/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.

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

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

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

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

The following interpretation satisfies the requirements:

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

Using this interpretation, the requirements translate to:

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

We can thus remove the following rules:

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

  minus(minus(X)) >? X 

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

The following interpretation satisfies the requirements:

  minus = \y0.3 + y0 

Using this interpretation, the requirements translate to:

  [[minus(minus(_x0))]] = 6 + x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  minus(minus(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.