let s be State of SCM+FSA; :: thesis: for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for f being FinSeq-Location
for I being Program of SCM+FSA st I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not f in UsedInt*Loc I holds
(IExec (I,p,s)) . f = (Initialized s) . f
let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for f being FinSeq-Location
for I being Program of SCM+FSA st I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not f in UsedInt*Loc I holds
(IExec (I,p,s)) . f = (Initialized s) . f
let f be FinSeq-Location ; :: thesis: for I being Program of SCM+FSA st I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not f in UsedInt*Loc I holds
(IExec (I,p,s)) . f = (Initialized s) . f
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not f in UsedInt*Loc I implies (IExec (I,p,s)) . f = (Initialized s) . f )
set a = f;
assume that
A1: I is_closed_on Initialized s,p and
A2: I is_halting_on Initialized s,p and
A3: not f in UsedInt*Loc I ; :: thesis: (IExec (I,p,s)) . f = (Initialized s) . f
set Is = Initialized s;
set sI = s +* (Initialized I);
set pI = p +* I;
A4: ProgramPart I = I by RELAT_1:209;
( Initialize I c= Initialized I & Initialized I c= s +* (Initialized I) ) by FUNCT_4:26, SCMFSA8C:19;
then A5: Initialize I c= s +* (Initialized I) by XBOOLE_1:1;
A6: p +* I halts_on (Initialized s) +* (Initialize I) by A2, SCMFSA7B:def 8, A4;
A7: dom (s | NAT) c= NAT by RELAT_1:87;
A8: dom (s | NAT) misses Data-Locations SCM+FSA by A7, XBOOLE_1:63, COMPOS_1:51;
A9: s +* (Initialized I) = (Initialized s) +* (Initialize I) by SCMFSA8A:13;
then for m being Element of NAT st m < LifeSpan ((p +* I),(s +* (Initialized I))) holds
IC (Comput ((p +* I),(s +* (Initialized I)),m)) in dom I by A1, SCMFSA7B:def 7, A4;
then A10: (Comput ((p +* I),(s +* (Initialized I)),(LifeSpan ((p +* I),(s +* (Initialized I)))))) . f = (s +* (Initialized I)) . f by A3, SF_MASTR:71, A5, FUNCT_4:26;
DataPart (s +* (Initialized I)) = DataPart (Initialized s) by SCMFSA8B:5;
then A11: (s +* (Initialized I)) . f = (Initialized s) . f by SCMFSA6A:38;
DataPart (IExec (I,p,s)) = DataPart ((Result ((p +* I),(s +* (Initialized I)))) +* (s | NAT)) by SCMFSA6B:def 1
.= DataPart (Result ((p +* I),(s +* (Initialized I)))) by A8, FUNCT_4:76
.= DataPart (Comput ((p +* I),(s +* (Initialized I)),(LifeSpan ((p +* I),(s +* (Initialized I)))))) by A9, A6, EXTPRO_1:23 ;
hence (IExec (I,p,s)) . f = (Initialized s) . f by A10, A11, SCMFSA6A:38; :: thesis: verum