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 the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] ex c(/\f.X(f)) =#> X(ex)   
    1] ex c(/\f.X(f)) =#> ex# {X : 1}  

  Rules R_0:

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

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 0, 1 
    * 1 :  

This graph has the following strongly connected components:

  P_1:

    ex c(/\f.X(f)) =#> X(ex)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f).

Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, formative).

This combination (P_1, R_0) has no formative rules!  We will name the empty set of rules:R_1.

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_1, R_1, minimal, formative).

Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

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

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  c = \G0.3 + G0(\y1.0) 
  ex = \y0.0 

Using this interpretation, the requirements translate to:

  [[ex c(/\f._x0(f))]] = 3 + F0(\y0.0) > F0(\y0.0) = [[_x0(ex)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_1) by ({}, R_1).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ 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.