We consider the system inlamb.

  Alphabet:

    a : [] --> B 
    b : [] --> B 
    c : [] --> B 
    f : [A -> B] --> B 

  Rules:

    f(/\x.a) => b 
    b => f(/\x.c) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

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.a) >? b 
  b >? f(/\x.c) 

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

Argument functions:

  [[c]] = _|_ 

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

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

  f(/\x.a) > b 
  b >= f(/\x._|_) 

With these choices, we have:

  1] f(/\x.a) > b  because [2], by definition 
  2] f*(/\x.a) >= b  because [3], by (Select) 
  3] a >= b  because a = b, by (Fun) 

  4] b >= f(/\x._|_)  because [5], by (Star) 
  5] b* >= f(/\x._|_)  because b > f and [6], by (Copy) 
  6] b* >= /\y._|_  because [7], by (F-Abs) 
  7] b*(x) >= _|_  by (Bot) 

We can thus remove the following rules:

  f(/\x.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]):

  b >? f(/\x.c) 

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

Argument functions:

  [[c]] = _|_ 

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

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

  b > f(/\x._|_) 

With these choices, we have:

  1] b > f(/\x._|_)  because [2], by definition 
  2] b* >= f(/\x._|_)  because b > f and [3], by (Copy) 
  3] b* >= /\y._|_  because [4], by (F-Abs) 
  4] b*(x) >= _|_  by (Bot) 

We can thus remove the following rules:

  b => f(/\x.c) 

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.