let s be State of SCM+FSA; :: thesis: for I being parahalting good Program of SCM+FSA
for a being read-write Int-Location st not I destroys a & s . (intloc 0) = 1 & s . a > 0 holds
ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = s +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) & k = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 & (Comput ((ProgramPart s2),s2,k)) . a = (s . a) - 1 & (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
set A = NAT ;
let I be parahalting good Program of SCM+FSA; :: thesis: for a being read-write Int-Location st not I destroys a & s . (intloc 0) = 1 & s . a > 0 holds
ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = s +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) & k = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 & (Comput ((ProgramPart s2),s2,k)) . a = (s . a) - 1 & (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
let a be read-write Int-Location ; :: thesis: ( not I destroys a & s . (intloc 0) = 1 & s . a > 0 implies ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = s +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) & k = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 & (Comput ((ProgramPart s2),s2,k)) . a = (s . a) - 1 & (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) ) )
assume A1: not I destroys a ; :: thesis: ( not s . (intloc 0) = 1 or not s . a > 0 or ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = s +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) & k = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 & (Comput ((ProgramPart s2),s2,k)) . a = (s . a) - 1 & (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) ) )
reconsider I1 = I ';' (SubFrom (a,(intloc 0))) as parahalting Program of SCM+FSA ;
set P = if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))));
set s1 = s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)));
A2: I1 is_halting_on s by SCMFSA7B:25;
set Is = (Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)));
A3: intloc 0 in dom s by SCMFSA_2:66;
assume s . (intloc 0) = 1 ; :: thesis: ( not s . a > 0 or ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = s +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) & k = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 & (Comput ((ProgramPart s2),s2,k)) . a = (s . a) - 1 & (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) ) )
then A4: s +* ((intloc 0) .--> 1) = s by A3, FUNCT_7:111;
A5: I1 is_closed_on Initialized s by SCMFSA7B:24;
A6: I1 is_halting_on Initialized s by SCMFSA7B:25;
assume A7: s . a > 0 ; :: thesis: ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = s +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) & k = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 & (Comput ((ProgramPart s2),s2,k)) . a = (s . a) - 1 & (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
then A8: (Initialized s) . a > 0 by SCMFSA6C:3;
then A9: if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_halting_on Initialized s by A5, A6, SCMFSA8B:18;
A10: if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_closed_on Initialized s by A8, A5, A6, SCMFSA8B:18;
take s2 = s +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))); :: thesis: ex k being Element of NAT st
( s2 = s +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) & k = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 & (Comput ((ProgramPart s2),s2,k)) . a = (s . a) - 1 & (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
take k = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1; :: thesis: ( s2 = s +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) & k = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 & (Comput ((ProgramPart s2),s2,k)) . a = (s . a) - 1 & (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
thus ( s2 = s +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) & k = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 ) ; :: thesis: ( (Comput ((ProgramPart s2),s2,k)) . a = (s . a) - 1 & (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
A11: dom (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) misses dom (Start-At (0,SCM+FSA)) by COMPOS_1:140;
A12: (Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))) = (s +* (Start-At (0,SCM+FSA))) +* ((Start-At (0,SCM+FSA)) +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by A4, A11, FUNCT_4:36
.= ((s +* (Start-At (0,SCM+FSA))) +* (Start-At (0,SCM+FSA))) +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) by FUNCT_4:15
.= (s +* ((Start-At (0,SCM+FSA)) +* (Start-At (0,SCM+FSA)))) +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) by FUNCT_4:15
.= s +* ((Start-At (0,SCM+FSA)) +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by FUNCT_4:15
.= s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))) by A11, FUNCT_4:36 ;
A13: I1 is_halting_on Initialized s by SCMFSA7B:25;
I1 is_closed_on Initialized s by SCMFSA7B:24;
then A14: IExec ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))),s) = (IExec (I1,s)) +* (Start-At ((((card (Goto 2)) + (card I1)) + 3),SCM+FSA)) by A7, A13, SCMFSA8B:19;
then A15: (IExec ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))),s)) . a = (IExec (I1,s)) . a by SCMFSA_3:11;
A16: I1 is_closed_on s by SCMFSA7B:24;
then A17: if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_closed_on s by A7, A2, SCMFSA8B:18;
A18: if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_halting_on s by A7, A16, A2, SCMFSA8B:18;
T: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))) by AMI_1:123;
A19: Comput ((ProgramPart s2),s2,((LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1)) = Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))))))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))))))),(Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))))),(Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))))))) by T ;
then A20: Comput ((ProgramPart s2),s2,((LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1)) = Exec ((goto 0),(Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))))))) by A17, A18, Lm2, T;

