We consider the system 449.

  Alphabet:

    a : [] --> o 
    f : [o] --> o 
    g : [o] --> o 
    h : [o] --> o 
    hprime : [o -> o] --> o 

  Rules:

    f(X) => X 
    hprime(/\x.h(f(g(x)))) => a 

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 
  hprime(/\x.h(f(g(x)))) >? a 

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

The following interpretation satisfies the requirements:

  a = 0 
  f = \y0.3 + 3y0 
  g = \y0.3 + y0 
  h = \y0.3 + 3y0 
  hprime = \G0.3 + 3G0(0) 

Using this interpretation, the requirements translate to:

  [[f(_x0)]] = 3 + 3x0 > x0 = [[_x0]] 
  [[hprime(/\x.h(f(g(x))))]] = 120 > 0 = [[a]] 

We can thus remove the following rules:

  f(X) => X 
  hprime(/\x.h(f(g(x)))) => a 

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.