/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] --> o 
  b : [o] --> o 
  c : [o] --> o 
  e : [] --> o 

  c(b(a(X))) => a(a(b(b(c(c(X)))))) 
  a(X) => e 
  b(X) => e 
  c(X) => e 

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] c#(b(a(X))) =#> a#(a(b(b(c(c(X))))))   
    1] c#(b(a(X))) =#> a#(b(b(c(c(X)))))   
    2] c#(b(a(X))) =#> b#(b(c(c(X))))   
    3] c#(b(a(X))) =#> b#(c(c(X)))   
    4] c#(b(a(X))) =#> c#(c(X))   
    5] c#(b(a(X))) =#> c#(X)   

  Rules R_0:

    c(b(a(X))) => a(a(b(b(c(c(X)))))) 
    a(X) => e 
    b(X) => e 
    c(X) => e 

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 : 0, 1, 2, 3, 4, 5 

This graph has the following strongly connected components:

  P_1:

    c#(b(a(X))) =#> c#(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(c#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(c#(b(a(X)))) = b(a(X)) |> X = nu(c#(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.