/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.

  plus : [o * o] --> o 
  s : [o] --> o 
  times : [o * o] --> o 

  plus(plus(X, Y), Z) => plus(X, plus(Y, Z)) 
  times(X, s(Y)) => plus(X, times(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(plus(X, Y), Z) >? plus(X, plus(Y, Z)) 
  times(X, s(Y)) >? plus(X, times(Y, X)) 

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

The following interpretation satisfies the requirements:

  plus = \y0y1.y0 + y1 
  s = \y0.3 + 3y0 
  times = \y0y1.y1 + 2y0 

Using this interpretation, the requirements translate to:

  [[plus(plus(_x0, _x1), _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[plus(_x0, plus(_x1, _x2))]] 
  [[times(_x0, s(_x1))]] = 3 + 2x0 + 3x1 > 2x0 + 2x1 = [[plus(_x0, times(_x1, _x0))]] 

We can thus remove the following rules:

  times(X, s(Y)) => plus(X, times(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(plus(X, Y), Z) >? plus(X, plus(Y, Z)) 

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

The following interpretation satisfies the requirements:

  plus = \y0y1.3 + y1 + 3y0 

Using this interpretation, the requirements translate to:

  [[plus(plus(_x0, _x1), _x2)]] = 12 + x2 + 3x1 + 9x0 > 6 + x2 + 3x0 + 3x1 = [[plus(_x0, plus(_x1, _x2))]] 

We can thus remove the following rules:

  plus(plus(X, Y), Z) => plus(X, plus(Y, Z)) 

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.