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


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YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  !770 : [o * o] --> o 
  !minus : [o * o] --> o 
  0 : [] --> o 
  gcd : [o * o] --> o 
  if : [o * o * o] --> o 
  s : [o] --> o 

  gcd(X, 0) => X 
  gcd(0, X) => X 
  gcd(s(X), s(Y)) => if(!770(X, Y), gcd(s(X), !minus(Y, X)), gcd(!minus(X, Y), s(Y))) 

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] gcd#(s(X), s(Y)) =#> gcd#(s(X), !minus(Y, X))   
    1] gcd#(s(X), s(Y)) =#> gcd#(!minus(X, Y), s(Y))   

  Rules R_0:

    gcd(X, 0) => X 
    gcd(0, X) => X 
    gcd(s(X), s(Y)) => if(!770(X, Y), gcd(s(X), !minus(Y, X)), gcd(!minus(X, Y), s(Y))) 

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 place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 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.