We consider the system 473.

  Alphabet:

    dots : [] --> o 
    f : [o -> o] --> o 
    g : [(o -> o) -> o] --> o 

  Rules:

    g(/\h.X(h)) => X(/\x.X(/\y.x)) 
    f(/\x.X(x)) => dots 

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(/\h.X(h)) >? X(/\x.X(/\y.x)) 
  f(/\x.X(x)) >? dots 

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

Argument functions:

  [[dots]] = _|_ 

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

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

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

With these choices, we have:

  1] g(/\f.X(f)) >= X(/\x.X(/\y.x))  because [2], by (Star) 
  2] g*(/\f.X(f)) >= X(/\x.X(/\y.x))  because [3], by (Select) 
  3] X(g*(/\f.X(f))) >= X(/\x.X(/\y.x))  because [4], by (Meta) 
  4] g*(/\f.X(f)) >= /\y.X(/\z.y)  because [5], by (F-Abs) 
  5] g*(/\f.X(f), x) >= X(/\y.x)  because [6], by (Select) 
  6] X(g*(/\f.X(f), x)) >= X(/\y.x)  because [7], by (Meta) 
  7] g*(/\f.X(f), x) >= /\z.x  because [8], by (F-Abs) 
  8] g*(/\f.X(f), x, y) >= x  because [9], by (Select) 
  9] x >= x  by (Var) 

  10] f(/\x.X(x)) > _|_  because [11], by definition 
  11] f*(/\x.X(x)) >= _|_  by (Bot) 

We can thus remove the following rules:

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

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(/\h.X(h)) >? X(/\x.X(/\y.x)) 

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

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

With these choices, we have:

  1] g(/\f.X(f)) > X(/\x.X(/\y.x))  because [2], by definition 
  2] g*(/\f.X(f)) >= X(/\x.X(/\y.x))  because [3], by (Select) 
  3] X(g*(/\f.X(f))) >= X(/\x.X(/\y.x))  because [4], by (Meta) 
  4] g*(/\f.X(f)) >= /\y.X(/\z.y)  because [5], by (F-Abs) 
  5] g*(/\f.X(f), x) >= X(/\y.x)  because [6], by (Select) 
  6] X(g*(/\f.X(f), x)) >= X(/\y.x)  because [7], by (Meta) 
  7] g*(/\f.X(f), x) >= /\z.x  because [8], by (F-Abs) 
  8] g*(/\f.X(f), x, y) >= x  because [9], by (Select) 
  9] x >= x  by (Var) 

We can thus remove the following rules:

  g(/\h.X(h)) => X(/\x.X(/\y.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.