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 (s +* (Initialized I)) = Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),n and
CurInstr (ProgramPart (Result (s +* (Initialized I)))),(Result (s +* (Initialized I))) = halt SCM+FSA by AMI_1:def 22;
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 (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