let k be Element of NAT ; :: thesis:
let p be FinPartState of SCM ; :: thesis:
assume A0: IC SCM in dom p ; :: thesis:
then A1: Start-At (IC p),SCM c= p by FUNCOP_1:99;
let s be State of SCM ; :: thesis:
assume that
A2: p c= s and
A3: Relocated p,k is autonomic ; :: thesis:
A4: Start-At (IC p),SCM c= s by A2, A1, XBOOLE_1:1;
defpred S₁[ Element of NAT ] means Comput (ProgramPart s),s,$1 = (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),$1) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),$1)) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p);
A5: for i being Element of NAT st S₁[i] holds
S₁[i + 1]

reconsider pp = ProgramPart p as preProgram of SCM ;
set sdom = s | (dom (ProgramPart (Relocated p,k)));
dom (ProgramPart (Relocated p,k)) c= the carrier of SCM by RELAT_1:def 18;
then dom (ProgramPart (Relocated p,k)) c= dom s by PARTFUN1:def 4;
then dom (ProgramPart (Relocated p,k)) = dom (s | (dom (ProgramPart (Relocated p,k)))) by RELAT_1:91;
then ( rng (s | (dom (ProgramPart (Relocated p,k)))) c= the Instructions of SCM & dom (s | (dom (ProgramPart (Relocated p,k)))) c= NAT ) by COMPOS_1:32, RELAT_1:87;
then reconsider sdom = s | (dom (ProgramPart (Relocated p,k))) as NAT -defined finite FinPartState of by RELSET_1:11;
let i be Element of NAT ; :: thesis:
assume A7: Comput (ProgramPart s),s,i = (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) ; :: thesis:
reconsider kk = IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) as Element of NAT ;
reconsider jk = IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) as Element of NAT ;
A8: ProgramPart (Relocated p,k) c= Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i by AMI_1:99, FUNCT_4:26;
IC SCM in dom (Relocated p,k) by AMISTD_2:72;
then not Relocated p,k is NAT -defined by COMPOS_1:19;
then A9: IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) in dom (ProgramPart (Relocated p,k)) by A3, AMI_5:86, FUNCT_4:26;
then IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) in dom (Reloc (ProgramPart p),k) by AMISTD_2:69;
then A10: jk in { (j + k) where j is Element of NAT : j in dom (ProgramPart p) } by AMISTD_2:70;
( dom sdom = (dom s) /\ (dom (ProgramPart (Relocated p,k))) & not IC SCM in dom (ProgramPart (Relocated p,k)) ) by COMPOS_1:12, RELAT_1:90;
then A11: not IC SCM in dom sdom by XBOOLE_0:def 4;
dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM ) = {(IC SCM )} by FUNCOP_1:19;
then A12: IC SCM in dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM ) by TARSKI:def 1;
consider j being Element of NAT such that
A13: jk = j + k and
A14: j in dom (ProgramPart p) by A10;
Y: (ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) /. (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) = (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) by COMPOS_1:38;
T: ProgramPart (s +* (Relocated p,k)) = ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) by AMI_1:123;
dom (Shift pp,k) = { (m + k) where m is Element of NAT : m in dom pp } by VALUED_1:def 12;
then A15: j + k in dom (Shift (ProgramPart p),k) by A14;
then A16: IncAddr ((Shift (ProgramPart p),k) /. kk),k = (Reloc (ProgramPart p),k) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) by A13, AMISTD_2:def 15
.= (ProgramPart (Relocated p,k)) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) by AMISTD_2:69
.= CurInstr (ProgramPart (s +* (Relocated p,k))),(Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) by A9, A8, Y, T, GRFUNC_1:8 ;
A17: (j + k) -' k = j by NAT_D:34;
Z: (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 in dom (ProgramPart p) by COMPOS_1:12;
then IC ((((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* sdom) +* (ProgramPart p)) = (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* sdom) . (IC SCM ) by FUNCT_4:12
.= ((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) . (IC SCM ) by A11, FUNCT_4:12
