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 a being read-write Int-Location st not a in UsedILoc I holds
(IExec (I,p,s)) . a = s . a
let p be Instruction-Sequence of SCM+FSA; for I being really-closed InitHalting Program of SCM+FSA
for a being read-write Int-Location st not a in UsedILoc I holds
(IExec (I,p,s)) . a = s . a
let I be really-closed InitHalting Program of SCM+FSA; for a being read-write Int-Location st not a in UsedILoc I holds
(IExec (I,p,s)) . a = s . a
let a be read-write Int-Location; ( not a in UsedILoc I implies (IExec (I,p,s)) . a = s . a )
( a <> intloc 0 & a <> IC )
by SCMFSA_2:56;
then A1:
not a in dom (Initialize ((intloc 0) .--> 1))
by SCMFSA_M:11, TARSKI:def 2;
A2:
(IExec (I,p,s)) . a = (Result ((p +* I),(Initialized s))) . a
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),(s +* (Initialize ((intloc 0) .--> 1))),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 A4, AMISTD_1:21;
assume
not a in UsedILoc I
; (IExec (I,p,s)) . a = s . a
hence (IExec (I,p,s)) . a =
(Initialized s) . a
by A2, A5, A6, FUNCT_4:25, SF_MASTR:61
.=
s . a
by A1, FUNCT_4:11
;
verum