let s be State of SCM+FSA; for p being Instruction-Sequence of SCM+FSA
for I being really-closed InitHalting Program of SCM+FSA
for f being FinSeq-Location st not f in UsedI*Loc I holds
(IExec (I,p,s)) . f = s . f
let p be Instruction-Sequence of SCM+FSA; for I being really-closed InitHalting Program of SCM+FSA
for f being FinSeq-Location st not f in UsedI*Loc I holds
(IExec (I,p,s)) . f = s . f
let I be really-closed InitHalting Program of SCM+FSA; for f being FinSeq-Location st not f in UsedI*Loc I holds
(IExec (I,p,s)) . f = s . f
let f be FinSeq-Location ; ( not f in UsedI*Loc I implies (IExec (I,p,s)) . f = s . f )
( f <> intloc 0 & f <> IC )
by SCMFSA_2:57, SCMFSA_2:58;
then A1:
not f in dom (Initialize ((intloc 0) .--> 1))
by SCMFSA_M:11, TARSKI:def 2;
A2:
(IExec (I,p,s)) . f = (Result ((p +* I),(Initialized s))) . f
by SCMFSA6B:def 1;
A3:
Initialize ((intloc 0) .--> 1) c= Initialized s
by FUNCT_4:25;
A4:
I c= p +* I
by FUNCT_4:25;
then
p +* I halts_on Initialized s
by Def1, A3;
then consider n being Nat such that
A5:
Result ((p +* I),(Initialized s)) = Comput ((p +* I),(Initialized s),n)
and
CurInstr ((p +* I),(Result ((p +* I),(Initialized s)))) = halt SCM+FSA
by EXTPRO_1:def 9;
IC (Initialized s) = 0
by MEMSTR_0:def 11;
then
IC (Initialized s) in dom I
by AFINSQ_1:65;
then A6:
for m being Nat st m < n holds
IC (Comput ((p +* I),(Initialized s),m)) in dom I
by AMISTD_1:21, A4;
assume
not f in UsedI*Loc I
; (IExec (I,p,s)) . f = s . f
hence (IExec (I,p,s)) . f =
(Initialized s) . f
by A2, A5, A6, FUNCT_4:25, SF_MASTR:63
.=
s . f
by A1, FUNCT_4:11
;
verum