let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA st I is_closed_on s & I is_halting_on s holds
for m being Element of NAT st m <= LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) holds
Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m, Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m equal_outside NAT
set A = NAT ;
let I be Program of SCM+FSA ; :: thesis: ( I is_closed_on s & I is_halting_on s implies for m being Element of NAT st m <= LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) holds
Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m, Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m equal_outside NAT )
set s1 = s +* (I +* (Start-At 0 ,SCM+FSA ));
set s2 = s +* ((loop I) +* (Start-At 0 ,SCM+FSA ));
assume A1: I is_closed_on s ; :: thesis: ( not I is_halting_on s or for m being Element of NAT st m <= LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) holds
Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m, Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m equal_outside NAT )
defpred S₁[ Nat] means ( $1 <= LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) implies Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),$1, Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),$1 equal_outside NAT );
assume I is_halting_on s ; :: thesis: for m being Element of NAT st m <= LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) holds
Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m, Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m equal_outside NAT
then A2: ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on s +* (I +* (Start-At 0 ,SCM+FSA )) 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 0 ,SCM+FSA ) by SCMFSA8A:9;
A5: I c= I +* (Start-At 0 ,SCM+FSA ) by SCMFSA8A:9;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( m <= LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) implies Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m, Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m equal_outside NAT ) ; :: thesis: S₁[m + 1]
(loop I) +* (Start-At 0 ,SCM+FSA ) c= s +* ((loop I) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
then loop I c= s +* ((loop I) +* (Start-At 0 ,SCM+FSA )) by A4, XBOOLE_1:1;
then A7: loop I c= Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m by AMI_1:81;
T: ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) by AMI_1:123;
A8: Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(m + 1) = Following (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) by T ;
T: ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m) by AMI_1:123;
A9: Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),(m + 1) = Following (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m) by T ;
A10: IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) in dom I by A1, SCMFSA7B:def 7;
Y: (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) /. (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) by COMPOS_1:38;
TX1: ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) by AMI_1:123;
I +* (Start-At 0 ,SCM+FSA ) c= s +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
then I c= s +* (I +* (Start-At 0 ,SCM+FSA )) by A5, XBOOLE_1:1;
then I c= Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m by AMI_1:81;
then A11: CurInstr (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) = I . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) by A10, Y, TX1, GRFUNC_1:8;
assume A12: m + 1 <= LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) ; :: thesis: Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(m + 1), Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),(m + 1) equal_outside NAT
then m < LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) by NAT_1:13;
then I . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) <> halt SCM+FSA by A2, A11, AMI_1:def 46;
then A13: I . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) = (loop I) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) by FUNCT_4:111;
Z: (ProgramPart (Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m)) /. (IC (Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m)) = (Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m) . (IC (Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m)) by COMPOS_1:38;
IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) in dom (loop I) by A10, FUNCT_4:105;
then CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) = (Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) by A7, A11, A13, TX1, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),m) by A6, A12, Z, COMPOS_1:24, NAT_1:13 ;
hence Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(m + 1), Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),(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 (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) ; :: thesis: Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 , Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),0 equal_outside NAT
A15: s,s +* (loop I) equal_outside NAT by FUNCT_7:132;
s +* I,s equal_outside NAT by FUNCT_7:28, FUNCT_7:132;
then s +* I,s +* (loop I) equal_outside NAT by A15, FUNCT_7:29;
then (s +* I) +* (Start-At 0 ,SCM+FSA ),(s +* (loop I)) +* (Start-At 0 ,SCM+FSA ) equal_outside NAT by FUNCT_7:106;
then (s +* I) +* (Start-At 0 ,SCM+FSA ),s +* ((loop I) +* (Start-At 0 ,SCM+FSA )) equal_outside NAT by FUNCT_4:15;
then s +* (I +* (Start-At 0 ,SCM+FSA )),s +* ((loop I) +* (Start-At 0 ,SCM+FSA )) equal_outside NAT by FUNCT_4:15;
then s +* (I +* (Start-At 0 ,SCM+FSA )), Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),0 equal_outside NAT by AMI_1:13;
hence Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 , Comput (ProgramPart (s +* ((loop I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((loop I) +* (Start-At 0 ,SCM+FSA ))),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