let k be Element of NAT ; :: thesis: for p being autonomic FinPartState of st IC SCM+FSA in dom p holds
for s being State of st p c= s holds
for i being Element of NAT holds Computation (s +* (Relocated p,k)),i = ((Computation s,i) +* (Start-At ((IC (Computation s,i)) + k))) +* (ProgramPart (Relocated p,k))
let p be autonomic FinPartState of ; :: thesis: ( IC SCM+FSA in dom p implies for s being State of st p c= s holds
for i being Element of NAT holds Computation (s +* (Relocated p,k)),i = ((Computation s,i) +* (Start-At ((IC (Computation s,i)) + k))) +* (ProgramPart (Relocated p,k)) )
assume A1: IC SCM+FSA in dom p ; :: thesis: for s being State of st p c= s holds
for i being Element of NAT holds Computation (s +* (Relocated p,k)),i = ((Computation s,i) +* (Start-At ((IC (Computation s,i)) + k))) +* (ProgramPart (Relocated p,k))
A2: dom (ProgramPart (Relocated p,k)) c= NAT by RELAT_1:def 18;
NAT misses dom (DataPart p) by Lm1, RELAT_1:87, SCMFSA_2:127, XBOOLE_1:63;
then dom (DataPart p) misses dom (ProgramPart (Relocated p,k)) by A2, XBOOLE_1:63;
then A3: (dom (DataPart p)) /\ (dom (ProgramPart (Relocated p,k))) = {} by XBOOLE_0:def 7;
dom (DataPart p) misses {(IC SCM+FSA )} by AMI_1:100, ZFMISC_1:56;
then dom (DataPart p) misses dom (Start-At ((IC p) + k)) by FUNCOP_1:19;
then ((dom (DataPart p)) /\ (dom (Start-At ((IC p) + k)))) \/ ((dom (DataPart p)) /\ (dom (ProgramPart (Relocated p,k)))) = {} by A3, XBOOLE_0:def 7;
then (dom (DataPart p)) /\ ((dom (Start-At ((IC p) + k))) \/ (dom (ProgramPart (Relocated p,k)))) = {} by XBOOLE_1:23;
then dom (DataPart p) misses (dom (Start-At ((IC p) + k))) \/ (dom (ProgramPart (Relocated p,k))) by XBOOLE_0:def 7;
then dom (DataPart p) misses dom ((Start-At ((IC p) + k)) +* (ProgramPart (Relocated p,k))) by FUNCT_4:def 1;
then A4: ((Start-At ((IC p) + k)) +* (ProgramPart (Relocated p,k))) +* (DataPart p) = (DataPart p) +* ((Start-At ((IC p) + k)) +* (ProgramPart (Relocated p,k))) by FUNCT_4:36;
let s be State of ; :: thesis: ( p c= s implies for i being Element of NAT holds Computation (s +* (Relocated p,k)),i = ((Computation s,i) +* (Start-At ((IC (Computation s,i)) + k))) +* (ProgramPart (Relocated p,k)) )
assume A5: p c= s ; :: thesis: for i being Element of NAT holds Computation (s +* (Relocated p,k)),i = ((Computation s,i) +* (Start-At ((IC (Computation s,i)) + k))) +* (ProgramPart (Relocated p,k))
defpred S₁[ Element of NAT ] means Computation (s +* (Relocated p,k)),$1 = ((Computation s,$1) +* (Start-At ((IC (Computation s,$1)) + k))) +* (ProgramPart (Relocated p,k));
A6: for i being Element of NAT st S₁[i] holds
S₁[i + 1]

