We consider the system 461.

  Alphabet:

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

  Rules:

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

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

The following interpretation satisfies the requirements:

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

Using this interpretation, the requirements translate to:

  [[f(_x0, _x0)]] = 3 + 4x0 > 0 = [[a]] 
  [[f(_x0, s(_x0))]] = 12 + 4x0 > 0 = [[b]] 

We can thus remove the following rules:

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

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.