let k be Element of NAT ; :: thesis: for p being autonomic FinPartState of st IC SCM 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 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 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))
dom (DataPart p) misses {(IC SCM )} by AMI_1:100, ZFMISC_1:56;
then (dom (DataPart p)) /\ {(IC SCM )} = {} by XBOOLE_0:def 7;
then A2: (dom (DataPart p)) /\ (dom (Start-At ((IC p) + k))) = {} by FUNCOP_1:19;
NAT misses dom (DataPart p) by AMI_2:29, AMI_3:72, RELAT_1:87, XBOOLE_1:63;
then dom (DataPart p) misses dom (ProgramPart (Relocated p,k)) by RELAT_1:87, XBOOLE_1:63;
then ((dom (DataPart p)) /\ (dom (Start-At ((IC p) + k)))) \/ ((dom (DataPart p)) /\ (dom (ProgramPart (Relocated p,k)))) = {} by A2, 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)) /\ (dom ((Start-At ((IC p) + k)) +* (ProgramPart (Relocated p,k)))) = {} by FUNCT_4:def 1;
then dom (DataPart p) misses dom ((Start-At ((IC p) + k)) +* (ProgramPart (Relocated p,k))) by XBOOLE_0:def 7;
then A3: ((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 A4: 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));
A5: 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 A6: 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 kk = IC (Computation s,i) as Element of NAT by ORDINAL1:def 13;
dom (Start-At ((IC (Computation s,i)) + k)) = {(IC SCM )} by FUNCOP_1:19;
then A7: IC SCM in dom (Start-At ((IC (Computation s,i)) + k)) by TARSKI:def 1;
not IC SCM in dom (ProgramPart (Relocated p,k)) by AMI_1:101;
then A8: 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 ) by FUNCT_4:12
.= (Start-At ((IC (Computation s,i)) + k)) . (IC SCM ) by A7, FUNCT_4:14
.= (IC (Computation s,i)) + k by FUNCOP_1:87 ;
A9: ProgramPart p c= Computation s,i by A4, AMI_1:99;
not p is NAT -defined by A1, AMI_1:109;
then A10: IC (Computation s,i) in dom (ProgramPart p) by A4, AMI_5:86;
then A11: IC (Computation s,i) in dom (IncAddr (ProgramPart p),k) by Def5;
A12: (ProgramPart p) /. kk = (ProgramPart p) . (IC (Computation s,i)) by A10, PARTFUN1:def 8
.= (Computation s,i) . (IC (Computation s,i)) by A10, A9, GRFUNC_1:8 ;
reconsider kk = IC (Computation s,i) as Element of NAT by ORDINAL1:def 13;
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 A10, Th24;
then (IC (Computation s,i)) + k in dom (ProgramPart (Relocated p,k)) by AMI_1:106;
then A13: CurInstr (Computation (s +* (Relocated p,k)),i) = (ProgramPart (Relocated p,k)) . ((IC (Computation s,i)) + k) by A6, A8, FUNCT_4:14
.= (IncAddr (Shift (ProgramPart p),k),k) . ((IC (Computation s,i)) + k) by Th22
.= (Shift (IncAddr (ProgramPart p),k),k) . ((IC (Computation s,i)) + k) by Th19
.= (IncAddr (ProgramPart p),k) . kk by A11, VALUED_1:def 12
.= IncAddr (CurInstr (Computation s,i)),k by A10, A12, Def5 ;
A14: ( Computation s,(i + 1) = Following (Computation s,i) & 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 Th31, AMI_1:14;
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 A6, A13, A14, AMI_5:77
.= ((Computation s,(i + 1)) +* (Start-At ((IC (Computation s,(i + 1))) + k))) +* (ProgramPart (Relocated p,k)) ; :: thesis: verum

end;

A15: Computation s,0 = s by AMI_1:13;
A16: IC p = p . (IC SCM ) by A1, AMI_1:def 43
.= IC s by A1, A4, GRFUNC_1:8 ;
DataPart p c= p by RELAT_1:88;
then A17: DataPart p c= s by A4, 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 Th22
.= (s +* (DataPart p)) +* ((Start-At ((IC p) + k)) +* (ProgramPart (Relocated p,k))) by A3, 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 A16, A17, A15, FUNCT_4:79 ;
then A18: S₁[ 0 ] ;
thus for i being Element of NAT holds S₁[i] from NAT_1:sch 1(A18, A5); :: thesis: verum