We consider the system 453.

  Alphabet:

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

  Rules:

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

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

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

The following interpretation satisfies the requirements:

  a = 3 
  b = 0 
  c = 0 
  f = \y0y1.y1 + 3y0 

Using this interpretation, the requirements translate to:

  [[f(a, _x0)]] = 9 + x0 > x0 = [[f(b, _x0)]] 
  [[a]] = 3 > 0 = [[b]] 
  [[b]] = 0 >= 0 = [[c]] 

We can thus remove the following rules:

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

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

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

  b >? c 

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

The following interpretation satisfies the requirements:

  b = 3 
  c = 0 

Using this interpretation, the requirements translate to:

  [[b]] = 3 > 0 = [[c]] 

We can thus remove the following rules:

  b => c 

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.