We consider the system 447.

  Alphabet:

    a : [] --> A 
    b : [] --> A 
    f : [A] --> A 
    g : [A] --> A 
    h : [A] --> A 

  Rules:

    f(X) => g(X) 
    g(X) => h(X) 
    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(X) >? g(X) 
  g(X) >? h(X) 
  a >? b 

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

Argument functions:

  [[b]] = _|_ 

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

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

  f(X) >= g(X) 
  g(X) >= h(X) 
  a > _|_ 

With these choices, we have:

  1] f(X) >= g(X)  because f = g, f in Mul and [2], by (Fun) 
  2] X >= X  by (Meta) 

  3] g(X) >= h(X)  because g = h, g in Mul and [4], by (Fun) 
  4] X >= X  by (Meta) 

  5] a > _|_  because [6], by definition 
  6] a* >= _|_  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(X) >? g(X) 
  g(X) >? h(X) 

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

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

With these choices, we have:

  1] f(X) > g(X)  because [2], by definition 
  2] f*(X) >= g(X)  because f > g and [3], by (Copy) 
  3] f*(X) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 

  5] g(X) >= h(X)  because g = h, g in Mul and [6], by (Fun) 
  6] X >= X  by (Meta) 

We can thus remove the following rules:

  f(X) => g(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]):

  g(X) >? h(X) 

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

Argument functions:

  [[h(x_1)]] = x_1 

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

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

  g(X) > X 

With these choices, we have:

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

We can thus remove the following rules:

  g(X) => h(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.