/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.

  !6220!6220 : [o * o] --> o 
  U11 : [o] --> o 
  U12 : [o] --> o 
  activate : [o] --> o 
  isNePal : [o] --> o 
  nil : [] --> o 
  tt : [] --> o 

  !6220!6220(!6220!6220(X, Y), Z) => !6220!6220(X, !6220!6220(Y, Z)) 
  !6220!6220(X, nil) => X 
  !6220!6220(nil, X) => X 
  U11(tt) => U12(tt) 
  U12(tt) => tt 
  isNePal(!6220!6220(X, !6220!6220(Y, X))) => U11(tt) 
  activate(X) => 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]):

  !6220!6220(!6220!6220(X, Y), Z) >? !6220!6220(X, !6220!6220(Y, Z)) 
  !6220!6220(X, nil) >? X 
  !6220!6220(nil, X) >? X 
  U11(tt) >? U12(tt) 
  U12(tt) >? tt 
  isNePal(!6220!6220(X, !6220!6220(Y, X))) >? U11(tt) 
  activate(X) >? X 

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

The following interpretation satisfies the requirements:

  !6220!6220 = \y0y1.3 + y1 + 3y0 
  U11 = \y0.2 + 3y0 
  U12 = \y0.2 + y0 
  activate = \y0.3 + y0 
  isNePal = \y0.3 + 3y0 
  nil = 3 
  tt = 2 

Using this interpretation, the requirements translate to:

  [[!6220!6220(!6220!6220(_x0, _x1), _x2)]] = 12 + x2 + 3x1 + 9x0 > 6 + x2 + 3x0 + 3x1 = [[!6220!6220(_x0, !6220!6220(_x1, _x2))]] 
  [[!6220!6220(_x0, nil)]] = 6 + 3x0 > x0 = [[_x0]] 
  [[!6220!6220(nil, _x0)]] = 12 + x0 > x0 = [[_x0]] 
  [[U11(tt)]] = 8 > 4 = [[U12(tt)]] 
  [[U12(tt)]] = 4 > 2 = [[tt]] 
  [[isNePal(!6220!6220(_x0, !6220!6220(_x1, _x0)))]] = 21 + 9x1 + 12x0 > 8 = [[U11(tt)]] 
  [[activate(_x0)]] = 3 + x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  !6220!6220(!6220!6220(X, Y), Z) => !6220!6220(X, !6220!6220(Y, Z)) 
  !6220!6220(X, nil) => X 
  !6220!6220(nil, X) => X 
  U11(tt) => U12(tt) 
  U12(tt) => tt 
  isNePal(!6220!6220(X, !6220!6220(Y, X))) => U11(tt) 
  activate(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.