let s be State of ; :: thesis: for I being Program of st I is_closed_on s & I is_halting_on s holds
for m being Element of NAT st m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
Computation (s +* (I +* (Start-At (insloc 0 )))),m, Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m equal_outside NAT
set A = NAT ;
let I be Program of ; :: thesis: ( I is_closed_on s & I is_halting_on s implies for m being Element of NAT st m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
Computation (s +* (I +* (Start-At (insloc 0 )))),m, Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m equal_outside NAT )
set s1 = s +* (I +* (Start-At (insloc 0 )));
set s2 = s +* ((loop I) +* (Start-At (insloc 0 )));
assume A1: I is_closed_on s ; :: thesis: ( not I is_halting_on s or for m being Element of NAT st m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
Computation (s +* (I +* (Start-At (insloc 0 )))),m, Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m equal_outside NAT )
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) implies Computation (s +* (I +* (Start-At (insloc 0 )))),$1, Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),$1 equal_outside NAT );
assume I is_halting_on s ; :: thesis: for m being Element of NAT st m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
Computation (s +* (I +* (Start-At (insloc 0 )))),m, Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m equal_outside NAT
then A2: ProgramPart (s +* (I +* (Start-At (insloc 0 )))) halts_on s +* (I +* (Start-At (insloc 0 ))) by SCMFSA7B:def 8;
A3: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

A4: loop I c= (loop I) +* (Start-At (insloc 0 )) by SCMFSA8A:9;
A5: I c= I +* (Start-At (insloc 0 )) by SCMFSA8A:9;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) implies Computation (s +* (I +* (Start-At (insloc 0 )))),m, Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m equal_outside NAT ) ; :: thesis: S₁[m + 1]
(loop I) +* (Start-At (insloc 0 )) c= s +* ((loop I) +* (Start-At (insloc 0 ))) by FUNCT_4:26;
then loop I c= s +* ((loop I) +* (Start-At (insloc 0 ))) by A4, XBOOLE_1:1;
then A7: loop I c= Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m by AMI_1:81;
A8: Computation (s +* (I +* (Start-At (insloc 0 )))),(m + 1) = Following (Computation (s +* (I +* (Start-At (insloc 0 )))),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m)),(Computation (s +* (I +* (Start-At (insloc 0 )))),m) ;
A9: Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),(m + 1) = Following (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m)),(Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) ;
A10: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) in dom I by A1, SCMFSA7B:def 7;
I +* (Start-At (insloc 0 )) c= s +* (I +* (Start-At (insloc 0 ))) by FUNCT_4:26;
then I c= s +* (I +* (Start-At (insloc 0 ))) by A5, XBOOLE_1:1;
then I c= Computation (s +* (I +* (Start-At (insloc 0 )))),m by AMI_1:81;
then A11: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A10, GRFUNC_1:8;
assume A12: m + 1 <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: Computation (s +* (I +* (Start-At (insloc 0 )))),(m + 1), Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),(m + 1) equal_outside NAT
then m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) by NAT_1:13;
then I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) <> halt SCM+FSA by A2, A11, AMI_1:def 46;
then A13: I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) = (loop I) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by FUNCT_4:111;
IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) in dom (loop I) by A10, FUNCT_4:105;
then CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A7, A11, A13, GRFUNC_1:8
.= CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) by A6, A12, AMI_1:121, NAT_1:13 ;
hence Computation (s +* (I +* (Start-At (insloc 0 )))),(m + 1), Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),(m + 1) equal_outside NAT by A6, A12, A8, A9, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

A14: S₁[ 0 ]

proof

assume 0 <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: Computation (s +* (I +* (Start-At (insloc 0 )))),0 , Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),0 equal_outside NAT
A15: s,s +* (loop I) equal_outside NAT by AMI_1:120;
s +* I,s equal_outside NAT by AMI_1:120, FUNCT_7:28;
then s +* I,s +* (loop I) equal_outside NAT by A15, FUNCT_7:29;
then (s +* I) +* (Start-At (insloc 0 )),(s +* (loop I)) +* (Start-At (insloc 0 )) equal_outside NAT by FUNCT_7:106;
then (s +* I) +* (Start-At (insloc 0 )),s +* ((loop I) +* (Start-At (insloc 0 ))) equal_outside NAT by FUNCT_4:15;
then s +* (I +* (Start-At (insloc 0 ))),s +* ((loop I) +* (Start-At (insloc 0 ))) equal_outside NAT by FUNCT_4:15;
then s +* (I +* (Start-At (insloc 0 ))), Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),0 equal_outside NAT by AMI_1:13;
hence Computation (s +* (I +* (Start-At (insloc 0 )))),0 , Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),0 equal_outside NAT by AMI_1:13; :: thesis: verum

end;

thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A14, A3); :: thesis: verum