5, A14, S, AMI_1:127
.=
((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+FSA))))) +* sdom) +* (ProgramPart (Relocated (p,k)))
by AMI_1:127
.=
(((Comput ((ProgramPart (s +* p)),(s +* p),(i + 1))) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),(i + 1)))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by A15, T, SCMFSA_4:28
;
verum
end;
set IP = Start-At ((IC p),SCM+FSA);
A16:
dom (Start-At ((IC p),SCM+FSA)) = {(IC SCM+FSA)}
by FUNCOP_1:19;
A17:
Start-At (((IC p) + k),SCM+FSA) c= Relocated (p,k)
by A1, COMPOS_1:123;
IC (Comput ((ProgramPart (s +* p)),(s +* p),0)) =
(s +* p) . (IC SCM+FSA)
by EXTPRO_1:3
.=
IC p
by A1, FUNCT_4:14
;
then A18:
Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA) c= s
by A2, A17, XBOOLE_1:1;
set DP = DataPart p;
A19:
{(IC SCM+FSA)} misses dom (DataPart p)
by COMPOS_1:13;
set PP = ProgramPart p;
A20:
dom (DataPart p) misses dom (ProgramPart p)
by COMPOS_1:15;
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 RELAT_1:def 18;
then
dom (ProgramPart p) c= dom s
by PARTFUN1:def 4;
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 COMPOS_1:115;
set PR = ProgramPart (Relocated (p,k));
set IS = Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA);
A25:
dom (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA)) = {(IC SCM+FSA)}
by FUNCOP_1:19;
Comput ((ProgramPart s),s,0) =
s
by EXTPRO_1:3
.=
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 (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (ProgramPart p)) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by A18, FUNCT_4:79
.=
((s +* ((Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA)) +* (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+FSA)))) +* (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+FSA))) +* (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+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by A24, FUNCT_4:79
.=
(((s +* ((DataPart p) +* (ProgramPart p))) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (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+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by A20, FUNCT_4:36
.=
((((s +* (ProgramPart p)) +* (DataPart p)) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by FUNCT_4:15
.=
(((((s +* (ProgramPart p)) +* (DataPart p)) +* (Start-At ((IC p),SCM+FSA))) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by A25, A16, FUNCT_4:78
.=
((((s +* ((ProgramPart p) +* (DataPart p))) +* (Start-At ((IC p),SCM+FSA))) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by FUNCT_4:15
.=
(((s +* (((ProgramPart p) +* (DataPart p)) +* (Start-At ((IC p),SCM+FSA)))) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by FUNCT_4:15
.=
(((s +* ((ProgramPart p) +* ((DataPart p) +* (Start-At ((IC p),SCM+FSA))))) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by FUNCT_4:15
.=
(((s +* ((ProgramPart p) +* ((Start-At ((IC p),SCM+FSA)) +* (DataPart p)))) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by A16, A19, FUNCT_4:36
.=
(((s +* (((ProgramPart p) +* (Start-At ((IC p),SCM+FSA))) +* (DataPart p))) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by FUNCT_4:15
.=
(((s +* (((Start-At ((IC p),SCM+FSA)) +* (ProgramPart p)) +* (DataPart p))) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by A16, COMPOS_1:14, FUNCT_4:36
.=
(((s +* p) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by A1, COMPOS_1:18
.=
(((Comput ((ProgramPart (s +* p)),(s +* p),0)) +* (Start-At (((IC (Comput ((ProgramPart (s +* p)),(s +* p),0))) + k),SCM+FSA))) +* (s | (dom (ProgramPart p)))) +* (ProgramPart (Relocated (p,k)))
by EXTPRO_1:3
;
then A26:
S1[ 0 ]
;
thus
for i being Element of NAT holds S1[i]
from NAT_1:sch 1(A26, A3); verum