let f be FinSeq-Location ; :: thesis: for s being State of SCM+FSA
for I being parahalting Program of SCM+FSA st not f in UsedInt*Loc I holds
(IExec (I,s)) . f = s . f
let s be State of SCM+FSA; :: thesis: for I being parahalting Program of SCM+FSA st not f in UsedInt*Loc I holds
(IExec (I,s)) . f = s . f
let I be parahalting Program of SCM+FSA; :: thesis: ( not f in UsedInt*Loc I implies (IExec (I,s)) . f = s . f )
assume A1: not f in UsedInt*Loc I ; :: thesis: (IExec (I,s)) . f = s . f
ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) by Th19, FUNCT_4:26;
then consider n being Element of NAT such that
A2: Result ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) = Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),n) and
CurInstr ((ProgramPart (s +* (Initialized I))),(Result ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) = halt SCM+FSA by EXTPRO_1:def 8;
A3: not f in dom (Initialized I) by SCMFSA6A:49;
f is not Element of NAT by SCMFSA_2:85;
then not f in NAT ;
then not f in dom (s | NAT) by RELAT_1:86;
then A4: (IExec (I,s)) . f = (Result ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) . f by FUNCT_4:12;
A5: I +* (Start-At (0,SCM+FSA)) c= s +* (Initialized I) by Th8, FUNCT_4:26;
then for m being Element of NAT st m < n holds
IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) in dom I by Def2;
hence (IExec (I,s)) . f = (s +* (Initialized I)) . f by A1, A4, A2, A5, SF_MASTR:71
.= s . f by A3, FUNCT_4:12 ;
:: thesis: verum