/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  !870 : [o * o] --> o 
  and : [o * o] --> o 
  implies : [o * o] --> o 
  not : [o] --> o 
  or : [o * o] --> o 
  true : [] --> o 
  xor : [o * o] --> o 

  not(X) => xor(X, true) 
  implies(X, Y) => xor(and(X, Y), xor(X, true)) 
  or(X, Y) => xor(and(X, Y), xor(X, Y)) 
  !870(X, Y) => xor(X, xor(Y, true)) 

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

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

  not(X) >? xor(X, true) 
  implies(X, Y) >? xor(and(X, Y), xor(X, true)) 
  or(X, Y) >? xor(and(X, Y), xor(X, Y)) 
  !870(X, Y) >? xor(X, xor(Y, true)) 

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

The following interpretation satisfies the requirements:

  !870 = \y0y1.3 + y0 + y1 
  and = \y0y1.y0 + y1 
  implies = \y0y1.3 + y1 + 3y0 
  not = \y0.3 + y0 
  or = \y0y1.3 + 3y0 + 3y1 
  true = 0 
  xor = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[not(_x0)]] = 3 + x0 > x0 = [[xor(_x0, true)]] 
  [[implies(_x0, _x1)]] = 3 + x1 + 3x0 > x1 + 2x0 = [[xor(and(_x0, _x1), xor(_x0, true))]] 
  [[or(_x0, _x1)]] = 3 + 3x0 + 3x1 > 2x0 + 2x1 = [[xor(and(_x0, _x1), xor(_x0, _x1))]] 
  [[!870(_x0, _x1)]] = 3 + x0 + x1 > x0 + x1 = [[xor(_x0, xor(_x1, true))]] 

We can thus remove the following rules:

  not(X) => xor(X, true) 
  implies(X, Y) => xor(and(X, Y), xor(X, true)) 
  or(X, Y) => xor(and(X, Y), xor(X, Y)) 
  !870(X, Y) => xor(X, xor(Y, true)) 

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.