let s be State of SCM+FSA; :: thesis: for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting Program of SCM+FSA
for f being FinSeq-Location 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 InitHalting Program of SCM+FSA
for f being FinSeq-Location st not f in UsedInt*Loc I holds
(IExec (I,p,s)) . f = s . f
let I be InitHalting Program of SCM+FSA; :: thesis: for f being FinSeq-Location st not f in UsedInt*Loc I holds
(IExec (I,p,s)) . f = s . f
let f be FinSeq-Location ; :: thesis: ( not f in UsedInt*Loc I implies (IExec (I,p,s)) . f = s . f )
J: NPP (Initialized I) = Initialize ((intloc 0) .--> 1) by SCMFSA6A:79;
then B0: dom (Initialize ((intloc 0) .--> 1)) c= dom (Initialized I) by RELAT_1:25;
not f in dom (Initialized I) by SCMFSA6A:49;
then B1: not f in dom (Initialize ((intloc 0) .--> 1)) by B0;
not f in NAT by SCMFSA_2:85;
then ( IExec (I,p,s) = (Result ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))))) +* (s | NAT) & not f in dom (s | NAT) ) by RELAT_1:86, SCMFSA6B:def 1;
then A2: (IExec (I,p,s)) . f = (Result ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))))) . f by FUNCT_4:12;
A3: Initialize ((intloc 0) .--> 1) c= s +* (Initialize ((intloc 0) .--> 1)) by FUNCT_4:26;
I c= p +* I by FUNCT_4:26;
then p +* I halts_on s +* (Initialize ((intloc 0) .--> 1)) by Th5, A3;
then consider n being Element of NAT such that
A4: 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 8;
A5: I c= p +* I by FUNCT_4:26;
A6: I c= p +* I by FUNCT_4:26;
A7: for m being Element of NAT st m < n holds
IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) in dom I by Def1, A5, A3;
Initialize I c= Initialized I by SCMFSA6A:80;
then NPP (Initialize I) c= NPP (Initialized I) by COMPOS_1:170;
then NPP (Initialize I) c= s +* (Initialize ((intloc 0) .--> 1)) by A3, J, XBOOLE_1:1;
then S1: Start-At (0,SCM+FSA) c= s +* (Initialize ((intloc 0) .--> 1)) by COMPOS_1:210;
assume not f in UsedInt*Loc I ; :: thesis: (IExec (I,p,s)) . f = s . f
hence (IExec (I,p,s)) . f = (s +* (Initialize ((intloc 0) .--> 1))) . f by A2, A4, A7, A6, S1, SF_MASTR:71
.= s . f by B1, FUNCT_4:12 ;
:: thesis: verum