let s be State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA
for I being parahalting Program of
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 parahalting Program of
for a being read-write Int-Location st not a in UsedIntLoc I holds
(IExec (I,P,s)) . a = s . a
let I be parahalting Program of ; :: 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 )
assume A1: not a in UsedIntLoc I ; :: thesis: (IExec (I,P,s)) . a = s . a
A2: I c= P +* I by FUNCT_4:25;
Initialize ((intloc 0) .--> 1) c= s +* (Initialize ((intloc 0) .--> 1)) by FUNCT_4:25;
then P +* I halts_on s +* (Initialize ((intloc 0) .--> 1)) by Th19, A2;
then consider n being Element of NAT such that
A3: Result ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) = Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),n) and
CurInstr ((P +* I),(Result ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))))) = halt SCM+FSA by EXTPRO_1:def 9;
XX: dom (Initialize ((intloc 0) .--> 1)) = (dom ((intloc 0) .--> 1)) \/ (dom (Start-At (0,SCM+FSA))) by FUNCT_4:def 1;
YY: not a in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
not a in {(intloc 0)} by TARSKI:def 1;
then not a in dom ((intloc 0) .--> 1) by FUNCOP_1:13;
then A4: not a in dom (Initialize ((intloc 0) .--> 1)) by XX, YY, XBOOLE_0:def 3;
for m being Element of NAT st m < n holds
IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) in dom I by A2, AMISTD_1:def 10;
hence (IExec (I,P,s)) . a = (s +* (Initialize ((intloc 0) .--> 1))) . a by A1, A3, FUNCT_4:25, SF_MASTR:61
.= s . a by A4, FUNCT_4:11 ;
:: thesis: verum