/export/starexec/sandbox2/solver/bin/starexec_run_HigherOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
We consider the system theBenchmark.

  Alphabet:

    cons : [nat -> nat * funlist] --> funlist 
    false : [] --> bool 
    head : [funlist] --> nat -> nat 
    if : [bool * nat -> string * nat -> string] --> nat -> string 
    nil : [] --> funlist 
    s : [nat] --> nat 
    tail : [funlist] --> funlist 
    test : [nat -> nat] --> bool 
    true : [] --> bool 

  Rules:

    if(true, f, g) => f 
    if(false, f, g) => g 
    test(/\x.s(x)) => true 
    head(cons(f, x)) => f 
    tail(cons(f, x)) => 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]).

In order to do so, we start by eta-expanding the system, which gives:

  if(true, F, G, X) => F X 
  if(false, F, G, X) => G X 
  test(/\x.s(x)) => true 
  head(cons(F, X), Y) => F Y 
  tail(cons(F, X)) => X 

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

  Dependency Pairs P_0:


  Rules R_0:

    if(true, F, G, X) => F X 
    if(false, F, G, X) => G X 
    test(/\x.s(x)) => true 
    head(cons(F, X), Y) => F Y 
    tail(cons(F, X)) => X 

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:


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  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.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.