.= (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM ) . (IC SCM ) by A12, FUNCT_4:14
.= (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k by FUNCOP_1:87 ;
then CurInstr (ProgramPart (Comput (ProgramPart s),s,i)),(Comput (ProgramPart s),s,i) = (ProgramPart p) . ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k) by A7, A13, A14, A17, Z, FUNCT_4:14
.= (Shift (ProgramPart p),k) . (IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) by A13, A14, A17, VALUED_1:def 12
.= (Shift (ProgramPart p),k) /. kk by A13, A15, PARTFUN1:def 8 ;
then A18: ( Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),(i + 1) = Following (ProgramPart (s +* (Relocated p,k))),(Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) & Exec (CurInstr (ProgramPart (Comput (ProgramPart s),s,i)),(Comput (ProgramPart s),s,i)),((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) = (Following (ProgramPart (s +* (Relocated p,k))),(Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) +* (Start-At ((IC (Following (ProgramPart (s +* (Relocated p,k))),(Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i))) -' k),SCM ) ) by A13, A16, Th32, 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 (CurInstr (ProgramPart s),(Comput (ProgramPart s),s,i)),(((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)) -' k),SCM )) +* sdom)) +* (ProgramPart p) by A7, AMI_5:77
.= (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),(i + 1)) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),(i + 1))) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by A18, S, AMI_5:77 ; :: thesis:

end;

A19: IC SCM in dom (Relocated p,k) by AMISTD_2:72;
A20: IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 ) = (s +* (Relocated p,k)) . (IC SCM ) by AMI_1:13
.= (Relocated p,k) . (IC SCM ) by A19, FUNCT_4:14
.= IC (Relocated p,k) ;
DataPart p c= p by RELAT_1:88;
then A21: DataPart p c= s by A2, XBOOLE_1:1;
ProgramPart p c= p by RELAT_1:88;
then A22: ProgramPart p c= s by A2, XBOOLE_1:1;
set DP = DataPart p;
set IP = Start-At ((IC p) + k),SCM ;
A23: dom (DataPart p) misses dom (ProgramPart (Relocated p,k)) by COMPOS_1:15;
set PP = ProgramPart p;
set IS = Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM ;
A24: dom (Start-At ((IC p) + k),SCM ) = {(IC SCM )} by FUNCOP_1:19;
set PR = ProgramPart (Relocated p,k);
set SD = s | (dom (ProgramPart (Relocated p,k)));
A25: {(IC SCM )} misses dom (DataPart p) by COMPOS_1:13;
dom (ProgramPart (Relocated p,k)) c= the carrier of SCM by RELAT_1:def 18;
then dom (ProgramPart (Relocated p,k)) c= dom s by PARTFUN1:def 4;
then A26: dom (ProgramPart (Relocated p,k)) = dom (s | (dom (ProgramPart (Relocated p,k)))) by RELAT_1:91;
{(IC SCM )} misses dom (ProgramPart (Relocated p,k)) by COMPOS_1:14;
then A27: {(IC SCM )} /\ (dom (ProgramPart (Relocated p,k))) = {} by XBOOLE_0:def 7;
A28: dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM ) = {(IC SCM )} by FUNCOP_1:19;
then (dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) /\ (dom (s | (dom (ProgramPart (Relocated p,k))))) = {(IC SCM )} /\ ((dom s) /\ (dom (ProgramPart (Relocated p,k)))) by RELAT_1:90
.= ({(IC SCM )} /\ (dom (ProgramPart (Relocated p,k)))) /\ (dom s) by XBOOLE_1:16
.= {} by A27 ;
then A29: dom (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM ) misses dom (s | (dom (ProgramPart (Relocated p,k)))) by XBOOLE_0:def 7;
T2: dom ((Start-At ((IC p) + k),SCM ) +* (Reloc (ProgramPart p),k)) = (dom (Start-At ((IC p) + k),SCM )) \/ (dom (Reloc (ProgramPart p),k)) by FUNCT_4:def 1;
X: NAT misses Data-Locations SCM by AMI_2:29, AMI_3:72;
dom (Reloc (ProgramPart p),k) c= NAT by RELAT_1:def 18;
then T3: dom (Reloc (ProgramPart p),k) misses Data-Locations SCM by X, XBOOLE_1:63;
dom (Start-At ((IC p) + k),SCM ) = {(IC SCM )} by FUNCOP_1:19;
