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 the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]).

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

  Dependency Pairs P_0:

    0] f#(/\x.a) =#> b#   
    1] b# =#> f#(/\x.c)   

  Rules R_0:

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

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 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:

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

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( b ) = #argfun-b#(f(/\x.c)) 
  pi( b# ) = #argfun-b##(f#(/\x.c)) 

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

The following interpretation satisfies the requirements:

  #argfun-b# = \y0.y0 
  #argfun-b## = \y0.1 + y0 
  a = 3 
  b = 0 
  b# = 0 
  c = 0 
  f = \G0.3G0(0) 
  f# = \G0.G0(0) 

Using this interpretation, the requirements translate to:

  [[f#(/\x.a)]] = 3 > 1 = [[#argfun-b##(f#(/\x.c))]] 
  [[#argfun-b##(f#(/\x.c))]] = 1 > 0 = [[f#(/\x.c)]] 
  [[f(/\x.a)]] = 9 >= 0 = [[#argfun-b#(f(/\x.c))]] 
  [[#argfun-b#(f(/\x.c))]] = 0 >= 0 = [[f(/\x.c)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_0, 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.
[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.