We consider the system AotoYamada_05__002.

  Alphabet:

    cons : [b * c] --> c 
    false : [] --> a 
    filter : [b -> a * c] --> c 
    filtersub : [a * b -> a * c] --> c 
    nil : [] --> c 
    true : [] --> a 

  Rules:

    filter(f, nil) => nil 
    filter(f, cons(x, y)) => filtersub(f x, f, cons(x, y)) 
    filtersub(true, f, cons(x, y)) => cons(x, filter(f, y)) 
    filtersub(false, f, cons(x, y)) => filter(f, y) 

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

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]).

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

  Dependency Pairs P_0:

    0] filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y))   
    1] filtersub#(true, F, cons(X, Y)) =#> filter#(F, Y)   
    2] filtersub#(false, F, cons(X, Y)) =#> filter#(F, Y)   

  Rules R_0:

    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filtersub(F X, F, cons(X, Y)) 
    filtersub(true, F, cons(X, Y)) => cons(X, filter(F, Y)) 
    filtersub(false, F, cons(X, Y)) => filter(F, Y) 

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

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

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  filter#(F, cons(X, Y)) >? filtersub#(F X, F, cons(X, Y)) 
  filtersub#(true, F, cons(X, Y)) >? filter#(F, Y) 
  filtersub#(false, F, cons(X, Y)) >? filter#(F, Y) 
  filter(F, nil) >= nil 
  filter(F, cons(X, Y)) >= filtersub(F X, F, cons(X, Y)) 
  filtersub(true, F, cons(X, Y)) >= cons(X, filter(F, Y)) 
  filtersub(false, F, cons(X, Y)) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  cons = \y0y1.1 + y1 
  false = 3 
  filter = \G0y1.y1 
  filter# = \G0y1.1 + y1 
  filtersub = \y0G1y2.y2 
  filtersub# = \y0G1y2.y2 
  nil = 0 
  true = 3 

Using this interpretation, the requirements translate to:

  [[filter#(_F0, cons(_x1, _x2))]] = 2 + x2 > 1 + x2 = [[filtersub#(_F0 _x1, _F0, cons(_x1, _x2))]] 
  [[filtersub#(true, _F0, cons(_x1, _x2))]] = 1 + x2 >= 1 + x2 = [[filter#(_F0, _x2)]] 
  [[filtersub#(false, _F0, cons(_x1, _x2))]] = 1 + x2 >= 1 + x2 = [[filter#(_F0, _x2)]] 
  [[filter(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 1 + x2 >= 1 + x2 = [[filtersub(_F0 _x1, _F0, cons(_x1, _x2))]] 
  [[filtersub(true, _F0, cons(_x1, _x2))]] = 1 + x2 >= 1 + x2 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filtersub(false, _F0, cons(_x1, _x2))]] = 1 + x2 >= x2 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_0, static, formative) by (P_1, R_0, static, formative), where P_1 consists of:

  filtersub#(true, F, cons(X, Y)) =#> filter#(F, Y)   
  filtersub#(false, F, cons(X, Y)) =#> filter#(F, Y)   

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

We consider the dependency pair problem (P_1, R_0, static, 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 :  
    * 1 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

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


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.