let s be State of SCM+FSA ; :: thesis: for I being paraclosed Program of SCM+FSA st ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on s +* (I +* (Start-At 0 ,SCM+FSA )) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= 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 ))),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),k equal_outside NAT
set SA0 = Start-At 0 ,SCM+FSA ;
let I be paraclosed Program of SCM+FSA ; :: thesis: ( ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on s +* (I +* (Start-At 0 ,SCM+FSA )) implies for J being Program of SCM+FSA
for k being Element of NAT st k <= 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 ))),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),k equal_outside NAT )
assume A1: ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on s +* (I +* (Start-At 0 ,SCM+FSA )) ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT st k <= 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 ))),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),k equal_outside NAT
set s1 = s +* (I +* (Start-At 0 ,SCM+FSA ));
let J be Program of SCM+FSA ; :: thesis: for k being Element of NAT st k <= 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 ))),k, Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),k equal_outside NAT
A2: I +* (Start-At 0 ,SCM+FSA ) c= s +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
set s2 = s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ));
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 +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),$1 equal_outside NAT );
A3: (I ';' J) +* (Start-At 0 ,SCM+FSA ) c= s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
A4: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

dom (I ';' J) = (dom (Directed I)) \/ (dom (ProgramPart (Relocated J,(card I)))) by FUNCT_4:def 1
.= (dom I) \/ (dom (ProgramPart (Relocated J,(card I)))) by FUNCT_4:105 ;
then A5: dom I c= dom (I ';' J) by XBOOLE_1:7;
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 +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m equal_outside NAT ) ; :: thesis: S₁[m + 1]
dom (I ';' J) misses dom (Start-At 0 ,SCM+FSA ) by SF_MASTR:64;
then I ';' J c= (I ';' J) +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:33;
then I ';' J c= s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )) by A3, XBOOLE_1:1;
then A7: I ';' J c= Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (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 +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m) by AMI_1:123;
A9: Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),(m + 1) = Following (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (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 A2, Def2;
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;
TX: 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;
dom I misses dom (Start-At 0 ,SCM+FSA ) by SF_MASTR:64;
then I c= I +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:33;
then I c= s +* (I +* (Start-At 0 ,SCM+FSA )) by A2, 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, TX, 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 +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),(m + 1) equal_outside NAT
then A13: IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) = IC (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m) by A6, COMPOS_1:24, NAT_1:13;
Y: (ProgramPart (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m)) /. (IC (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m)) = (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m) . (IC (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m)) by COMPOS_1:38;
m < LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) by A12, 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 A1, A11, AMI_1:def 46;
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) = (I ';' J) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) by A10, A11, TX, SCMFSA6A:54
.= CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),m) by A13, A10, A7, A5, Y, GRFUNC_1:8 ;
hence Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(m + 1), Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),(m + 1) equal_outside NAT by A6, A12, A8, A9, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

A14: ( Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 = s +* (I +* (Start-At 0 ,SCM+FSA )) & Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),0 = s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )) ) by AMI_1:13;
A15: ( (s +* (Start-At 0 ,SCM+FSA )) +* I,s +* (Start-At 0 ,SCM+FSA ) equal_outside NAT & s +* (Start-At 0 ,SCM+FSA ),(s +* (Start-At 0 ,SCM+FSA )) +* (I ';' J) equal_outside NAT ) by FUNCT_7:28, FUNCT_7:132;
A16: s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )) = (s +* (I ';' J)) +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:15
.= (s +* (Start-At 0 ,SCM+FSA )) +* (I ';' J) by Th14 ;
s +* (I +* (Start-At 0 ,SCM+FSA )) = (s +* I) +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:15
.= (s +* (Start-At 0 ,SCM+FSA )) +* I by Th14 ;
then A17: S₁[ 0 ] by A16, A15, A14, FUNCT_7:29;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A17, A4); :: thesis: verum