/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.

  0 : [] --> o 
  cons : [o * o] --> o 
  copy : [o * o * o] --> o 
  f : [o] --> o 
  n : [] --> o 
  nil : [] --> o 
  s : [o] --> o 

  f(cons(nil, X)) => X 
  f(cons(f(cons(nil, X)), Y)) => copy(n, X, Y) 
  copy(0, X, Y) => f(Y) 
  copy(s(X), Y, Z) => copy(X, Y, cons(f(Y), Z)) 

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] f#(cons(f(cons(nil, X)), Y)) =#> copy#(n, X, Y)   
    1] copy#(0, X, Y) =#> f#(Y)   
    2] copy#(s(X), Y, Z) =#> copy#(X, Y, cons(f(Y), Z))   
    3] copy#(s(X), Y, Z) =#> f#(Y)   

  Rules R_0:

    f(cons(nil, X)) => X 
    f(cons(f(cons(nil, X)), Y)) => copy(n, X, Y) 
    copy(0, X, Y) => f(Y) 
    copy(s(X), Y, Z) => copy(X, Y, cons(f(Y), Z)) 

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 : 0 
    * 2 : 1, 2, 3 
    * 3 : 0 

This graph has the following strongly connected components:

  P_1:

    copy#(s(X), Y, Z) =#> copy#(X, Y, cons(f(Y), Z))   

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(copy#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(copy#(s(X), Y, Z)) = s(X) |> X = nu(copy#(X, Y, cons(f(Y), Z))) 

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.