/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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 
  quot : [o * o * o] --> o 
  s : [o] --> o 

  quot(0, s(X), s(Y)) => 0 
  quot(s(X), s(Y), Z) => quot(X, Y, Z) 
  quot(X, 0, s(Y)) => s(quot(X, s(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] quot#(s(X), s(Y), Z) =#> quot#(X, Y, Z)   
    1] quot#(X, 0, s(Y)) =#> quot#(X, s(Y), s(Y))   

  Rules R_0:

    quot(0, s(X), s(Y)) => 0 
    quot(s(X), s(Y), Z) => quot(X, Y, Z) 
    quot(X, 0, s(Y)) => s(quot(X, s(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 apply the subterm criterion with the following projection function:

  nu(quot#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(quot#(s(X), s(Y), Z)) = s(X) |> X = nu(quot#(X, Y, Z)) 
  nu(quot#(X, 0, s(Y))) = X = X = nu(quot#(X, s(Y), s(Y))) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_0, R_0, minimal, f) by (P_1, R_0, minimal, f), where P_1 contains:

  quot#(X, 0, s(Y)) =#> quot#(X, s(Y), s(Y))   

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

    * 0 :  

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.