We consider the system 445.

  Alphabet:

    f : [(A -> A) -> A] --> A 

  Rules:

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

We can thus remove the following rules:

  f(/\g.X(g)) => X(/\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.