We consider the system 471.

  Alphabet:

    C : [] --> o 
    F : [o * o * o] --> o 
    g : [o -> o -> o] --> o 

  Rules:

    g(/\x./\y.F(x, y, X(x, y))) => C 
    g(/\x./\y.F(X(x, y), x, y)) => C 
    g(/\x./\y.F(y, X(x, y), x)) => C 

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

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

  g(/\x./\y.F(x, y, X(x, y))) >? C 
  g(/\x./\y.F(X(x, y), x, y)) >? C 
  g(/\x./\y.F(y, X(x, y), x)) >? C 

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

The following interpretation satisfies the requirements:

  C = 0 
  F = \y0y1y2.3 + y0 + y2 + 2y1 
  g = \G0.3 + 3G0(0,0) 

Using this interpretation, the requirements translate to:

  [[g(/\x./\y.F(x, y, _x0(x, y)))]] = 12 + 3F0(0,0) > 0 = [[C]] 
  [[g(/\x./\y.F(_x0(x, y), x, y))]] = 12 + 3F0(0,0) > 0 = [[C]] 
  [[g(/\x./\y.F(y, _x0(x, y), x))]] = 12 + 6F0(0,0) > 0 = [[C]] 

We can thus remove the following rules:

  g(/\x./\y.F(x, y, X(x, y))) => C 
  g(/\x./\y.F(X(x, y), x, y)) => C 
  g(/\x./\y.F(y, X(x, y), x)) => C 

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.