/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.

  2nd : [o] --> o 
  activate : [o] --> o 
  cons : [o * o] --> o 
  cons1 : [o * o] --> o 
  from : [o] --> o 
  n!6220!6220from : [o] --> o 
  s : [o] --> o 

  2nd(cons1(X, cons(Y, Z))) => Y 
  2nd(cons(X, Y)) => 2nd(cons1(X, activate(Y))) 
  from(X) => cons(X, n!6220!6220from(s(X))) 
  from(X) => n!6220!6220from(X) 
  activate(n!6220!6220from(X)) => from(X) 
  activate(X) => 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, formative):

  Dependency Pairs P_0:

    0] 2nd#(cons(X, Y)) =#> 2nd#(cons1(X, activate(Y)))   
    1] 2nd#(cons(X, Y)) =#> activate#(Y)   
    2] activate#(n!6220!6220from(X)) =#> from#(X)   

  Rules R_0:

    2nd(cons1(X, cons(Y, Z))) => Y 
    2nd(cons(X, Y)) => 2nd(cons1(X, activate(Y))) 
    from(X) => cons(X, n!6220!6220from(s(X))) 
    from(X) => n!6220!6220from(X) 
    activate(n!6220!6220from(X)) => from(X) 
    activate(X) => X 

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 : 2 
    * 2 :  

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.