let I be Program of SCM+FSA; :: thesis: ( I is InitHalting implies I is InitClosed )
set II = Initialized I;
assume I is InitHalting ; :: thesis: I is InitClosed
then A1: Initialized I is halting by Def2;
let s be State of SCM+FSA; :: according to SCM_HALT:def 1 :: thesis: for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT st I c= P holds
for n being Element of NAT st Initialized I c= s holds
IC (Comput (P,s,n)) in dom I
let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: ( I c= P implies for n being Element of NAT st Initialized I c= s holds
IC (Comput (P,s,n)) in dom I )
assume A2: I c= P ; :: thesis: for n being Element of NAT st Initialized I c= s holds
IC (Comput (P,s,n)) in dom I
let n be Element of NAT ; :: thesis: ( Initialized I c= s implies IC (Comput (P,s,n)) in dom I )
assume A3: Initialized I c= s ; :: thesis: IC (Comput (P,s,n)) in dom I
A4: ProgramPart (Initialized I) = I by SCMFSA6A:33;
defpred S₁[ Nat] means not IC (Comput (P,s,c₁)) in dom I;
assume not IC (Comput (P,s,n)) in dom I ; :: thesis: contradiction
then A5: ex n being Nat st S₁[n] ;
consider n being Nat such that
A6: S₁[n] and
A7: for m being Nat st S₁[m] holds
n <= m from NAT_1:sch 5(A5);
reconsider n = n as Element of NAT by ORDINAL1:def 13;
A8: for m being Element of NAT st m < n holds
IC (Comput (P,s,m)) in dom I by A7;
set s2 = Comput (P,s,n);
set p2 = P;
set s0 = s +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))));
set p0 = P +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))));
set s1 = (Comput (P,s,n)) +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))));
set p1 = P +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))));
A9: ProgramPart s = ProgramPart (Comput (P,s,n)) by AMI_1:123;
A10: ProgramPart (Comput ((P +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))))),(s +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))))),n)) = ProgramPart (s +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))))) by AMI_1:123
.= ((Comput (P,s,n)) +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))))) | NAT by FUNCT_7:95, A9 ;
A11: s +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))),s equal_outside NAT by FUNCT_7:28, FUNCT_7:93;
A12: Comput (P,s,n),(Comput (P,s,n)) +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) equal_outside NAT by FUNCT_7:93;
(Initialized I) | NAT = I by SCMFSA6A:33;
then dom I = (dom (Initialized I)) /\ NAT by RELAT_1:90;
then not IC (Comput (P,s,n)) in dom (Initialized I) by A6, XBOOLE_0:def 4;
then A13: Initialized I c= s +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) by A3, FUNCT_7:91;
not IC (Comput (P,s,n)) in dom I by A6;
then A14: I c= P +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) by FUNCT_7:91, A2;
then Comput ((P +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))))),(s +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))))),n), Comput (P,s,n) equal_outside NAT by A11, A8, AMISTD_2:66, A2;
then Comput ((P +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))))),(s +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))))),n),(Comput (P,s,n)) +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) equal_outside NAT by A12, FUNCT_7:29;
then A15: Comput ((P +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))))),(s +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n)))))),n) = (Comput (P,s,n)) +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) by FUNCT_7:92, A10;
A16: not P +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) halts_on (Comput (P,s,n)) +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) by SCMFSA6B:20;
P +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) halts_on s +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) by A1, A13, EXTPRO_1:def 10, A4, A14;
then P +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) halts_on (Comput (P,s,n)) +* ((IC (Comput (P,s,n))),(goto (IC (Comput (P,s,n))))) by A15, EXTPRO_1:22;
hence contradiction by A16; :: thesis: verum