We consider the system noneating.

  Alphabet:

    0 : [] --> o 
    a : [] --> o 
    f : [o -> o] --> o 
    g : [o] --> o 
    h : [o * o] --> o 
    s : [o] --> o 

  Rules:

    a => f(/\x.g(x)) 
    f(/\x.y) => a 
    g(x) => h(x, x) 
    h(0, x) => x 
    h(s(x), 0) => g(x) 

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

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

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

  a >? f(/\x.g(x)) 
  f(/\x.X) >? a 
  g(X) >? h(X, X) 
  h(0, X) >? X 
  h(s(X), 0) >? g(X) 

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

The following interpretation satisfies the requirements:

  0 = 3 
  a = 0 
  f = \G0.G0(0) 
  g = \y0.3y0 
  h = \y0y1.y0 + 2y1 
  s = \y0.3 + 3y0 

Using this interpretation, the requirements translate to:

  [[a]] = 0 >= 0 = [[f(/\x.g(x))]] 
  [[f(/\x._x0)]] = x0 >= 0 = [[a]] 
  [[g(_x0)]] = 3x0 >= 3x0 = [[h(_x0, _x0)]] 
  [[h(0, _x0)]] = 3 + 2x0 > x0 = [[_x0]] 
  [[h(s(_x0), 0)]] = 9 + 3x0 > 3x0 = [[g(_x0)]] 

We can thus remove the following rules:

  h(0, X) => X 
  h(s(X), 0) => g(X) 

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, all):

  Dependency Pairs P_0:

    0] a# =#> f#(/\x.g(x))   
    1] a# =#> g#(X)   
    2] f#(/\x.X) =#> a#   

  Rules R_0:

    a => f(/\x.g(x)) 
    f(/\x.X) => a 
    g(X) => h(X, X) 

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

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

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 :  
    * 2 : 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.
[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.