We consider the system 448.

  Alphabet:

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

  Rules:

    f(a, X) => X 
    f(X, b) => 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]):

  f(a, X) >? X 
  f(X, b) >? X 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

We choose Lex = {} and Mul = {a, b, f}, and the following precedence: b > f > a

With these choices, we have:

  1] f(a, X) > X  because [2], by definition 
  2] f*(a, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] f(X, b) >= X  because [5], by (Star) 
  5] f*(X, b) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 

We can thus remove the following rules:

  f(a, 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]):

  f(X, b) >? X 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

We choose Lex = {} and Mul = {b, f}, and the following precedence: b > f

With these choices, we have:

  1] f(X, b) > X  because [2], by definition 
  2] f*(X, b) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

We can thus remove the following rules:

  f(X, b) => 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.