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 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, all):

  Dependency Pairs P_0:

    0] a# =#> f#(/\x.g(x))   
    1] a# =#> g#(X)   
    2] f#(/\x.X) =#> a#   
    3] g#(X) =#> h#(X, X)   
    4] h#(s(X), 0) =#> g#(X)   

  Rules R_0:

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

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

We consider the dependency pair problem (P_0, R_0, static, 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 : 3 
    * 2 : 0, 1 
    * 3 : 4 
    * 4 : 3 

This graph has the following strongly connected components:

  P_1:

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

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, static, all) is finite.

We consider the dependency pair problem (P_1, R_0, static, all).

We apply the subterm criterion with the following projection function:

  nu(g#) = 1 
  nu(h#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(g#(X)) = X = X = nu(h#(X, X)) 
  nu(h#(s(X), 0)) = s(X) |> X = nu(g#(X)) 

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_1, R_0, static, f) by (P_2, R_0, static, f), where P_2 contains:

  g#(X) =#> h#(X, X)   

Thus, the original system is terminating if (P_2, R_0, static, all) is finite.

We consider the dependency pair problem (P_2, R_0, static, 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 :  

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.