/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 
  minus : [o] --> o 

  !times(X, !times(minus(Y), Y)) => !times(minus(!times(Y, Y)), 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(minus(Y), Y)) >? !times(minus(!times(Y, Y)), X) 

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

The following interpretation satisfies the requirements:

  !times = \y0y1.2y0 + 2y1 
  minus = \y0.2 + y0 

Using this interpretation, the requirements translate to:

  [[!times(_x0, !times(minus(_x1), _x1))]] = 8 + 2x0 + 8x1 > 4 + 2x0 + 8x1 = [[!times(minus(!times(_x1, _x1)), _x0)]] 

We can thus remove the following rules:

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