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


--------------------------------------------------------------------------------


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 
  n!6220!6220s : [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(n!6220!6220s(X))) 
  from(X) => n!6220!6220from(X) 
  s(X) => n!6220!6220s(X) 
  activate(n!6220!6220from(X)) => from(activate(X)) 
  activate(n!6220!6220s(X)) => s(activate(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#(activate(X))   
    3] activate#(n!6220!6220from(X)) =#> activate#(X)   
    4] activate#(n!6220!6220s(X)) =#> s#(activate(X))   
    5] activate#(n!6220!6220s(X)) =#> activate#(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(n!6220!6220s(X))) 
    from(X) => n!6220!6220from(X) 
    s(X) => n!6220!6220s(X) 
    activate(n!6220!6220from(X)) => from(activate(X)) 
    activate(n!6220!6220s(X)) => s(activate(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, 3, 4, 5 
    * 2 :  
    * 3 : 2, 3, 4, 5 
    * 4 :  
    * 5 : 2, 3, 4, 5 

This graph has the following strongly connected components:

  P_1:

    activate#(n!6220!6220from(X)) =#> activate#(X)   
    activate#(n!6220!6220s(X)) =#> activate#(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, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(activate#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(activate#(n!6220!6220from(X))) = n!6220!6220from(X) |> X = nu(activate#(X)) 
  nu(activate#(n!6220!6220s(X))) = n!6220!6220s(X) |> X = nu(activate#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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.