/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.

  implies : [o * o] --> o 
  not : [o] --> o 
  or : [o * o] --> o 

  implies(not(X), Y) => or(X, Y) 
  implies(not(X), or(Y, Z)) => implies(Y, or(X, Z)) 
  implies(X, or(Y, Z)) => or(Y, implies(X, Z)) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  implies(not(X), Y) >? or(X, Y) 
  implies(not(X), or(Y, Z)) >? implies(Y, or(X, Z)) 
  implies(X, or(Y, Z)) >? or(Y, implies(X, Z)) 

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

The following interpretation satisfies the requirements:

  implies = \y0y1.3 + 2y1 + 3y0 
  not = \y0.3 + 3y0 
  or = \y0y1.2 + y1 + 2y0 

Using this interpretation, the requirements translate to:

  [[implies(not(_x0), _x1)]] = 12 + 2x1 + 9x0 > 2 + x1 + 2x0 = [[or(_x0, _x1)]] 
  [[implies(not(_x0), or(_x1, _x2))]] = 16 + 2x2 + 4x1 + 9x0 > 7 + 2x2 + 3x1 + 4x0 = [[implies(_x1, or(_x0, _x2))]] 
  [[implies(_x0, or(_x1, _x2))]] = 7 + 2x2 + 3x0 + 4x1 > 5 + 2x1 + 2x2 + 3x0 = [[or(_x1, implies(_x0, _x2))]] 

We can thus remove the following rules:

  implies(not(X), Y) => or(X, Y) 
  implies(not(X), or(Y, Z)) => implies(Y, or(X, Z)) 
  implies(X, or(Y, Z)) => or(Y, implies(X, 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.