We consider the system 04arrow.

  Alphabet:

    and : [c * c] --> c 
    arrow : [t * t] --> t 
    lessthan : [t * t] --> c 

  Rules:

    lessthan(arrow(x, y), arrow(z, u)) => and(lessthan(z, x), lessthan(y, u)) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

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

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

  lessthan(arrow(X, Y), arrow(Z, U)) >? and(lessthan(Z, X), lessthan(Y, U)) 

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

The following interpretation satisfies the requirements:

  and = Lam[y0;y1].y0 + y1 
  arrow = Lam[y0;y1].3 + 3*y0 + 3*y1 
  lessthan = Lam[y0;y1].2*y1 + 3*y0 

Using this interpretation, the requirements translate to:

  [[lessthan(arrow(_x0, _x1), arrow(_x2, _x3))]] = 15 + 6*x2 + 6*x3 + 9*x0 + 9*x1 > 2*x0 + 2*x3 + 3*x1 + 3*x2 = [[and(lessthan(_x2, _x0), lessthan(_x1, _x3))]] 

We can thus remove the following rules:

  lessthan(arrow(X, Y), arrow(Z, U)) => and(lessthan(Z, X), lessthan(Y, U)) 

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.