let k be natural number ; :: thesis: for R being good Ring
for s being State of (SCM R) st not R is trivial holds
for p being autonomic FinPartState of (SCM R) st IC (SCM R) in dom p & p c= s holds
for i being Element of NAT holds Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i = ((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k))
let R be good Ring; :: thesis: for s being State of (SCM R) st not R is trivial holds
for p being autonomic FinPartState of (SCM R) st IC (SCM R) in dom p & p c= s holds
for i being Element of NAT holds Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i = ((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k))
let s be State of (SCM R); :: thesis: ( not R is trivial implies for p being autonomic FinPartState of (SCM R) st IC (SCM R) in dom p & p c= s holds
for i being Element of NAT holds Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i = ((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k)) )
assume A1: not R is trivial ; :: thesis: for p being autonomic FinPartState of (SCM R) st IC (SCM R) in dom p & p c= s holds
for i being Element of NAT holds Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i = ((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k))
let p be autonomic FinPartState of (SCM R); :: thesis: ( IC (SCM R) in dom p & p c= s implies for i being Element of NAT holds Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i = ((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k)) )
assume that
A2: IC (SCM R) in dom p and
A3: p c= s ; :: thesis: for i being Element of NAT holds Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i = ((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k))
A4: IC p = p . (IC (SCM R))
.= IC s by A2, A3, GRFUNC_1:8 ;
defpred S₁[ Element of NAT ] means Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),$1 = ((Comput (ProgramPart s),s,$1) +* (Start-At ((IC (Comput (ProgramPart s),s,$1)) + k,(SCM R)),(SCM R))) +* (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: Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i = ((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k)) ; :: thesis: S₁[i + 1]
reconsider ii = IC (Comput (ProgramPart s),s,i) as Element of NAT ;
dom (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R)) = {(IC (SCM R))} by FUNCOP_1:19;
then A7: IC (SCM R) in dom (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R)) by TARSKI:def 1;
not IC (SCM R) in dom (ProgramPart (Relocated p,k)) by AMI_1:101;
then A8: IC (((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k))) = ((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R))) . (IC (SCM R)) by FUNCT_4:12
.= (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R)) . (IC (SCM R)) by A7, FUNCT_4:14
.= (IC (Comput (ProgramPart s),s,i)) + k,(SCM R) by FUNCOP_1:87 ;
A9: ProgramPart p c= Comput (ProgramPart s),s,i by A3, AMI_1:99;
not p is NAT -defined by A2, AMI_1:109;
then A10: IC (Comput (ProgramPart s),s,i) in dom (ProgramPart p) by A1, A3, Th27;
then A11: IC (Comput (ProgramPart s),s,i) in dom (IncAddr (ProgramPart p),k) by AMISTD_2:def 15;
A12: pi (ProgramPart p),ii = (ProgramPart p) . ii by A10, AMI_1:def 47
.= (Comput (ProgramPart s),s,i) . (IC (Comput (ProgramPart s),s,i)) by A10, A9, GRFUNC_1:8 ;
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 AMI_1:150;
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 AMI_1:150;
ProgramPart p c= p by RELAT_1:88;
then dom (ProgramPart p) c= dom p by GRFUNC_1:8;
then (IC (Comput (ProgramPart s),s,i)) + k,(SCM R) in dom (Relocated p,k) by A10, AMISTD_2:71;
then (IC (Comput (ProgramPart s),s,i)) + k,(SCM R) in dom (ProgramPart (Relocated p,k)) by AMI_1:106;
