/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.

  !minus : [o * o] --> o 
  neg : [o] --> o 

  !minus(!minus(neg(X), neg(X)), !minus(neg(Y), neg(Y))) => !minus(!minus(X, Y), !minus(X, Y)) 

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

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

  !minus(!minus(neg(X), neg(X)), !minus(neg(Y), neg(Y))) >? !minus(!minus(X, Y), !minus(X, Y)) 

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

The following interpretation satisfies the requirements:

  !minus = \y0y1.y1 + 2y0 
  neg = \y0.3 + 3y0 

Using this interpretation, the requirements translate to:

  [[!minus(!minus(neg(_x0), neg(_x0)), !minus(neg(_x1), neg(_x1)))]] = 27 + 9x1 + 18x0 > 3x1 + 6x0 = [[!minus(!minus(_x0, _x1), !minus(_x0, _x1))]] 

We can thus remove the following rules:

  !minus(!minus(neg(X), neg(X)), !minus(neg(Y), neg(Y))) => !minus(!minus(X, Y), !minus(X, Y)) 

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.