/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.

  f : [o * o * o] --> o 
  g : [o * o] --> o 
  s : [o] --> o 

  g(X, Y) => X 
  g(X, Y) => Y 
  f(s(X), Y, Y) => f(Y, X, s(X)) 

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

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

  g(X, Y) >? X 
  g(X, Y) >? Y 
  f(s(X), Y, Y) >? f(Y, X, s(X)) 

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

The following interpretation satisfies the requirements:

  f = \y0y1y2.y2 + 3y0 + 3y1 
  g = \y0y1.3 + y0 + y1 
  s = \y0.3 + 3y0 

Using this interpretation, the requirements translate to:

  [[g(_x0, _x1)]] = 3 + x0 + x1 > x0 = [[_x0]] 
  [[g(_x0, _x1)]] = 3 + x0 + x1 > x1 = [[_x1]] 
  [[f(s(_x0), _x1, _x1)]] = 9 + 4x1 + 9x0 > 3 + 3x1 + 6x0 = [[f(_x1, _x0, s(_x0))]] 

We can thus remove the following rules:

  g(X, Y) => X 
  g(X, Y) => Y 
  f(s(X), Y, Y) => f(Y, X, s(X)) 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.