We consider the system 447.

  Alphabet:

    a : [] --> A 
    b : [] --> A 
    f : [A] --> A 
    g : [A] --> A 
    h : [A] --> A 

  Rules:

    f(X) => g(X) 
    g(X) => h(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]):

  f(X) >? g(X) 
  g(X) >? h(X) 
  a >? b 

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

The following interpretation satisfies the requirements:

  a = 3 
  b = 0 
  f = \y0.3 + 3y0 
  g = \y0.y0 
  h = \y0.y0 

Using this interpretation, the requirements translate to:

  [[f(_x0)]] = 3 + 3x0 > x0 = [[g(_x0)]] 
  [[g(_x0)]] = x0 >= x0 = [[h(_x0)]] 
  [[a]] = 3 > 0 = [[b]] 

We can thus remove the following rules:

  f(X) => g(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]):

  g(X) >? h(X) 

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

The following interpretation satisfies the requirements:

  g = \y0.3 + 3y0 
  h = \y0.y0 

Using this interpretation, the requirements translate to:

  [[g(_x0)]] = 3 + 3x0 > x0 = [[h(_x0)]] 

We can thus remove the following rules:

  g(X) => h(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.