/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.

  a : [] --> o 
  f : [o * o] --> o 
  g : [o] --> o 
  h : [o * o * o] --> o 

  f(a, g(X)) => g(g(X)) 
  f(g(X), a) => f(X, g(a)) 
  f(g(X), g(Y)) => h(g(Y), X, g(Y)) 
  h(g(X), Y, Z) => f(Y, h(X, Y, Z)) 
  h(a, X, Y) => 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] f#(g(X), a) =#> f#(X, g(a))   
    1] f#(g(X), g(Y)) =#> h#(g(Y), X, g(Y))   
    2] h#(g(X), Y, Z) =#> f#(Y, h(X, Y, Z))   
    3] h#(g(X), Y, Z) =#> h#(X, Y, Z)   

  Rules R_0:

    f(a, g(X)) => g(g(X)) 
    f(g(X), a) => f(X, g(a)) 
    f(g(X), g(Y)) => h(g(Y), X, g(Y)) 
    h(g(X), Y, Z) => f(Y, h(X, Y, Z)) 
    h(a, X, Y) => 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(f#) = 1 
  nu(h#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(f#(g(X), a)) = g(X) |> X = nu(f#(X, g(a))) 
  nu(f#(g(X), g(Y))) = g(X) |> X = nu(h#(g(Y), X, g(Y))) 
  nu(h#(g(X), Y, Z)) = Y = Y = nu(f#(Y, h(X, Y, Z))) 
  nu(h#(g(X), Y, Z)) = Y = Y = nu(h#(X, Y, Z)) 

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:

  h#(g(X), Y, Z) =#> f#(Y, h(X, Y, Z))   
  h#(g(X), Y, Z) =#> h#(X, Y, Z)   

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 :  
    * 1 : 0, 1 

This graph has the following strongly connected components:

  P_2:

    h#(g(X), Y, Z) =#> h#(X, Y, Z)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_1, R_0, m, f) by (P_2, R_0, m, f).

Thus, the original system is terminating if (P_2, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(h#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(h#(g(X), Y, Z)) = g(X) |> X = nu(h#(X, Y, Z)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, 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.