let f be FinSeq-Location ; :: thesis: for s being State of SCM+FSA
for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being parahalting Program of SCM+FSA st not f in UsedInt*Loc I holds
(IExec (I,P,s)) . f = s . f
let s be State of SCM+FSA; :: thesis: for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being parahalting Program of SCM+FSA 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 parahalting Program of SCM+FSA st not f in UsedInt*Loc I holds
(IExec (I,P,s)) . f = s . f
let I be parahalting Program of SCM+FSA; :: thesis: ( not f in UsedInt*Loc I implies (IExec (I,P,s)) . f = s . f )
assume A1: not f in UsedInt*Loc I ; :: thesis: (IExec (I,P,s)) . f = s . f
A2: I c= P +* I by FUNCT_4:26;
Initialized I c= s +* (Initialized I) by FUNCT_4:26;
then P +* I halts_on s +* (Initialized I) by Th19, A2;
then consider n being Element of NAT such that
A3: 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;
A4: not f in dom (Initialized I) by SCMFSA6A:49;
dom (ProgramPart s) = NAT by COMPOS_1:34;
then not f in dom (s | NAT) by SCMFSA_2:85;
then A5: (IExec (I,P,s)) . f = (Result ((P +* I),(s +* (Initialized I)))) . f by FUNCT_4:12;
A6: Initialize I c= s +* (Initialized I) by Th8, FUNCT_4:26;
for m being Element of NAT st m < n holds
IC (Comput ((P +* I),(s +* (Initialized I)),m)) in dom I by Def2, A6, FUNCT_4:26;
hence (IExec (I,P,s)) . f = (s +* (Initialized I)) . f by A1, A5, A3, A6, SF_MASTR:71, FUNCT_4:26
.= s . f by A4, FUNCT_4:12 ;
:: thesis: verum