let s be State of SCMPDS ; :: thesis: for I being parahalting Program of SCMPDS
for J being Program of SCMPDS
for k being Element of NAT st k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) holds
Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (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 be parahalting Program of SCMPDS ; :: thesis: for J being Program of SCMPDS
for k being Element of NAT st k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) holds
Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),k equal_outside NAT
let J be Program of SCMPDS ; :: thesis: for k being Element of NAT st k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) holds
Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (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: ( k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) implies Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (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 = (Initialize s) +* (stop I);
set s2 = s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ));
set n = LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I));
I1: s +* (Initialize (stop I)) = (Initialize s) +* (stop I) by SCMPDS_4:5;
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) implies Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),$1, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),$1 equal_outside NAT );
A2: 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 ;
A3: Initialize (stop I) c= (Initialize s) +* (stop I) by I1, FUNCT_4:26;
A4: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A5: ( m <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) implies Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (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 +* ((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_1:123;
A6: 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 ;
A7: IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m) in dom (stop I) by A3, SCMPDS_4:def 9;
T: ProgramPart ((Initialize s) +* (stop I)) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m) by AMI_1:123;
A8: Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(m + 1) = Following (ProgramPart ((Initialize s) +* (stop I))),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m) by T ;
assume A9: m + 1 <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) ; :: thesis: Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (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 A10: IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m) = IC (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),m) by A5, COMPOS_1:24, NAT_1:13;
m < LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) by A9, NAT_1:13;
then A11: IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m) in dom I by Th28;
then A12: IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m) in dom (I ';' J) by FUNCT_4:13;
Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)) = (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)) by COMPOS_1:38;
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 COMPOS_1:38;
CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m) = ((Initialize s) +* (stop I)) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)) by Y, AMI_1:54
.= (stop I) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)) by A7, FUNCT_4:14
.= I . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)) by A11, SCMPDS_4:37
.= (I ';' J) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)) by A11, SCMPDS_4:37
.= (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),m)) by A2, A12, FUNCT_4:14
.= 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 A10, Z, AMI_1:54 ;
hence Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (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 A5, A9, A8, A6, NAT_1:13, SCMPDS_4:15; :: thesis: verum

end;

A13: 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;
A14: Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),0 = (Initialize s) +* (stop I) by AMI_1:13;
A15: s +* (Start-At 0 ,SCMPDS ),(s +* (Start-At 0 ,SCMPDS )) +* (I ';' J) equal_outside NAT by FUNCT_7:132;
(s +* (Start-At 0 ,SCMPDS )) +* (stop I),s +* (Start-At 0 ,SCMPDS ) equal_outside NAT by FUNCT_7:28, FUNCT_7:132;
then A16: S₁[ 0 ] by A2, A15, A14, A13, FUNCT_7:29;
A17: for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A16, A4);
assume k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) ; :: thesis: Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),k equal_outside NAT
hence Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCMPDS )))),(s +* ((I ';' J) +* (Start-At 0 ,SCMPDS ))),k equal_outside NAT by A17; :: thesis: verum