let s be State of SCM+FSA; :: thesis: for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting Program of SCM+FSA
for f being FinSeq-Location st not f in UsedInt*Loc I holds
(IExec (I,p,s)) . f = s . f
let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for I being InitHalting Program of SCM+FSA
for f being FinSeq-Location st not f in UsedInt*Loc I holds
(IExec (I,p,s)) . f = s . f
let I be InitHalting Program of SCM+FSA; :: thesis: for f being FinSeq-Location st not f in UsedInt*Loc I holds
(IExec (I,p,s)) . f = s . f
let f be FinSeq-Location ; :: thesis: ( not f in UsedInt*Loc I implies (IExec (I,p,s)) . f = s . f )
A1: not f in dom (Initialized I) by SCMFSA6A:49;
not f in NAT by SCMFSA_2:85;
then ( IExec (I,p,s) = (Result ((p +* I),(s +* (Initialized I)))) +* (s | NAT) & not f in dom (s | NAT) ) by RELAT_1:86, SCMFSA6B:def 1;
then A2: (IExec (I,p,s)) . f = (Result ((p +* I),(s +* (Initialized I)))) . f by FUNCT_4:12;
A3: Initialized I c= s +* (Initialized I) by FUNCT_4:26;
I c= p +* I by FUNCT_4:26;
then p +* I halts_on s +* (Initialized I) by Th5, A3;
then consider n being Element of NAT such that
A4: Result ((p +* I),(s +* (Initialized I))) = Comput ((p +* I),(s +* (Initialized I)),n) and
CurInstr ((p +* I),(Result ((p +* I),(s +* (Initialized I))))) = halt SCM+FSA by EXTPRO_1:def 8;
A5: I c= p +* I by FUNCT_4:26;
A6: I c= p +* I by FUNCT_4:26;
Initialized I c= s +* (Initialized I) by FUNCT_4:26;
then A7: ( Initialize I c= s +* (Initialized I) & ( for m being Element of NAT st m < n holds
IC (Comput ((p +* I),(s +* (Initialized I)),m)) in dom I ) ) by Def1, SCMFSA6B:8, A5;
assume not f in UsedInt*Loc I ; :: thesis: (IExec (I,p,s)) . f = s . f
hence (IExec (I,p,s)) . f = (s +* (Initialized I)) . f by A2, A4, A7, SF_MASTR:71, A6
.= s . f by A1, FUNCT_4:12 ;
:: thesis: verum