We consider the system 09ex.

  Alphabet:

    c : [] --> ((C -> L) -> L) -> C 
    d : [] --> C 
    ex : [] --> C -> L 
    nil : [] --> L 

  Rules:

    ex d => nil 
    ex (c (/\f.g f)) => g ex 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    c : [(C -> L) -> L] --> C 
    d : [] --> C 
    ex : [] --> C -> L 
    nil : [] --> L 
    ~AP1 : [(C -> L) -> L * C -> L] --> L 

  Rules:

    ex d => nil 
    ex c(/\f.~AP1(F, f)) => ~AP1(F, ex) 
    ~AP1(F, G) => F G 

Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:

  Alphabet:

    c : [(C -> L) -> L] --> C 
    d : [] --> C 
    ex : [] --> C -> L 
    nil : [] --> L 

  Rules:

    ex d => nil 
    ex c(/\f.X(f)) => X(ex) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  ex d >? nil 
  ex c(/\f.X(f)) >? X(ex) 

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

Argument functions:

  [[ex]] = _|_ 
  [[nil]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, c, d}, and the following precedence: c > @_{o -> o} > d

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

  @_{o -> o}(_|_, d) > _|_ 
  @_{o -> o}(_|_, c(/\f.X(f))) > X(_|_) 

With these choices, we have:

  1] @_{o -> o}(_|_, d) > _|_  because [2], by definition 
  2] @_{o -> o}*(_|_, d) >= _|_  by (Bot) 

  3] @_{o -> o}(_|_, c(/\f.X(f))) > X(_|_)  because [4], by definition 
  4] @_{o -> o}*(_|_, c(/\f.X(f))) >= X(_|_)  because [5], by (Select) 
  5] c(/\f.X(f)) >= X(_|_)  because [6], by (Star) 
  6] c*(/\f.X(f)) >= X(_|_)  because [7], by (Select) 
  7] X(c*(/\f.X(f))) >= X(_|_)  because [8], by (Meta) 
  8] c*(/\f.X(f)) >= _|_  by (Bot) 

We can thus remove the following rules:

  ex d => nil 
  ex c(/\f.X(f)) => X(ex) 

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 +++

[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.