let s be State of SCM+FSA; :: thesis: for b being Int-Location
for I being Program of SCM+FSA st I is_closed_on Initialized s & I is_halting_on Initialized s & not b in UsedIntLoc I holds
(IExec (I,s)) . b = (Initialized s) . b
let b be Int-Location ; :: thesis: for I being Program of SCM+FSA st I is_closed_on Initialized s & I is_halting_on Initialized s & not b in UsedIntLoc I holds
(IExec (I,s)) . b = (Initialized s) . b
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on Initialized s & I is_halting_on Initialized s & not b in UsedIntLoc I implies (IExec (I,s)) . b = (Initialized s) . b )
set a = b;
assume that
A1: I is_closed_on Initialized s and
A2: I is_halting_on Initialized s and
A3: not b in UsedIntLoc I ; :: thesis: (IExec (I,s)) . b = (Initialized s) . b
set Is = Initialized s;
set sI = s +* (Initialized I);
A4: ( I +* (Start-At (0,SCM+FSA)) c= Initialized I & Initialized I c= s +* (Initialized I) ) by FUNCT_4:26, SCMFSA8C:19;
A5: ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))) halts_on (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by A2, SCMFSA7B:def 8;
NAT misses Int-Locations \/ FinSeq-Locations by SCMFSA_2:13, SCMFSA_2:14, XBOOLE_1:70;
then A6: dom (ProgramPart s) misses Int-Locations \/ FinSeq-Locations by COMPOS_1:34;
A7: s +* (Initialized I) = (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by SCMFSA8A:13;
then for m being Element of NAT st m < LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) holds
IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) in dom I by A1, SCMFSA7B:def 7;
then A8: (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))))) . b = (s +* (Initialized I)) . b by A3, A4, SF_MASTR:69, XBOOLE_1:1;
DataPart (s +* (Initialized I)) = DataPart (Initialized s) by SCMFSA8B:5;
then A9: (s +* (Initialized I)) . b = (Initialized s) . b by SCMFSA6A:38;
DataPart (IExec (I,s)) = DataPart ((Result ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) +* (s | NAT)) by SCMFSA6B:def 1
.= DataPart (Result ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) by A6, FUNCT_4:76, SCMFSA_2:127
.= DataPart (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))))) by A7, A5, EXTPRO_1:23 ;
hence (IExec (I,s)) . b = (Initialized s) . b by A8, A9, SCMFSA6A:38; :: thesis: verum