then T1: dom (Start-At ((IC p) + k),SCM ) misses dom (DataPart p) by A25;
dom (DataPart p) c= Data-Locations SCM by RELAT_1:87;
then dom (Reloc (ProgramPart p),k) misses dom (DataPart p) by T3, XBOOLE_1:63;
then T: dom ((Start-At ((IC p) + k),SCM ) +* (Reloc (ProgramPart p),k)) misses dom (DataPart p) by T1, T2, XBOOLE_1:70;
RR: IC SCM in dom p by A0;
X: ((Start-At ((IC p) + k),SCM ) +* (ProgramPart (Relocated p,k))) +* (DataPart p) = ((Start-At ((IC p) + k),SCM ) +* (Reloc (ProgramPart p),k)) +* (DataPart p) by AMISTD_2:69
.= (DataPart p) +* ((Start-At ((IC p) + k),SCM ) +* (Reloc (ProgramPart p),k)) by T, FUNCT_4:36
.= ((DataPart p) +* (Start-At ((IC p) + k),SCM )) +* (Reloc (ProgramPart p),k) by FUNCT_4:15
.= Relocated p,k by RR, COMPOS_1:75 ;
Comput (ProgramPart s),s,0 = s by AMI_1:13
.= s +* (ProgramPart p) by A22, FUNCT_4:79
.= (s +* (Start-At (IC p),SCM )) +* (ProgramPart p) by A4, FUNCT_4:79
.= (s +* (Start-At (((IC p) + k) -' k),SCM )) +* (ProgramPart p) by NAT_D:34
.= (s +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (ProgramPart p) by A20, A0, Th26
.= ((s +* (s | (dom (ProgramPart (Relocated p,k))))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (ProgramPart p) by FUNCT_4:80
.= (((s +* (ProgramPart (Relocated p,k))) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (ProgramPart p) by A26, FUNCT_4:78
.= ((s +* (ProgramPart (Relocated p,k))) +* ((s | (dom (ProgramPart (Relocated p,k)))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM ))) +* (ProgramPart p) by FUNCT_4:15
.= ((s +* (ProgramPart (Relocated p,k))) +* ((Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM ) +* (s | (dom (ProgramPart (Relocated p,k)))))) +* (ProgramPart p) by A29, FUNCT_4:36
.= (((s +* (ProgramPart (Relocated p,k))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by FUNCT_4:15
.= ((((s +* (DataPart p)) +* (ProgramPart (Relocated p,k))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by A21, FUNCT_4:79
.= (((s +* ((DataPart p) +* (ProgramPart (Relocated p,k)))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by FUNCT_4:15
.= (((s +* ((ProgramPart (Relocated p,k)) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by A23, FUNCT_4:36
.= ((((s +* (ProgramPart (Relocated p,k))) +* (DataPart p)) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by FUNCT_4:15
.= (((((s +* (ProgramPart (Relocated p,k))) +* (DataPart p)) +* (Start-At ((IC p) + k),SCM )) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by A28, A24, FUNCT_4:78
.= ((((s +* ((ProgramPart (Relocated p,k)) +* (DataPart p))) +* (Start-At ((IC p) + k),SCM )) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by FUNCT_4:15
.= (((s +* (((ProgramPart (Relocated p,k)) +* (DataPart p)) +* (Start-At ((IC p) + k),SCM ))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by FUNCT_4:15
.= (((s +* ((ProgramPart (Relocated p,k)) +* ((DataPart p) +* (Start-At ((IC p) + k),SCM )))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by FUNCT_4:15
.= (((s +* ((ProgramPart (Relocated p,k)) +* ((Start-At ((IC p) + k),SCM ) +* (DataPart p)))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by A24, A25, FUNCT_4:36
.= (((s +* (((ProgramPart (Relocated p,k)) +* (Start-At ((IC p) + k),SCM )) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by FUNCT_4:15
.= (((s +* (((Start-At ((IC p) + k),SCM ) +* (ProgramPart (Relocated p,k))) +* (DataPart p))) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by A24, COMPOS_1:14, FUNCT_4:36
.= (((s +* (Relocated p,k)) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by X
.= (((Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 ) +* (Start-At ((IC (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 )) -' k),SCM )) +* (s | (dom (ProgramPart (Relocated p,k))))) +* (ProgramPart p) by AMI_1:13 ;
then A30: S₁[ 0 ] ;
thus for i being Element of NAT holds S₁[i] from NAT_1:sch 1(A30, A5); :: thesis: