/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.

  !times : [o * o] --> o 

  !times(X, !times(Y, Z)) => !times(!times(X, Y), Z) 
  !times(X, X) => 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]):

  !times(X, !times(Y, Z)) >? !times(!times(X, Y), Z) 
  !times(X, X) >? X 

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

The following interpretation satisfies the requirements:

  !times = \y0y1.3 + y0 + 3y1 

Using this interpretation, the requirements translate to:

  [[!times(_x0, !times(_x1, _x2))]] = 12 + x0 + 3x1 + 9x2 > 6 + x0 + 3x1 + 3x2 = [[!times(!times(_x0, _x1), _x2)]] 
  [[!times(_x0, _x0)]] = 3 + 4x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  !times(X, !times(Y, Z)) => !times(!times(X, Y), Z) 
  !times(X, X) => 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.