A21: now

let b be Int-Location ; :: thesis: (Comput ((ProgramPart s2),s2,((LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1))) . b = (Comput ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))) . b
(Comput ((ProgramPart s2),s2,((LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1))) . b = (Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))) . b by A20, SCMFSA_2:95;
hence (Comput ((ProgramPart s2),s2,((LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1))) . b = (Comput ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))) . b by A17, A18, A12, Th109, SCMFSA10:92; :: thesis: verum

end;

then (Comput ((ProgramPart s2),s2,((LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1))) . a = (Comput ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))) . a
.= (IExec ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))),s)) . a by A9, Th87 ;
hence (Comput ((ProgramPart s2),s2,k)) . a = (Comput ((ProgramPart ((Initialized s) +* (I1 +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I1 +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* (I1 +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I1 +* (Start-At (0,SCM+FSA)))))))) . a by A13, A15, Th87
.= (s . a) - 1 by A1, Th98 ;
:: thesis: ( (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
A22: if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is good by Th115;
(Comput ((ProgramPart s2),s2,((LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1))) . (intloc 0) = (Comput ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))) . (intloc 0) by A21
.= 1 by A9, A10, A22, Th96 ;
hence (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 ; :: thesis: ( ( for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )

hereby :: thesis: ( ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )

let b be read-write Int-Location ; :: thesis: ( b <> a implies (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b )
assume A23: b <> a ; :: thesis: (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b
thus (Comput ((ProgramPart s2),s2,k)) . b = (Comput ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))) . b by A21
.= (IExec ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))),s)) . b by A9, Th87
.= (IExec (I1,s)) . b by A14, SCMFSA_3:11
.= (Exec ((SubFrom (a,(intloc 0))),(IExec (I,s)))) . b by SCMFSA6C:7
.= (IExec (I,s)) . b by A23, SCMFSA_2:91 ; :: thesis: verum

end;

hereby :: thesis: ( IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )

let f be FinSeq-Location ; :: thesis: (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f
(Comput ((ProgramPart s2),s2,((LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1))) . f = (Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))) . f by A20, SCMFSA_2:95;
hence (Comput ((ProgramPart s2),s2,k)) . f = (Comput ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))) . f by A17, A18, A12, Th109, SCMFSA10:93
.= (IExec ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))),s)) . f by A9, Th87
.= (IExec (I1,s)) . f by A14, SCMFSA_3:12
.= (Exec ((SubFrom (a,(intloc 0))),(IExec (I,s)))) . f by SCMFSA6C:8
.= (IExec (I,s)) . f by SCMFSA_2:91 ;
:: thesis: verum

end;

A24: IC (Comput ((ProgramPart s2),s2,((LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1))) = (Exec ((goto 0),(Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))))))) . (IC SCM+FSA) by A17, A18, A19, Lm2, T
.= 0 by SCMFSA_2:95 ;
hence IC (Comput ((ProgramPart s2),s2,k)) = 0 ; :: thesis: for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))

hereby :: thesis: verum

let n be Element of NAT ; :: thesis: ( n <= k implies IC (Comput ((ProgramPart s2),s2,b₁)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) )
assume A25: n <= k ; :: thesis: IC (Comput ((ProgramPart s2),s2,b₁)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))

per cases ( n <= LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))) or n = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 ) by A25, NAT_1:8;

suppose A26: n <= LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))) ; :: thesis: IC (Comput ((ProgramPart s2),s2,b₁)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))

A27: I1 is_halting_on s by SCMFSA7B:25;
A28: I1 is_closed_on s by SCMFSA7B:24;
then A29: if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_closed_on s by A7, A27, SCMFSA8B:18;
if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_halting_on s by A7, A28, A27, SCMFSA8B:18;
then Comput ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))),n), Comput ((ProgramPart s2),s2,n) equal_outside NAT by A26, A29, Th109;
then A30: IC (Comput ((ProgramPart s2),s2,n)) = IC (Comput ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))),n)) by SCMFSA8A:6;
IC (Comput ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))),n)) in dom (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) by A29, SCMFSA7B:def 7;
hence IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by A30, FUNCT_4:105; :: thesis: verum

end;

suppose A31: n = (LifeSpan ((ProgramPart (s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),(s +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 ; :: thesis: IC (Comput ((ProgramPart s2),s2,b₁)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))

card (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) = card (dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))))
.= card (dom (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by FUNCT_4:105
.= card (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) ;
hence IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by A24, A31, AFINSQ_1:70; :: thesis: verum

end;

end;

end;