let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for I being Program of SCM+FSA
for a being Int-Location
for k being Element of NAT st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P & not I destroys a holds
(IExec (I,P,s)) . a = (Comput ((P +* I),((Initialized s) +* (Initialize I)),k)) . a
let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA
for a being Int-Location
for k being Element of NAT st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P & not I destroys a holds
(IExec (I,P,s)) . a = (Comput ((P +* I),((Initialized s) +* (Initialize I)),k)) . a
set A = NAT ;
let I be Program of SCM+FSA; :: thesis: for a being Int-Location
for k being Element of NAT st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P & not I destroys a holds
(IExec (I,P,s)) . a = (Comput ((P +* I),((Initialized s) +* (Initialize I)),k)) . a
let a be Int-Location ; :: thesis: for k being Element of NAT st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P & not I destroys a holds
(IExec (I,P,s)) . a = (Comput ((P +* I),((Initialized s) +* (Initialize I)),k)) . a
let k be Element of NAT ; :: thesis: ( I is_closed_on Initialized s,P & I is_halting_on Initialized s,P & not I destroys a implies (IExec (I,P,s)) . a = (Comput ((P +* I),((Initialized s) +* (Initialize I)),k)) . a )
assume A1: I is_closed_on Initialized s,P ; :: thesis: ( not I is_halting_on Initialized s,P or I destroys a or (IExec (I,P,s)) . a = (Comput ((P +* I),((Initialized s) +* (Initialize I)),k)) . a )
not a in NAT by SCMFSA_2:84;
then not a in (dom s) /\ NAT by XBOOLE_0:def 4;
then A2: not a in dom (s | NAT) by RELAT_1:90;
set s0 = Initialized s;
set s1 = (Initialized s) +* (Initialize I);
set P1 = P +* I;
A3: ProgramPart I = I by RELAT_1:209;
assume I is_halting_on Initialized s,P ; :: thesis: ( I destroys a or (IExec (I,P,s)) . a = (Comput ((P +* I),((Initialized s) +* (Initialize I)),k)) . a )
then A4: P +* I halts_on (Initialized s) +* (Initialize I) by SCMFSA7B:def 8, A3;
assume A5: not I destroys a ; :: thesis: (IExec (I,P,s)) . a = (Comput ((P +* I),((Initialized s) +* (Initialize I)),k)) . a
s +* (Initialized I) = (Initialized s) +* (Initialize I) by SCMFSA8A:13;
hence (IExec (I,P,s)) . a = ((Result ((P +* I),((Initialized s) +* (Initialize I)))) +* (s | NAT)) . a
.= (Result ((P +* I),((Initialized s) +* (Initialize I)))) . a by A2, FUNCT_4:12
.= (Comput ((P +* I),((Initialized s) +* (Initialize I)),(LifeSpan ((P +* I),((Initialized s) +* (Initialize I)))))) . a by A4, EXTPRO_1:23
.= (Initialized s) . a by A1, A5, SCMFSA7B:27
.= (Comput ((P +* I),((Initialized s) +* (Initialize I)),k)) . a by A1, A5, SCMFSA7B:27 ;
:: thesis: verum