We consider the system 449.

  Alphabet:

    a : [] --> o 
    f : [o] --> o 
    g : [o] --> o 
    h : [o] --> o 
    hprime : [o -> o] --> o 

  Rules:

    f(X) => X 
    hprime(/\x.h(f(g(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) >? X 
  hprime(/\x.h(f(g(x)))) >? a 

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

Argument functions:

  [[a]] = _|_ 
  [[f(x_1)]] = x_1 
  [[g(x_1)]] = x_1 
  [[h(x_1)]] = x_1 

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

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

  X >= X 
  hprime(/\x.x) > _|_ 

With these choices, we have:

  1] X >= X  by (Meta) 

  2] hprime(/\x.x) > _|_  because [3], by definition 
  3] hprime*(/\x.x) >= _|_  by (Bot) 

We can thus remove the following rules:

  hprime(/\x.h(f(g(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) >? X 

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

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

With these choices, we have:

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

We can thus remove the following rules:

  f(X) => 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.