/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 
  b : [] --> o 
  f : [o] --> o 
  g : [o] --> o 
  h : [o] --> o 
  i : [o] --> o 

  f(h(X)) => f(i(X)) 
  g(i(X)) => g(h(X)) 
  h(a) => b 
  i(a) => b 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  f(h(X)) >? f(i(X)) 
  g(i(X)) >? g(h(X)) 
  h(a) >? b 
  i(a) >? b 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  a = 3 
  b = 0 
  f = \y0.y0 
  g = \y0.2y0 
  h = \y0.y0 
  i = \y0.y0 

Using this interpretation, the requirements translate to:

  [[f(h(_x0))]] = x0 >= x0 = [[f(i(_x0))]] 
  [[g(i(_x0))]] = 2x0 >= 2x0 = [[g(h(_x0))]] 
  [[h(a)]] = 3 > 0 = [[b]] 
  [[i(a)]] = 3 > 0 = [[b]] 

We can thus remove the following rules:

  h(a) => b 
  i(a) => b 

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#(h(X)) =#> f#(i(X))   
    1] g#(i(X)) =#> g#(h(X))   

  Rules R_0:

    f(h(X)) => f(i(X)) 
    g(i(X)) => g(h(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 :  

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.