/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.

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

  !plus(0, X) => X 
  !plus(s(X), Y) => s(!plus(X, Y)) 
  !minus(0, X) => 0 
  !minus(X, 0) => X 
  !minus(s(X), s(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]):

  !plus(0, X) >? X 
  !plus(s(X), Y) >? s(!plus(X, Y)) 
  !minus(0, X) >? 0 
  !minus(X, 0) >? X 
  !minus(s(X), s(Y)) >? !minus(X, Y) 

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.y1 + 3y0 
  0 = 0 
  s = \y0.3 + y0 

Using this interpretation, the requirements translate to:

  [[!plus(0, _x0)]] = x0 >= x0 = [[_x0]] 
  [[!plus(s(_x0), _x1)]] = 9 + x1 + 3x0 > 3 + x1 + 3x0 = [[s(!plus(_x0, _x1))]] 
  [[!minus(0, _x0)]] = 3 + x0 > 0 = [[0]] 
  [[!minus(_x0, 0)]] = 3 + x0 > x0 = [[_x0]] 
  [[!minus(s(_x0), s(_x1))]] = 9 + x0 + x1 > 3 + x0 + x1 = [[!minus(_x0, _x1)]] 

We can thus remove the following rules:

  !plus(s(X), Y) => s(!plus(X, Y)) 
  !minus(0, X) => 0 
  !minus(X, 0) => X 
  !minus(s(X), s(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]):

  !plus(0, X) >? X 

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

The following interpretation satisfies the requirements:

  !plus = \y0y1.3 + y0 + y1 
  0 = 3 

Using this interpretation, the requirements translate to:

  [[!plus(0, _x0)]] = 6 + x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  !plus(0, 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.