let s be State of SCM+FSA; :: thesis: 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; :: thesis: 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; :: thesis: 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 ; :: thesis: ( 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 ; :: thesis: (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 ;
:: thesis: verum