let I be Program of ; :: thesis: for s being State of st I is_closed_on s & I is_halting_on s holds
( IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) = insloc (card I) & DataPart (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = DataPart (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) & ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) halts_on s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) & LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) = (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1 & I ';' (Stop SCM+FSA ) is_closed_on s & I ';' (Stop SCM+FSA ) is_halting_on s )
let s be State of ; :: thesis: ( I is_closed_on s & I is_halting_on s implies ( IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) = insloc (card I) & DataPart (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = DataPart (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) & ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) halts_on s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) & LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) = (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1 & I ';' (Stop SCM+FSA ) is_closed_on s & I ';' (Stop SCM+FSA ) is_halting_on s ) )
assume A1: I is_closed_on s ; :: thesis: ( not I is_halting_on s or ( IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) = insloc (card I) & DataPart (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = DataPart (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) & ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) halts_on s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) & LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) = (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1 & I ';' (Stop SCM+FSA ) is_closed_on s & I ';' (Stop SCM+FSA ) is_halting_on s ) )
card (I ';' (Stop SCM+FSA )) = (card I) + 1 by Th17, SCMFSA6A:61;
then card I < card (I ';' (Stop SCM+FSA )) by NAT_1:13;
then A2: insloc (card I) in dom (I ';' (Stop SCM+FSA )) by SCMFSA6A:15;
A3: ProgramPart (Relocated (Stop SCM+FSA ),(card I)) c= Relocated (Stop SCM+FSA ),(card I) by RELAT_1:88;
A4: dom (ProgramPart (Stop SCM+FSA )) = dom (Stop SCM+FSA ) by AMI_1:105;
then insloc (0 + (card I)) in { (m + (card I)) where m is Element of NAT : m in dom (ProgramPart (Stop SCM+FSA )) } by Th15;
then A5: insloc (0 + (card I)) in dom (ProgramPart (Relocated (Stop SCM+FSA ),(card I))) by SCMFSA_5:3;
set s2 = s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )));
set s1 = s +* (I +* (Start-At (insloc 0 )));
assume A6: I is_halting_on s ; :: thesis: ( IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) = insloc (card I) & DataPart (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = DataPart (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) & ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) halts_on s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) & LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) = (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1 & I ';' (Stop SCM+FSA ) is_closed_on s & I ';' (Stop SCM+FSA ) is_halting_on s )
then A7: IC (Computation (s +* ((Directed I) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) = IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) by A1, Th42;
I ';' (Stop SCM+FSA ) c= (I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )) by Th9;
then dom (I ';' (Stop SCM+FSA )) c= dom ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
then A8: (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) . (insloc (card I)) = ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) . (insloc (card I)) by A2, FUNCT_4:14
.= (I ';' (Stop SCM+FSA )) . (insloc (card I)) by A2, SCMFSA6B:7
.= (ProgramPart (Relocated (Stop SCM+FSA ),(card I))) . (insloc (card I)) by A5, FUNCT_4:14
.= (Relocated (Stop SCM+FSA ),(card I)) . (insloc (0 + (card I))) by A5, A3, GRFUNC_1:8
.= IncAddr (halt SCM+FSA ),(card I) by A4, Th16, SCMFSA_5:7, SCMNORM:2
.= halt SCM+FSA by SCMFSA_4:8 ;
DataPart (Computation (s +* ((Directed I) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) = DataPart (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) by A1, A6, Th42;
hence ( IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) = insloc (card I) & DataPart (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = DataPart (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) ) by A1, A6, A7, Th36; :: thesis: ( ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) halts_on s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) & LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) = (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1 & I ';' (Stop SCM+FSA ) is_closed_on s & I ';' (Stop SCM+FSA ) is_halting_on s )
A9: CurInstr (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) = (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1)) . (insloc (card I)) by A1, A6, A7, Th36
.= halt SCM+FSA by A8, AMI_1:54 ;
hence A10: ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) halts_on s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) by AMI_1:146; :: thesis: ( LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) = (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1 & I ';' (Stop SCM+FSA ) is_closed_on s & I ';' (Stop SCM+FSA ) is_halting_on s )

