/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.

  !3220 : [o * o] --> o 
  !div : [o * o] --> o 
  !dot : [o * o] --> o 
  e : [] --> o 

  !3220(X, X) => e 
  !div(X, X) => e 
  !dot(e, X) => X 
  !dot(X, e) => X 
  !3220(e, X) => X 
  !div(X, e) => X 
  !dot(X, !3220(X, Y)) => Y 
  !dot(!div(X, Y), Y) => X 
  !3220(X, !dot(X, Y)) => Y 
  !div(!dot(X, Y), Y) => X 
  !div(X, !3220(Y, X)) => Y 
  !3220(!div(X, Y), 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]):

  !3220(X, X) >? e 
  !div(X, X) >? e 
  !dot(e, X) >? X 
  !dot(X, e) >? X 
  !3220(e, X) >? X 
  !div(X, e) >? X 
  !dot(X, !3220(X, Y)) >? Y 
  !dot(!div(X, Y), Y) >? X 
  !3220(X, !dot(X, Y)) >? Y 
  !div(!dot(X, Y), Y) >? X 
  !div(X, !3220(Y, X)) >? Y 
  !3220(!div(X, Y), X) >? Y 

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

The following interpretation satisfies the requirements:

  !3220 = \y0y1.3 + y0 + y1 
  !div = \y0y1.3 + y0 + y1 
  !dot = \y0y1.3 + y0 + y1 
  e = 0 

Using this interpretation, the requirements translate to:

  [[!3220(_x0, _x0)]] = 3 + 2x0 > 0 = [[e]] 
  [[!div(_x0, _x0)]] = 3 + 2x0 > 0 = [[e]] 
  [[!dot(e, _x0)]] = 3 + x0 > x0 = [[_x0]] 
  [[!dot(_x0, e)]] = 3 + x0 > x0 = [[_x0]] 
  [[!3220(e, _x0)]] = 3 + x0 > x0 = [[_x0]] 
  [[!div(_x0, e)]] = 3 + x0 > x0 = [[_x0]] 
  [[!dot(_x0, !3220(_x0, _x1))]] = 6 + x1 + 2x0 > x1 = [[_x1]] 
  [[!dot(!div(_x0, _x1), _x1)]] = 6 + x0 + 2x1 > x0 = [[_x0]] 
  [[!3220(_x0, !dot(_x0, _x1))]] = 6 + x1 + 2x0 > x1 = [[_x1]] 
  [[!div(!dot(_x0, _x1), _x1)]] = 6 + x0 + 2x1 > x0 = [[_x0]] 
  [[!div(_x0, !3220(_x1, _x0))]] = 6 + x1 + 2x0 > x1 = [[_x1]] 
  [[!3220(!div(_x0, _x1), _x0)]] = 6 + x1 + 2x0 > x1 = [[_x1]] 

We can thus remove the following rules:

  !3220(X, X) => e 
  !div(X, X) => e 
  !dot(e, X) => X 
  !dot(X, e) => X 
  !3220(e, X) => X 
  !div(X, e) => X 
  !dot(X, !3220(X, Y)) => Y 
  !dot(!div(X, Y), Y) => X 
  !3220(X, !dot(X, Y)) => Y 
  !div(!dot(X, Y), Y) => X 
  !div(X, !3220(Y, X)) => Y 
  !3220(!div(X, Y), 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.