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 & I is_halting_on Initialized s & not I destroysdestroy a holds
(IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),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 & I is_halting_on Initialized s & not I destroysdestroy a holds
(IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a
let a be Int-Location ; :: thesis: for k being Element of NAT st I is_closed_on Initialized s & I is_halting_on Initialized s & not I destroysdestroy a holds
(IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a
let k be Element of NAT ; :: thesis: ( I is_closed_on Initialized s & I is_halting_on Initialized s & not I destroysdestroy a implies (IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a )
assume A1: I is_closed_on Initialized s ; :: thesis: ( not I is_halting_on Initialized s or I destroysdestroy a or (IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a )
not a in NAT by SCMFSA_2:84;
then not a in (dom s) /\ NAT by XBOOLE_0:def 4;
then A3: not a in dom (s | NAT ) by RELAT_1:90;
set s0 = Initialized s;
set s1 = (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ));
assume I is_halting_on Initialized s ; :: thesis: ( I destroysdestroy a or (IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a )
then A4: ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )) by SCMFSA7B:def 8;
assume A5: not I destroysdestroy a ; :: thesis: (IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a
s +* (Initialized I) = (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )) by SCMFSA8A:13;
hence (IExec I,s) . a = ((Result (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))) +* (s | NAT )) . a by SCMFSA6B:def 1
.= (Result (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))) . a by A3, FUNCT_4:12
.= (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . a by A4, AMI_1:122
.= (Initialized s) . a by A1, A5, SCMFSA7B:27
.= (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a by A1, A5, SCMFSA7B:27 ;
:: thesis: verum