We consider the system 455.

  Alphabet:

    pi : [term * term] --> term 
    pi1 : [term] --> term 
    pi2 : [term] --> term 

  Rules:

    pi1(pi(X, Y)) => X 
    pi2(pi(X, Y)) => Y 
    pi(pi1(X), pi2(X)) => X 

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

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

  pi1(pi(X, Y)) >? X 
  pi2(pi(X, Y)) >? Y 
  pi(pi1(X), pi2(X)) >? X 

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

The following interpretation satisfies the requirements:

  pi = \y0y1.3 + y0 + y1 
  pi1 = \y0.3 + y0 
  pi2 = \y0.3 + y0 

Using this interpretation, the requirements translate to:

  [[pi1(pi(_x0, _x1))]] = 6 + x0 + x1 > x0 = [[_x0]] 
  [[pi2(pi(_x0, _x1))]] = 6 + x0 + x1 > x1 = [[_x1]] 
  [[pi(pi1(_x0), pi2(_x0))]] = 9 + 2x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  pi1(pi(X, Y)) => X 
  pi2(pi(X, Y)) => Y 
  pi(pi1(X), pi2(X)) => 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.