We consider the system 453.

  Alphabet:

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

  Rules:

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

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

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

Argument functions:

  [[b]] = _|_ 
  [[c]] = _|_ 

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

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

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

With these choices, we have:

  1] f(a, X) >= f(_|_, X)  because f in Mul, [2] and [3], by (Fun) 
  2] a >= _|_  by (Bot) 
  3] X >= X  by (Meta) 

  4] a > _|_  because [5], by definition 
  5] a* >= _|_  by (Bot) 

  6] _|_ >= _|_  by (Bot) 

We can thus remove the following rules:

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

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

Argument functions:

  [[c]] = _|_ 

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

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

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

With these choices, we have:

  1] f(a, X) >= f(b, X)  because [2], by (Star) 
  2] f*(a, X) >= f(b, X)  because f in Mul, [3] and [5], by (Stat) 
  3] a > b  because [4], by definition 
  4] a* >= b  because a > b, by (Copy) 
  5] X >= X  by (Meta) 

  6] b > _|_  because [7], by definition 
  7] b* >= _|_  by (Bot) 

We can thus remove the following rules:

  b => 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]):

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

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

Argument functions:

  [[b]] = _|_ 
  [[f(x_1, x_2)]] = f(x_2, x_1) 

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

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

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

With these choices, we have:

  1] f(a, X) > f(_|_, X)  because [2], by definition 
  2] f*(a, X) >= f(_|_, X)  because [3], [5], [6] and [7], by (Stat) 
  3] a > _|_  because [4], by definition 
  4] a* >= _|_  by (Bot) 
  5] X >= X  by (Meta) 
  6] f*(a, X) >= _|_  by (Bot) 
  7] f*(a, X) >= X  because [5], by (Select) 

We can thus remove the following rules:

  f(a, X) => f(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.