We consider the system 730.

  Alphabet:

    get : [S] --> V 
    put : [S * V] --> S 

  Rules:

    put(X, get(X)) => X 
    get(put(X, Y)) => Y 
    put(put(X, Y), Y) => put(X, Y) 

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

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

  put(X, get(X)) >? X 
  get(put(X, Y)) >? Y 
  put(put(X, Y), Y) >? put(X, Y) 

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

The following interpretation satisfies the requirements:

  get = \y0.3 + y0 
  put = \y0y1.3 + y1 + 2y0 

Using this interpretation, the requirements translate to:

  [[put(_x0, get(_x0))]] = 6 + 3x0 > x0 = [[_x0]] 
  [[get(put(_x0, _x1))]] = 6 + x1 + 2x0 > x1 = [[_x1]] 
  [[put(put(_x0, _x1), _x1)]] = 9 + 3x1 + 4x0 > 3 + x1 + 2x0 = [[put(_x0, _x1)]] 

We can thus remove the following rules:

  put(X, get(X)) => X 
  get(put(X, Y)) => Y 
  put(put(X, Y), Y) => put(X, Y) 

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.