let s be State of SCM+FSA; :: thesis: for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for b being Int-Location
for I being Program of SCM+FSA st I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not b in UsedIntLoc I holds
(IExec (I,p,s)) . b = (Initialized s) . b
let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for b being Int-Location
for I being Program of SCM+FSA st I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not b in UsedIntLoc I holds
(IExec (I,p,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,p & I is_halting_on Initialized s,p & not b in UsedIntLoc I holds
(IExec (I,p,s)) . b = (Initialized s) . b
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not b in UsedIntLoc I implies (IExec (I,p,s)) . b = (Initialized s) . b )
set a = b;
assume that
A1: I is_closed_on Initialized s,p and
A2: I is_halting_on Initialized s,p and
A3: not b in UsedIntLoc I ; :: thesis: (IExec (I,p,s)) . b = (Initialized s) . b
set Is = Initialized s;
A4: ProgramPart I = I by RELAT_1:209;
set sI = s +* (Initialized I);
set pI = p +* I;
( 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)))))) . b = (s +* (Initialized I)) . b by A3, SF_MASTR:69, FUNCT_4:26, A5;
DataPart (s +* (Initialized I)) = DataPart (Initialized s) by SCMFSA8B:5;
then A11: (s +* (Initialized I)) . b = (Initialized s) . b 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)) . b = (Initialized s) . b by A10, A11, SCMFSA6A:38; :: thesis: verum