We consider the system 443.

  Alphabet:

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

  Rules:

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

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

Argument functions:

  [[g(x_1)]] = x_1 
  [[h(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(x), Y) > X(X(Y)) 
  X >= X 

With these choices, we have:

  1] f(/\x.X(x), Y) > X(X(Y))  because [2], by definition 
  2] f*(/\x.X(x), Y) >= X(X(Y))  because [3], by (Select) 
  3] X(f*(/\x.X(x), Y)) >= X(X(Y))  because [4], by (Meta) 
  4] f*(/\x.X(x), Y) >= X(Y)  because [5], by (Select) 
  5] X(f*(/\x.X(x), Y)) >= X(Y)  because [6], by (Meta) 
  6] f*(/\x.X(x), Y) >= Y  because [7], by (Select) 
  7] Y >= Y  by (Meta) 

  8] X >= X  by (Meta) 

We can thus remove the following rules:

  f(/\x.X(x), Y) => X(X(Y)) 

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 orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  g = \y0.3 + 3y0 
  h = \y0.y0 

Using this interpretation, the requirements translate to:

  [[g(_x0)]] = 3 + 3x0 > x0 = [[h(_x0)]] 

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.