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 a being read-write Int-Location st not a in UsedIntLoc I holds
(IExec (I,p,s)) . a = s . a
let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for I being InitHalting Program of SCM+FSA
for a being read-write Int-Location st not a in UsedIntLoc I holds
(IExec (I,p,s)) . a = s . a
let I be InitHalting Program of SCM+FSA; :: thesis: for a being read-write Int-Location st not a in UsedIntLoc I holds
(IExec (I,p,s)) . a = s . a
let a be read-write Int-Location ; :: thesis: ( not a in UsedIntLoc I implies (IExec (I,p,s)) . a = s . a )
A1: not a in dom (Initialized I) by SCMFSA6A:48;
not a in NAT by SCMFSA_2:84;
then ( IExec (I,p,s) = (Result ((p +* I),(s +* (Initialized I)))) +* (s | NAT) & not a in dom (s | NAT) ) by RELAT_1:86, SCMFSA6B:def 1;
then A2: (IExec (I,p,s)) . a = (Result ((p +* I),(s +* (Initialized I)))) . a 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;
Initialized I c= s +* (Initialized I) by FUNCT_4:26;
then A6: ( 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 a in UsedIntLoc I ; :: thesis: (IExec (I,p,s)) . a = s . a
hence (IExec (I,p,s)) . a = (s +* (Initialized I)) . a by A2, A4, A6, SF_MASTR:69, A5
.= s . a by A1, FUNCT_4:12 ;
:: thesis: verum