let I be Program of SCM+FSA ; :: thesis: ( I is parahalting implies I is paraclosed )
set IAt = I +* (Start-At 0 ,SCM+FSA );
dom I misses dom (Start-At 0 ,SCM+FSA ) by SF_MASTR:64;
then A1: I c= I +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:33;
assume I is parahalting ; :: thesis: I is paraclosed
then A2: I +* (Start-At 0 ,SCM+FSA ) is halting by Def3;
let s be State of SCM+FSA ; :: according to SCMFSA6B:def 2 :: thesis: for n being Element of NAT st I +* (Start-At 0 ,SCM+FSA ) c= s holds
IC (Comput (ProgramPart s),s,n) in dom I
let n be Element of NAT ; :: thesis: ( I +* (Start-At 0 ,SCM+FSA ) c= s implies IC (Comput (ProgramPart s),s,n) in dom I )
defpred S₁[ Nat] means not IC (Comput (ProgramPart s),s,c₁) in dom I;
assume A3: I +* (Start-At 0 ,SCM+FSA ) c= s ; :: thesis: IC (Comput (ProgramPart s),s,n) in dom I
then A4: I c= s by A1, XBOOLE_1:1;
assume not IC (Comput (ProgramPart s),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 (ProgramPart s),s,m) in dom I by A7;
set s2 = Comput (ProgramPart s),s,n;
set s0 = s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)));
set s1 = (Comput (ProgramPart s),s,n) +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)));
A9: s | NAT = (Comput (ProgramPart s),s,n) | NAT by AMI_1:123;
A10: (Comput (ProgramPart (s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))))),(s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)))),n) | NAT = (s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)))) | NAT by AMI_1:123
.= ((Comput (ProgramPart s),s,n) +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)))) | NAT by A9, FUNCT_7:95 ;
A12: s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))),s equal_outside NAT by FUNCT_7:28, FUNCT_7:93;
A13: Comput (ProgramPart s),s,n,(Comput (ProgramPart s),s,n) +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))) equal_outside NAT by FUNCT_7:93;
(I +* (Start-At 0 ,SCM+FSA )) | NAT = I by Th6;
then dom I = (dom (I +* (Start-At 0 ,SCM+FSA ))) /\ NAT by RELAT_1:90;
then not IC (Comput (ProgramPart s),s,n) in dom (I +* (Start-At 0 ,SCM+FSA )) by A6, XBOOLE_0:def 4;
then A14: I +* (Start-At 0 ,SCM+FSA ) c= s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))) by A3, FUNCT_7:91;
then I c= s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))) by A1, XBOOLE_1:1;
then Comput (ProgramPart (s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))))),(s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)))),n, Comput (ProgramPart s),s,n equal_outside NAT by A12, A4, A8, Th21;
then A15: Comput (ProgramPart (s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))))),(s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)))),n = (Comput (ProgramPart s),s,n) +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))) by A13, A10, FUNCT_7:29, FUNCT_7:92;
A16: not ProgramPart ((Comput (ProgramPart s),s,n) +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)))) halts_on (Comput (ProgramPart s),s,n) +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))) by Th20;
ProgramPart (s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)))) halts_on s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))) by A2, A14, AMI_1:def 26;
then ProgramPart (s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)))) halts_on Comput (ProgramPart (s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n))))),(s +* (IC (Comput (ProgramPart s),s,n)),(goto (IC (Comput (ProgramPart s),s,n)))),n by AMI_1:93;
hence contradiction by A15, A16, AMI_1:144; :: thesis: verum