/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.

  div : [o * o] --> o 
  e : [] --> o 
  i : [o] --> o 

  div(X, e) => i(X) 
  i(div(X, Y)) => div(Y, X) 
  div(div(X, Y), Z) => div(Y, div(i(X), Z)) 

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

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

  div(X, e) >? i(X) 
  i(div(X, Y)) >? div(Y, X) 
  div(div(X, Y), Z) >? div(Y, div(i(X), Z)) 

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

The following interpretation satisfies the requirements:

  div = \y0y1.y1 + 2y0 
  e = 3 
  i = \y0.2y0 

Using this interpretation, the requirements translate to:

  [[div(_x0, e)]] = 3 + 2x0 > 2x0 = [[i(_x0)]] 
  [[i(div(_x0, _x1))]] = 2x1 + 4x0 >= x0 + 2x1 = [[div(_x1, _x0)]] 
  [[div(div(_x0, _x1), _x2)]] = x2 + 2x1 + 4x0 >= x2 + 2x1 + 4x0 = [[div(_x1, div(i(_x0), _x2))]] 

We can thus remove the following rules:

  div(X, e) => i(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]):

  i(div(X, Y)) >? div(Y, X) 
  div(div(X, Y), Z) >? div(Y, div(i(X), Z)) 

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

The following interpretation satisfies the requirements:

  div = \y0y1.1 + y1 + 2y0 
  i = \y0.2y0 

Using this interpretation, the requirements translate to:

  [[i(div(_x0, _x1))]] = 2 + 2x1 + 4x0 > 1 + x0 + 2x1 = [[div(_x1, _x0)]] 
  [[div(div(_x0, _x1), _x2)]] = 3 + x2 + 2x1 + 4x0 > 2 + x2 + 2x1 + 4x0 = [[div(_x1, div(i(_x0), _x2))]] 

We can thus remove the following rules:

  i(div(X, Y)) => div(Y, X) 
  div(div(X, Y), Z) => div(Y, div(i(X), Z)) 

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.