We consider the system 479.

  Alphabet:

    a : [] --> o 
    apply : [o -> o * o] --> o 
    b : [] --> o 

  Rules:

    apply(/\x.X(x), Y) => X(Y) 
    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]):

  apply(/\x.X(x), Y) >? X(Y) 
  a >? b 

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

Argument functions:

  [[b]] = _|_ 

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

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

  apply(/\x.X(x), Y) >= X(Y) 
  a > _|_ 

With these choices, we have:

  1] apply(/\x.X(x), Y) >= X(Y)  because [2], by (Star) 
  2] apply*(/\x.X(x), Y) >= X(Y)  because [3], by (Select) 
  3] X(apply*(/\x.X(x), Y)) >= X(Y)  because [4], by (Meta) 
  4] apply*(/\x.X(x), Y) >= Y  because [5], by (Select) 
  5] Y >= Y  by (Meta) 

  6] a > _|_  because [7], by definition 
  7] 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]):

  apply(/\x.X(x), Y) >? X(Y) 

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

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

With these choices, we have:

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

We can thus remove the following rules:

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

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.