let s be State of SCM+FSA ; :: thesis: for I, J being Program of SCM+FSA st I is_pseudo-closed_on s holds
for k being Element of NAT st k <= pseudo-LifeSpan s,I 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
let I, J be Program of SCM+FSA ; :: thesis: ( I is_pseudo-closed_on s implies for k being Element of NAT st k <= pseudo-LifeSpan s,I 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 ));
set s2 = s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ));
set I1 = I +* (Start-At 0 ,SCM+FSA );
set I2 = (I ';' J) +* (Start-At 0 ,SCM+FSA );
defpred S₁[ Nat] means ( $1 <= pseudo-LifeSpan s,I 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 );
assume A1: I is_pseudo-closed_on s ; :: thesis: for k being Element of NAT st k <= pseudo-LifeSpan s,I 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: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
thus S₁[k + 1] :: thesis: verum

proof

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 ))),k) by AMI_1:144;
A4: Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),(k + 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 ))),k) 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 ))),k)),(Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (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) by T ;
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 ))),k) by AMI_1:144;
A5: Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 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 ))),k) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) by T ;
A6: dom I c= dom (I ';' J) by SCMFSA6A:56;
( I ';' J c= (I ';' J) +* (Start-At 0 ,SCM+FSA ) & (I ';' J) +* (Start-At 0 ,SCM+FSA ) c= s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )) ) by Th9, FUNCT_4:26;
then I ';' J c= s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )) by 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 ))),k by AMI_1:81;
A8: k + 0 < k + 1 by XREAL_1:8;
assume A9: k + 1 <= pseudo-LifeSpan s,I ; :: thesis: Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1), Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),(k + 1) equal_outside NAT
then A10: k < pseudo-LifeSpan s,I by A8, XXREAL_0:2;
then A11: IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom I by A1, Th31;
Y: (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) /. (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by AMI_1:150;
Z: (ProgramPart (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),k)) /. (IC (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (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) . (IC (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),k)) by AMI_1:150;
( I c= I +* (Start-At 0 ,SCM+FSA ) & I +* (Start-At 0 ,SCM+FSA ) c= s +* (I +* (Start-At 0 ,SCM+FSA )) ) by Th9, FUNCT_4:26;
then I c= s +* (I +* (Start-At 0 ,SCM+FSA )) by XBOOLE_1:1;
then I c= Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k by AMI_1:81;
then A12: CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) = I . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by A11, GRFUNC_1:8, Y;
then I . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) <> halt SCM+FSA by A1, A10, Th31;
then CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) = (I ';' J) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by A11, A12, SCMFSA6A:54
.= (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),k) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by A7, A11, A6, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (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) by A3, A9, A8, AMI_1:121, XXREAL_0:2, Z ;
hence Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1), Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),(k + 1) equal_outside NAT by A3, A9, A8, A5, A4, SCMFSA6A:32, XXREAL_0:2; :: thesis: verum

end;

end;

A13: S₁[ 0 ]

proof

assume 0 <= pseudo-LifeSpan s,I ; :: thesis: Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 , Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),0 equal_outside NAT
s +* (I +* (Start-At 0 ,SCM+FSA )),s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )) equal_outside NAT by Th14;
then Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ,s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )) 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 +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),0 equal_outside NAT by AMI_1:13; :: thesis: verum

end;

thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A13, A2); :: thesis: verum