let p be autonomic halting FinPartState of SCM+FSA ; :: thesis: ( IC SCM+FSA in dom p implies for k being Element of NAT holds DataPart (Result p) = DataPart (Result (Relocated p,k)) )
assume A1: IC SCM+FSA in dom p ; :: thesis: for k being Element of NAT holds DataPart (Result p) = DataPart (Result (Relocated p,k))
let k be Element of NAT ; :: thesis: DataPart (Result p) = DataPart (Result (Relocated p,k))
consider s being State of SCM+FSA such that
A2: p c= s by PBOOLE:156;
A3: Relocated p,k is autonomic by A1, Th16;
A4: Relocated p,k is halting by A1, Th13;
ProgramPart s halts_on s by A2, AMI_1:def 26;
then consider j1 being Element of NAT such that
A5: Result s = Comput (ProgramPart s),s,j1 and
A6: CurInstr (ProgramPart (Result s)),(Result s) = halt SCM+FSA by AMI_1:def 22;
consider t being State of SCM+FSA such that
A7: Relocated p,k c= t by PBOOLE:156;
Y: (ProgramPart (Comput (ProgramPart t),t,j1)) /. (IC (Comput (ProgramPart t),t,j1)) = (Comput (ProgramPart t),t,j1) . (IC (Comput (ProgramPart t),t,j1)) by AMI_1:150;
t . (IC (Comput (ProgramPart t),t,j1)) = CurInstr (ProgramPart (Comput (ProgramPart t),t,j1)),(Comput (ProgramPart t),t,j1) by AMI_1:54, Y
.= IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s),s,j1)),(Comput (ProgramPart s),s,j1)),k by A1, A2, A7, Th12
.= halt SCM+FSA by A5, A6, SCMFSA_4:8 ;
then A8: Result t = Comput (ProgramPart t),t,j1 by AMI_1:56;
thus DataPart (Result p) = DataPart ((Result s) | (dom p)) by A2, AMI_1:def 28
.= (Result s) | ((dom p) /\ (Int-Locations \/ FinSeq-Locations )) by RELAT_1:100, SCMFSA_2:127
.= (Result s) | (dom (DataPart p)) by RELAT_1:90, SCMFSA_2:127
.= (Result t) | (dom (DataPart (Relocated p,k))) by A1, A2, A5, A7, A8, Th12
.= (Result t) | ((dom (Relocated p,k)) /\ (Int-Locations \/ FinSeq-Locations )) by RELAT_1:90, SCMFSA_2:127
.= DataPart ((Result t) | (dom (Relocated p,k))) by RELAT_1:100, SCMFSA_2:127
.= DataPart (Result (Relocated p,k)) by A7, A4, A3, AMI_1:def 28 ; :: thesis: verum
reconsider s3 = s +* (DataPart t) as State of SCM+FSA ;