let s be State of ; :: thesis: for I, J being Program of
for k being Element of NAT st I is_closed_on s & I is_halting_on s & k <= LifeSpan (s +* (Initialized (stop I))) holds
Computation (s +* (Initialized (stop I))),k, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),k equal_outside NAT
let I, J be Program of ; :: thesis: for k being Element of NAT st I is_closed_on s & I is_halting_on s & k <= LifeSpan (s +* (Initialized (stop I))) holds
Computation (s +* (Initialized (stop I))),k, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),k equal_outside NAT
let k be Element of NAT ; :: thesis: ( I is_closed_on s & I is_halting_on s & k <= LifeSpan (s +* (Initialized (stop I))) implies Computation (s +* (Initialized (stop I))),k, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),k equal_outside NAT )
set SA0 = Start-At (inspos 0 );
set spI = stop I;
set s1 = s +* (Initialized (stop I));
set s2 = s +* ((I ';' J) +* (Start-At (inspos 0 )));
set n = LifeSpan (s +* (Initialized (stop I)));
assume that
A1: I is_closed_on s and
A2: I is_halting_on s and
A3: k <= LifeSpan (s +* (Initialized (stop I))) ; :: thesis: Computation (s +* (Initialized (stop I))),k, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),k equal_outside NAT
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan (s +* (Initialized (stop I))) implies Computation (s +* (Initialized (stop I))),$1, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),$1 equal_outside NAT );
A4: s +* ((I ';' J) +* (Start-At (inspos 0 ))) = (s +* (I ';' J)) +* (Start-At (inspos 0 )) by FUNCT_4:15
.= (s +* (Start-At (inspos 0 ))) +* (I ';' J) by SCMPDS_4:62 ;
A5: s +* (Initialized (stop I)) = (s +* (stop I)) +* (Start-At (inspos 0 )) by FUNCT_4:15
.= (s +* (Start-At (inspos 0 ))) +* (stop I) by SCMPDS_4:62 ;
A6: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A7: ( m <= LifeSpan (s +* (Initialized (stop I))) implies Computation (s +* (Initialized (stop I))),m, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m equal_outside NAT ) ; :: thesis: S₁[m + 1]
A8: Computation (s +* (Initialized (stop I))),(m + 1) = Following (Computation (s +* (Initialized (stop I))),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* (Initialized (stop I))),m)),(Computation (s +* (Initialized (stop I))),m) ;
A9: IC (Computation (s +* (Initialized (stop I))),m) in dom (stop I) by A1, SCMPDS_6:def 2;
A10: Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),(m + 1) = Following (Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m)),(Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m) ;
assume A11: m + 1 <= LifeSpan (s +* (Initialized (stop I))) ; :: thesis: Computation (s +* (Initialized (stop I))),(m + 1), Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),(m + 1) equal_outside NAT
then m < LifeSpan (s +* (Initialized (stop I))) by NAT_1:13;
then A12: IC (Computation (s +* (Initialized (stop I))),m) in dom I by A1, A2, SCMPDS_6:40;
then A13: IC (Computation (s +* (Initialized (stop I))),m) in dom (I ';' J) by FUNCT_4:13;
CurInstr (Computation (s +* (Initialized (stop I))),m) = (s +* (Initialized (stop I))) . (IC (Computation (s +* (Initialized (stop I))),m)) by AMI_1:54
.= (stop I) . (IC (Computation (s +* (Initialized (stop I))),m)) by A5, A9, FUNCT_4:14
.= I . (IC (Computation (s +* (Initialized (stop I))),m)) by A12, SCMPDS_4:37
.= (I ';' J) . (IC (Computation (s +* (Initialized (stop I))),m)) by A12, SCMPDS_4:37
.= (s +* ((I ';' J) +* (Start-At (inspos 0 )))) . (IC (Computation (s +* (Initialized (stop I))),m)) by A4, A13, FUNCT_4:14
.= (Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m) . (IC (Computation (s +* (Initialized (stop I))),m)) by AMI_1:54
.= CurInstr (Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m) by A7, A11, AMI_1:121, NAT_1:13 ;
hence Computation (s +* (Initialized (stop I))),(m + 1), Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),(m + 1) equal_outside NAT by A7, A11, A10, A8, NAT_1:13, SCMPDS_4:15; :: thesis: verum

end;

A14: Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),0 = s +* ((I ';' J) +* (Start-At (inspos 0 ))) by AMI_1:13;
A15: Computation (s +* (Initialized (stop I))),0 = s +* (Initialized (stop I)) by AMI_1:13;
A16: s +* (Start-At (inspos 0 )),(s +* (Start-At (inspos 0 ))) +* (I ';' J) equal_outside NAT by AMI_1:120;
(s +* (Start-At (inspos 0 ))) +* (stop I),s +* (Start-At (inspos 0 )) equal_outside NAT by AMI_1:120, FUNCT_7:28;
then A17: S₁[ 0 ] by A5, A4, A16, A15, A14, FUNCT_7:29;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A17, A6);
hence Computation (s +* (Initialized (stop I))),k, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),k equal_outside NAT by A3; :: thesis: verum