A11: now

let k be Element of NAT ; :: thesis: ( k <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) implies IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),k) in dom (I ';' (Stop SCM+FSA )) )
assume A12: k <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),k) in dom (I ';' (Stop SCM+FSA ))
then IC (Computation (s +* (I +* (Start-At (insloc 0 )))),k) = IC (Computation (s +* ((Directed I) +* (Start-At (insloc 0 )))),k) by A1, A6, Th35, AMI_1:121;
then A13: IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),k) = IC (Computation (s +* (I +* (Start-At (insloc 0 )))),k) by A1, A6, A12, Th42;
( IC (Computation (s +* (I +* (Start-At (insloc 0 )))),k) in dom I & dom I c= dom (I ';' (Stop SCM+FSA )) ) by A1, SCMFSA6A:56, SCMFSA7B:def 7;
hence IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),k) in dom (I ';' (Stop SCM+FSA )) by A13; :: thesis: verum

end;

defpred S₁[ Element of NAT ] means ( ( LifeSpan (s +* (I +* (Start-At (insloc 0 )))) < $1 implies IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),$1) = insloc (card I) ) & IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),$1) in dom (I ';' (Stop SCM+FSA )) );
card (I ';' (Stop SCM+FSA )) = (card I) + 1 by Th17, SCMFSA6A:61;
then A14: (card I) + 0 < card (I ';' (Stop SCM+FSA )) by XREAL_1:8;
then A15: insloc (card I) in dom (I ';' (Stop SCM+FSA )) by SCMFSA6A:15;

A16: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[b₁ + 1] )
assume A17: S₁[k] ; :: thesis: S₁[b₁ + 1]

per cases ( k < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) or k = LifeSpan (s +* (I +* (Start-At (insloc 0 )))) or k > LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ) by XXREAL_0:1;

suppose k < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: S₁[b₁ + 1]

then k + 1 <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) by NAT_1:13;
hence S₁[k + 1] by A11; :: thesis: verum

end;

suppose k = LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: S₁[b₁ + 1]

hence S₁[k + 1] by A1, A6, A7, A15, Th36; :: thesis: verum

end;

suppose A18: k > LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: S₁[b₁ + 1]

A19: now

assume k + 1 > LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),(k + 1)) = insloc (card I)
A20: CurInstr (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),k) = halt SCM+FSA by A8, A17, A18, AMI_1:54;
thus IC (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),(k + 1)) = IC (Following (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),k)) by AMI_1:14
.= insloc (card I) by A17, A18, A20, AMI_1:def 8 ; :: thesis: verum

end;

k + 1 > k + 0 by XREAL_1:8;
hence S₁[k + 1] by A14, A18, A19, SCMFSA6A:15, XXREAL_0:2; :: thesis: verum

end;

end;

end;

now

let k be Element of NAT ; :: thesis: ( k < (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1 implies CurInstr (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),k) <> halt SCM+FSA )
assume k < (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1 ; :: thesis: CurInstr (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),k) <> halt SCM+FSA
then A21: k <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) by NAT_1:13;
then CurInstr (Computation (s +* ((Directed I) +* (Start-At (insloc 0 )))),k) <> halt SCM+FSA by A1, A6, Th35;
hence CurInstr (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),k) <> halt SCM+FSA by A1, A6, A21, Th42; :: thesis: verum

end;

then for k being Element of NAT st CurInstr (Computation (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),k) = halt SCM+FSA holds
(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1 <= k ;
hence LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) = (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1 by A9, A10, AMI_1:def 46; :: thesis: ( I ';' (Stop SCM+FSA ) is_closed_on s & I ';' (Stop SCM+FSA ) is_halting_on s )
A22: S₁[ 0 ] by A11, NAT_1:2;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A22, A16);
hence I ';' (Stop SCM+FSA ) is_closed_on s by SCMFSA7B:def 7; :: thesis: I ';' (Stop SCM+FSA ) is_halting_on s
thus I ';' (Stop SCM+FSA ) is_halting_on s by A10, SCMFSA7B:def 8; :: thesis: verum