We consider the system 479.

  Alphabet:

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

  Rules:

    apply(/\x.X(x), Y) => X(Y) 
    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]):

  apply(/\x.X(x), Y) >? X(Y) 
  a >? b 

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

The following interpretation satisfies the requirements:

  a = 3 
  apply = \G0y1.3 + y1 + G0(y1) 
  b = 0 

Using this interpretation, the requirements translate to:

  [[apply(/\x._x0(x), _x1)]] = 3 + x1 + F0(x1) > F0(x1) = [[_x0(_x1)]] 
  [[a]] = 3 > 0 = [[b]] 

We can thus remove the following rules:

  apply(/\x.X(x), Y) => X(Y) 
  a => 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.