We consider the system 461.

  Alphabet:

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

  Rules:

    f(X, X) => a 
    f(X, s(X)) => b 

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

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

Argument functions:

  [[a]] = _|_ 
  [[b]] = _|_ 
  [[s(x_1)]] = x_1 

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

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  f(X, X) > _|_ 
  f(X, X) >= _|_ 

With these choices, we have:

  1] f(X, X) > _|_  because [2], by definition 
  2] f*(X, X) >= _|_  by (Bot) 

  3] f(X, X) >= _|_  by (Bot) 

We can thus remove the following rules:

  f(X, X) => a 

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, s(X)) >? b 

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

Argument functions:

  [[b]] = _|_ 
  [[s(x_1)]] = x_1 

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

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  f(X, X) > _|_ 

With these choices, we have:

  1] f(X, X) > _|_  because [2], by definition 
  2] f*(X, X) >= _|_  by (Bot) 

We can thus remove the following rules:

  f(X, s(X)) => b 

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.