We consider the system filter.

  Alphabet:

    0 : [] --> nat 
    bool : [nat] --> boolean 
    cons : [nat * list] --> list 
    consif : [boolean * nat * list] --> list 
    false : [] --> boolean 
    filter : [nat -> boolean * list] --> list 
    nil : [] --> list 
    rand : [nat] --> nat 
    s : [nat] --> nat 
    true : [] --> boolean 

  Rules:

    rand(x) => x 
    rand(s(x)) => rand(x) 
    bool(0) => false 
    bool(s(0)) => true 
    filter(f, nil) => nil 
    filter(f, cons(x, y)) => consif(f x, x, filter(f, y)) 
    consif(true, x, y) => cons(x, y) 
    consif(false, x, y) => 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] rand#(s(X)) =#> rand#(X)   
    1] filter#(F, cons(X, Y)) =#> consif#(F X, X, filter(F, Y))   
    2] filter#(F, cons(X, Y)) =#> filter#(F, Y)   

  Rules R_0:

    rand(X) => X 
    rand(s(X)) => rand(X) 
    bool(0) => false 
    bool(s(0)) => true 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => consif(F X, X, filter(F, Y)) 
    consif(true, X, Y) => cons(X, Y) 
    consif(false, X, Y) => 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 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 :  
    * 2 : 1, 2 

This graph has the following strongly connected components:

  P_1:

    rand#(s(X)) =#> rand#(X)   

  P_2:

    filter#(F, cons(X, Y)) =#> filter#(F, Y)   

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) and (P_2, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite.

We consider the dependency pair problem (P_2, R_0, static, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_2, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  filter#(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 = \y0y1.3 + 2y1 
  filter# = \G0y1.y1 

Using this interpretation, the requirements translate to:

  [[filter#(_F0, cons(_x1, _x2))]] = 3 + 2x2 > x2 = [[filter#(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_2, R_0) by ({}, R_0).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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 will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_1, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  rand#(s(X)) >? rand#(X) 

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

The following interpretation satisfies the requirements:

  rand# = \y0.y0 
  s = \y0.1 + 2y0 

Using this interpretation, the requirements translate to:

  [[rand#(s(_x0))]] = 1 + 2x0 > x0 = [[rand#(_x0)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_0) by ({}, R_0).  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 +++

[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.