proof

let i be Element of NAT ; :: thesis: ( S₁[i] implies S₁[i + 1] )
assume A7: Computation (s +* (Relocated p,k)),i = ((Computation s,i) +* (Start-At ((IC (Computation s,i)) + k))) +* (ProgramPart (Relocated p,k)) ; :: thesis: S₁[i + 1]
reconsider ii = IC (Computation s,i) as Element of NAT by ORDINAL1:def 13;
dom (Start-At ((IC (Computation s,i)) + k)) = {(IC SCM+FSA )} by FUNCOP_1:19;
then A8: IC SCM+FSA in dom (Start-At ((IC (Computation s,i)) + k)) by TARSKI:def 1;
not IC SCM+FSA in dom (ProgramPart (Relocated p,k)) by AMI_1:101;
then A9: IC (((Computation s,i) +* (Start-At ((IC (Computation s,i)) + k))) +* [(ProgramPart (Relocated p,k))]) = ((Computation s,i) +* (Start-At ((IC (Computation s,i)) + k))) . (IC SCM+FSA ) by FUNCT_4:12
.= (Start-At ((IC (Computation s,i)) + k)) . (IC SCM+FSA ) by A8, FUNCT_4:14
.= (IC (Computation s,i)) + k by FUNCOP_1:87 ;
A10: ProgramPart p c= Computation s,i by A5, AMI_1:99;
not p is NAT -defined by A1, AMI_1:109;
then A11: IC (Computation s,i) in dom [(ProgramPart p)] by A5, SCMFSA_3:17;
then A12: IC (Computation s,i) in dom (IncAddr [(ProgramPart p)],k) by SCMFSA_4:def 6;
A13: pi [(ProgramPart p)],ii = [(ProgramPart p)] . (IC (Computation s,i)) by A11, AMI_1:def 47
.= (Computation s,i) . (IC (Computation s,i)) by A11, A10, GRFUNC_1:8 ;
ProgramPart p c= p by RELAT_1:88;
then dom (ProgramPart p) c= dom p by GRFUNC_1:8;
then (IC (Computation s,i)) + k in dom (Relocated p,k) by A11, Th4;
then (IC (Computation s,i)) + k in dom (ProgramPart (Relocated p,k)) by AMI_1:106;
then A14: CurInstr (Computation (s +* (Relocated p,k)),i) = (ProgramPart (Relocated p,k)) . ((IC (Computation s,i)) + k) by A7, A9, FUNCT_4:14
.= (IncAddr [(Shift (ProgramPart p),k)],k) . ((IC (Computation s,i)) + k) by Th2
.= (Shift (IncAddr [(ProgramPart p)],k),k) . ((IC (Computation s,i)) + k) by SCMFSA_4:35
.= (IncAddr [(ProgramPart p)],k) . ii by A12, VALUED_1:def 12
.= IncAddr (CurInstr (Computation s,i)),k by A11, A13, SCMFSA_4:24 ;
A15: Computation s,(i + 1) = Following (Computation s,i) by AMI_1:14;
A16: Exec (IncAddr (CurInstr (Computation s,i)),k),((Computation s,i) +* (Start-At ((IC (Computation s,i)) + k))) = (Following (Computation s,i)) +* (Start-At ((IC (Following (Computation s,i))) + k)) by SCMFSA_4:28;
thus Computation (s +* (Relocated p,k)),(i + 1) = Following (Computation (s +* (Relocated p,k)),i) by AMI_1:14
.= ((Computation s,(i + 1)) +* (Start-At ((IC (Computation s,(i + 1))) + k))) +* [(ProgramPart (Relocated p,k))] by A7, A15, A14, A16, SCMFSA_3:10
.= ((Computation s,(i + 1)) +* (Start-At ((IC (Computation s,(i + 1))) + k))) +* (ProgramPart (Relocated p,k)) ; :: thesis: verum

end;

A17: Computation s,0 = s by AMI_1:13;
A18: IC p = p . (IC SCM+FSA ) by A1, AMI_1:def 43
.= IC s by A1, A5, GRFUNC_1:8 ;
DataPart p c= p by RELAT_1:88;
then A19: DataPart p c= s by A5, XBOOLE_1:1;
Computation (s +* (Relocated p,k)),0 = s +* (((Start-At ((IC p) + k)) +* (IncAddr [(Shift (ProgramPart p),k)],k)) +* (DataPart p)) by AMI_1:13
.= s +* (((Start-At ((IC p) + k)) +* (ProgramPart (Relocated p,k))) +* (DataPart p)) by Th2
.= (s +* (DataPart p)) +* ((Start-At ((IC p) + k)) +* (ProgramPart (Relocated p,k))) by A4, FUNCT_4:15
.= ((s +* (DataPart p)) +* (Start-At ((IC p) + k))) +* (ProgramPart (Relocated p,k)) by FUNCT_4:15
.= ((Computation s,0 ) +* (Start-At ((IC (Computation s,0 )) + k))) +* (ProgramPart (Relocated p,k)) by A18, A19, A17, FUNCT_4:79 ;
then A20: S₁[ 0 ] ;
thus for i being Element of NAT holds S₁[i] from NAT_1:sch 1(A20, A6); :: thesis: verum