/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

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

  !plus(!minus(X, Y), Z) => !minus(!plus(X, Z), Y) 
  !minus(!plus(X, 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]):

  !plus(!minus(X, Y), Z) >? !minus(!plus(X, Z), Y) 
  !minus(!plus(X, Y), Y) >? X 

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

The following interpretation satisfies the requirements:

  !minus = \y0y1.3 + y0 + y1 
  !plus = \y0y1.3 + y1 + 3y0 

Using this interpretation, the requirements translate to:

  [[!plus(!minus(_x0, _x1), _x2)]] = 12 + x2 + 3x0 + 3x1 > 6 + x1 + x2 + 3x0 = [[!minus(!plus(_x0, _x2), _x1)]] 
  [[!minus(!plus(_x0, _x1), _x1)]] = 6 + 2x1 + 3x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  !plus(!minus(X, Y), Z) => !minus(!plus(X, Z), Y) 
  !minus(!plus(X, 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.