let s be State of SCMPDS ; :: thesis: for I, J being Program of SCMPDS
for k being Element of NAT st I is_closed_on s & I is_halting_on s & k <= LifeSpan (s +* (Initialized (stop I))) holds
Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),k equal_outside NAT
let I, J be Program of SCMPDS ; :: 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
Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),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 Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),k equal_outside NAT )
set SA0 = Start-At 0 ,SCMPDS ;
set spI = stop I;
set s1 = s +* (Initialized (stop I));
set s2 = s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ));
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: Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),k equal_outside NAT
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan (s +* (Initialized (stop I))) implies Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),$1, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),$1 equal_outside NAT );
A4: s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )) = (s +* (I ';' J)) +* (Start-At 0 ,SCMPDS ) by FUNCT_4:15
.= (s +* (Start-At 0 ,SCMPDS )) +* (I ';' J) by SCMPDS_4:62 ;
A5: s +* (Initialized (stop I)) = (s +* (stop I)) +* (Start-At 0 ,SCMPDS ) by FUNCT_4:15
.= (s +* (Start-At 0 ,SCMPDS )) +* (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 Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m equal_outside NAT ) ; :: thesis: S₁[m + 1]
T: ProgramPart (s +* (Initialized (stop I))) = ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) by AMI_1:144;
A8: Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(m + 1) = Following (ProgramPart (s +* (Initialized (stop I)))),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) by T ;
A9: IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) in dom (stop I) by A1, SCMPDS_6:def 2;
T: ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))) = ProgramPart (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m) by AMI_1:144;
A10: Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),(m + 1) = Following (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m)),(Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m)),(Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m) by T ;
assume A11: m + 1 <= LifeSpan (s +* (Initialized (stop I))) ; :: thesis: Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(m + 1), Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),(m + 1) equal_outside NAT
then m < LifeSpan (s +* (Initialized (stop I))) by NAT_1:13;
then A12: IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) in dom I by A1, A2, SCMPDS_6:40;
then A13: IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) in dom (I ';' J) by FUNCT_4:13;
Y: (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) /. (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) = (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by AMI_1:150;
Z: (ProgramPart (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m)) /. (IC (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m)) = (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m) . (IC (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m)) by AMI_1:150;
CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m) = (s +* (Initialized (stop I))) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by AMI_1:54, Y
.= (stop I) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by A5, A9, FUNCT_4:14
.= I . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by A12, SCMPDS_4:37
.= (I ';' J) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by A12, SCMPDS_4:37
.= (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by A4, A13, FUNCT_4:14
.= (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),m)) by AMI_1:54
.= CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m)),(Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m) by A7, A11, AMI_1:121, NAT_1:13, Z ;
hence Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(m + 1), Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),(m + 1) equal_outside NAT by A7, A11, A10, A8, NAT_1:13, SCMPDS_4:15; :: thesis: verum

end;

A14: Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),0 = s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )) by AMI_1:13;
A15: Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),0 = s +* (Initialized (stop I)) by AMI_1:13;
A16: s +* (Start-At 0 ,SCMPDS ),(s +* (Start-At 0 ,SCMPDS )) +* (I ';' J) equal_outside NAT by AMI_1:120;
(s +* (Start-At 0 ,SCMPDS )) +* (stop I),s +* (Start-At 0 ,SCMPDS ) 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 Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),k equal_outside NAT by A3; :: thesis: verum