We consider the system 452.

  Alphabet:

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

  Rules:

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

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

Argument functions:

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

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

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

  f(/\x.X(x)) >= X(X(_|_)) 
  g(X) > X 

With these choices, we have:

  1] f(/\x.X(x)) >= X(X(_|_))  because [2], by (Star) 
  2] f*(/\x.X(x)) >= X(X(_|_))  because [3], by (Select) 
  3] X(f*(/\x.X(x))) >= X(X(_|_))  because [4], by (Meta) 
  4] f*(/\x.X(x)) >= X(_|_)  because [5], by (Select) 
  5] X(f*(/\x.X(x))) >= X(_|_)  because [6], by (Meta) 
  6] f*(/\x.X(x)) >= _|_  by (Bot) 

  7] g(X) > X  because [8], by definition 
  8] g*(X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 

We can thus remove the following rules:

  g(X) => h(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.X(x)) >? X(X(a)) 

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

Argument functions:

  [[a]] = _|_ 

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(x)) > X(X(_|_)) 

With these choices, we have:

  1] f(/\x.X(x)) > X(X(_|_))  because [2], by definition 
  2] f*(/\x.X(x)) >= X(X(_|_))  because [3], by (Select) 
  3] X(f*(/\x.X(x))) >= X(X(_|_))  because [4], by (Meta) 
  4] f*(/\x.X(x)) >= X(_|_)  because [5], by (Select) 
  5] X(f*(/\x.X(x))) >= X(_|_)  because [6], by (Meta) 
  6] f*(/\x.X(x)) >= _|_  by (Bot) 

We can thus remove the following rules:

  f(/\x.X(x)) => X(X(a)) 

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.