then A13: CurInstr (ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i)),(Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) = (ProgramPart (Relocated p,k)) . ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)) by A6, A8, FUNCT_4:14, Y
.= (IncAddr (Shift (ProgramPart p),k),k) . ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)) by AMISTD_2:69
.= (Shift (IncAddr (ProgramPart p),k),k) . ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)) by AMISTD_2:75
.= (IncAddr (ProgramPart p),k) . (IC (Comput (ProgramPart s),s,i)) by A11, AMISTD_2:65
.= IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s),s,i)),(Comput (ProgramPart s),s,i)),k by A10, A12, AMISTD_2:74, Z ;
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,i) by AMI_1:144;
A14: Comput (ProgramPart s),s,(i + 1) = Following (ProgramPart s),(Comput (ProgramPart s),s,i) by AMI_1:14;
S: ProgramPart (s +* (Relocated p,k)) = ProgramPart (Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i) by AMI_1:144;
thus 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) by AMI_1:14
.= (Exec (IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s),s,i)),(Comput (ProgramPart s),s,i)),k),((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R)))) +* (ProgramPart (Relocated p,k)) by A6, A13, AMISTD_2:67, S
.= ((Comput (ProgramPart s),s,(i + 1)) +* (Start-At ((IC (Comput (ProgramPart s),s,(i + 1))) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k)) by A14, Th22, T ; :: thesis: verum

end;

A15: Comput (ProgramPart s),s,0 = s by AMI_1:13;
NAT misses dom (DataPart p) by Th13, AMI_2:29, XBOOLE_1:63;
then dom (DataPart p) misses dom (ProgramPart (Relocated p,k)) by RELAT_1:87, XBOOLE_1:63;
then A16: (dom (DataPart p)) /\ (dom (ProgramPart (Relocated p,k))) = {} by XBOOLE_0:def 7;
(dom (DataPart p)) /\ {(IC (SCM R))} = {} by Lm1, AMI_1:100;
then ((dom (DataPart p)) /\ (dom (Start-At ((IC p) + k,(SCM R)),(SCM R)))) \/ ((dom (DataPart p)) /\ (dom (ProgramPart (Relocated p,k)))) = {} by A16, FUNCOP_1:19;
then (dom (DataPart p)) /\ ((dom (Start-At ((IC p) + k,(SCM R)),(SCM R))) \/ (dom (ProgramPart (Relocated p,k)))) = {} by XBOOLE_1:23;
then (dom (DataPart p)) /\ (dom ((Start-At ((IC p) + k,(SCM R)),(SCM R)) +* (ProgramPart (Relocated p,k)))) = {} by FUNCT_4:def 1;
then A17: dom (DataPart p) misses dom ((Start-At ((IC p) + k,(SCM R)),(SCM R)) +* (ProgramPart (Relocated p,k))) by XBOOLE_0:def 7;
DataPart p c= p by RELAT_1:88;
then A18: DataPart p c= s by A3, XBOOLE_1:1;
Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),0 = s +* (((Start-At ((IC p) + k,(SCM R)),(SCM R)) +* (IncAddr (Shift (ProgramPart p),k),k)) +* (DataPart p)) by AMI_1:13
.= s +* (((Start-At ((IC p) + k,(SCM R)),(SCM R)) +* (ProgramPart (Relocated p,k))) +* (DataPart p)) by AMISTD_2:69
.= s +* ((DataPart p) +* ((Start-At ((IC p) + k,(SCM R)),(SCM R)) +* (ProgramPart (Relocated p,k)))) by A17, FUNCT_4:36
.= (s +* (DataPart p)) +* ((Start-At ((IC p) + k,(SCM R)),(SCM R)) +* (ProgramPart (Relocated p,k))) by FUNCT_4:15
.= ((s +* (DataPart p)) +* (Start-At ((IC p) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k)) by FUNCT_4:15
.= ((Comput (ProgramPart s),s,0 ) +* (Start-At ((IC (Comput (ProgramPart s),s,0 )) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k)) by A4, A18, A15, FUNCT_4:79 ;
then A19: S₁[ 0 ] ;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A19, A5);
hence for i being Element of NAT holds Comput (ProgramPart (s +* (Relocated p,k))),(s +* (Relocated p,k)),i = ((Comput (ProgramPart s),s,i) +* (Start-At ((IC (Comput (ProgramPart s),s,i)) + k,(SCM R)),(SCM R))) +* (ProgramPart (Relocated p,k)) ; :: thesis: verum