let p be Instruction-Sequence of SCM+FSA; for I being InitHalting keepInt0_1 Program of SCM+FSA
for s being State of SCM+FSA holds DataPart (Initialized (IExec (I,p,s))) = DataPart (IExec (I,p,s))
set IF = Data-Locations ;
let I be InitHalting keepInt0_1 Program of SCM+FSA; for s being State of SCM+FSA holds DataPart (Initialized (IExec (I,p,s))) = DataPart (IExec (I,p,s))
let s be State of SCM+FSA; DataPart (Initialized (IExec (I,p,s))) = DataPart (IExec (I,p,s))
set IE = IExec (I,p,s);
now ( dom (DataPart (Initialized (IExec (I,p,s)))) = (dom (IExec (I,p,s))) /\ (Data-Locations ) & ( for x being object st x in dom (DataPart (Initialized (IExec (I,p,s)))) holds
(DataPart (Initialized (IExec (I,p,s)))) . x = (IExec (I,p,s)) . x ) )A1:
dom (Initialized (IExec (I,p,s))) = the
carrier of
SCM+FSA
by PARTFUN1:def 2;
A2:
dom (Initialized (IExec (I,p,s))) = (Data-Locations ) \/ {(IC )}
by MEMSTR_0:13;
A3:
dom (IExec (I,p,s)) = the
carrier of
SCM+FSA
by PARTFUN1:def 2;
hence
dom (DataPart (Initialized (IExec (I,p,s)))) = (dom (IExec (I,p,s))) /\ (Data-Locations )
by A1, RELAT_1:61;
for x being object st x in dom (DataPart (Initialized (IExec (I,p,s)))) holds
(DataPart (Initialized (IExec (I,p,s)))) . b2 = (IExec (I,p,s)) . b2then A4:
dom (DataPart (Initialized (IExec (I,p,s)))) = Data-Locations
by A1, A3, A2, XBOOLE_1:21;
let x be
object ;
( x in dom (DataPart (Initialized (IExec (I,p,s)))) implies (DataPart (Initialized (IExec (I,p,s)))) . b1 = (IExec (I,p,s)) . b1 )assume A5:
x in dom (DataPart (Initialized (IExec (I,p,s))))
;
(DataPart (Initialized (IExec (I,p,s)))) . b1 = (IExec (I,p,s)) . b1per cases
( x in Int-Locations or x in FinSeq-Locations )
by A5, A4, SCMFSA_2:100, XBOOLE_0:def 3;
suppose
x in Int-Locations
;
(DataPart (Initialized (IExec (I,p,s)))) . b1 = (IExec (I,p,s)) . b1then reconsider x9 =
x as
Int-Location by AMI_2:def 16;
hereby verum
per cases
( x9 is read-write or x9 is read-only )
;
suppose
x9 is
read-only
;
(DataPart (Initialized (IExec (I,p,s)))) . x = (IExec (I,p,s)) . xthen A7:
x9 = intloc 0
;
thus (DataPart (Initialized (IExec (I,p,s)))) . x =
(Initialized (IExec (I,p,s))) . x9
by A5, A4, FUNCT_1:49
.=
1
by A7, SCMFSA_M:9
.=
(IExec (I,p,s)) . x
by A7, Th7
;
verum end; end;
end; end; end; end;
hence
DataPart (Initialized (IExec (I,p,s))) = DataPart (IExec (I,p,s))
by FUNCT_1:46; verum