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 (s | NAT ) misses Int-Locations \/ FinSeq-Locations by SCMFSA8A:3;
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, AMI_1:122 ;
hence (IExec I,s) . b = (Initialized s) . b by A8, A9, SCMFSA6A:38; :: thesis: verum