/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.

  a!6220!6220f : [o * o] --> o 
  f : [o * o] --> o 
  g : [o] --> o 
  mark : [o] --> o 

  a!6220!6220f(g(X), Y) => a!6220!6220f(mark(X), f(g(X), Y)) 
  mark(f(X, Y)) => a!6220!6220f(mark(X), Y) 
  mark(g(X)) => g(mark(X)) 
  a!6220!6220f(X, Y) => f(X, 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] a!6220!6220f#(g(X), Y) =#> a!6220!6220f#(mark(X), f(g(X), Y))   
    1] a!6220!6220f#(g(X), Y) =#> mark#(X)   
    2] mark#(f(X, Y)) =#> a!6220!6220f#(mark(X), Y)   
    3] mark#(f(X, Y)) =#> mark#(X)   
    4] mark#(g(X)) =#> mark#(X)   

  Rules R_0:

    a!6220!6220f(g(X), Y) => a!6220!6220f(mark(X), f(g(X), Y)) 
    mark(f(X, Y)) => a!6220!6220f(mark(X), Y) 
    mark(g(X)) => g(mark(X)) 
    a!6220!6220f(X, Y) => f(X, 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 will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220f#(g(X), Y) >? a!6220!6220f#(mark(X), f(g(X), Y)) 
  a!6220!6220f#(g(X), Y) >? mark#(X) 
  mark#(f(X, Y)) >? a!6220!6220f#(mark(X), Y) 
  mark#(f(X, Y)) >? mark#(X) 
  mark#(g(X)) >? mark#(X) 
  a!6220!6220f(g(X), Y) >= a!6220!6220f(mark(X), f(g(X), Y)) 
  mark(f(X, Y)) >= a!6220!6220f(mark(X), Y) 
  mark(g(X)) >= g(mark(X)) 
  a!6220!6220f(X, Y) >= f(X, Y) 

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

The following interpretation satisfies the requirements:

  a!6220!6220f = \y0y1.2y0 
  a!6220!6220f# = \y0y1.y0 
  f = \y0y1.2y0 
  g = \y0.1 + 2y0 
  mark = \y0.2y0 
  mark# = \y0.y0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220f#(g(_x0), _x1)]] = 1 + 2x0 > 2x0 = [[a!6220!6220f#(mark(_x0), f(g(_x0), _x1))]] 
  [[a!6220!6220f#(g(_x0), _x1)]] = 1 + 2x0 > x0 = [[mark#(_x0)]] 
  [[mark#(f(_x0, _x1))]] = 2x0 >= 2x0 = [[a!6220!6220f#(mark(_x0), _x1)]] 
  [[mark#(f(_x0, _x1))]] = 2x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(g(_x0))]] = 1 + 2x0 > x0 = [[mark#(_x0)]] 
  [[a!6220!6220f(g(_x0), _x1)]] = 2 + 4x0 >= 4x0 = [[a!6220!6220f(mark(_x0), f(g(_x0), _x1))]] 
  [[mark(f(_x0, _x1))]] = 4x0 >= 4x0 = [[a!6220!6220f(mark(_x0), _x1)]] 
  [[mark(g(_x0))]] = 2 + 4x0 >= 1 + 4x0 = [[g(mark(_x0))]] 
  [[a!6220!6220f(_x0, _x1)]] = 2x0 >= 2x0 = [[f(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_1, R_0, minimal, formative), where P_1 consists of:

  mark#(f(X, Y)) =#> a!6220!6220f#(mark(X), Y)   
  mark#(f(X, Y)) =#> mark#(X)   

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:

    mark#(f(X, Y)) =#> mark#(X)   

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(mark#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(mark#(f(X, Y))) = f(X, Y) |> X = nu(mark#(X)) 

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.