Example 9 in [ Lucas - IC'06]

Summary: Ex9_Luc06

• TPDB format
• OBJ format
• Negative results:
  1. Lucas'
  2. Ferreira and Ribeiro's transformation
  3. Zantema's transformation
• Positive results:
  1. Giesl and Middeldorp's transformation

Ex9_Luc06 in TPDB format ( Ex9_Luc06.trs):

(VAR X)
(STRATEGY CONTEXTSENSITIVE 
  (f 2)
  (a)
  (b)
)
(RULES 
f(a,X,X) -> f(X,b,b)
b -> a
)

The CS-TRS in OBJ format (file Ex9_Luc06.obj):

obj Ex9_Luc06 is
  sort S .
  op f : S S S -> S [strat (2 0)] .
  op a : -> S .
  op b : -> S .
  vars X : S .
  eq f(a,X,X) = f(X,b,b) .
  eq b = a .
endo

Negative results

1. The µ-termination of Ex9_Luc06 cannot be proved by using Lucas' transformation. The TRS Ex9_Luc06_L:

   	f(X) -> f(b)
   	b -> a
   	
   

   is not terminating (AProVE).
   
2. The µ-termination of Ex9_Luc06 cannot be proved by using Ferreira and Ribeiro's transformation. The TRS Ex9_Luc06_FR:

   						
   	f(n__a,X,X) -> f(activate(X),b,n__b)
   	b -> a
   	a -> n__a
   	b -> n__b
   	activate(n__a) -> a
   	activate(n__b) -> b
   	activate(X) -> X
   
   

   is not terminating (AProVE).
   
3. The µ-termination of Ex9_Luc06 cannot be proved by using Zantema' transformation .

   	f(n__a,X,X) -> f(activate(X),b,n__b)
   	b -> a
   	a -> n__a
   	b -> n__b
   	activate(n__a) -> a
   	activate(n__b) -> b
   	activate(X) -> X
   
   				

   is not terminating (AProVE).

Positive results

1. The µ-termination of Ex9_Luc06 can be proved by using Giesl and Middeldorp's transformation. The TRS Ex9_Luc06_GM:

   	a__f(a,X,X) -> a__f(X,a__b,b)
   	a__b -> a
   	mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
   	mark(b) -> a__b
   	mark(a) -> a
   	a__f(X1,X2,X3) -> f(X1,X2,X3)
   	a__b -> b
   
   

   can be proved terminating by AProVE