let s be State of SCM+FSA; :: thesis: for p being Instruction-Sequence of SCM+FSA
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 Instruction-Sequence of SCM+FSA; :: 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 )
( a <> intloc 0 & a <> IC ) by SCMFSA_2:56;
then B1: not a in dom (Initialize ((intloc 0) .--> 1)) by SCMFSA6A:42, 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;
I c= p +* I by FUNCT_4:25;
then p +* I halts_on Initialized s by Def2, A3;
then consider n being Element of NAT such that
A4: 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;
A5: I c= p +* I by FUNCT_4:25;
A6: for m being Element of NAT st m < n holds
IC (Comput ((p +* I),(Initialized s),m)) in dom I by Def1, A5, A3;
assume not a in UsedIntLoc I ; :: thesis: (IExec (I,p,s)) . a = s . a
hence (IExec (I,p,s)) . a = (Initialized s) . a by A2, A4, A6, FUNCT_4:25, SF_MASTR:61
.= s . a by B1, FUNCT_4:11 ;
:: thesis: verum