We consider the system 448.

  Alphabet:

    a : [] --> o 
    b : [] --> o 
    f : [o * o] --> o 

  Rules:

    f(a, X) => X 
    f(X, b) => X 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  f(a, X) >? X 
  f(X, b) >? X 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  a = 3 
  b = 3 
  f = \y0y1.3 + y0 + y1 

Using this interpretation, the requirements translate to:

  [[f(a, _x0)]] = 6 + x0 > x0 = [[_x0]] 
  [[f(_x0, b)]] = 6 + x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  f(a, X) => X 
  f(X, b) => X 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.