/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.

  cons : [o * o] --> o 
  empty : [] --> o 
  f : [o * o] --> o 

  f(X, empty) => X 
  f(empty, cons(X, Y)) => f(cons(X, Y), Y) 
  f(cons(X, Y), Z) => f(Z, 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]):

  f(X, empty) >? X 
  f(empty, cons(X, Y)) >? f(cons(X, Y), Y) 
  f(cons(X, Y), Z) >? f(Z, Y) 

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

The following interpretation satisfies the requirements:

  cons = \y0y1.y0 + 3y1 
  empty = 3 
  f = \y0y1.3 + 2y0 + 3y1 

Using this interpretation, the requirements translate to:

  [[f(_x0, empty)]] = 12 + 2x0 > x0 = [[_x0]] 
  [[f(empty, cons(_x0, _x1))]] = 9 + 3x0 + 9x1 > 3 + 2x0 + 9x1 = [[f(cons(_x0, _x1), _x1)]] 
  [[f(cons(_x0, _x1), _x2)]] = 3 + 2x0 + 3x2 + 6x1 >= 3 + 2x2 + 3x1 = [[f(_x2, _x1)]] 

We can thus remove the following rules:

  f(X, empty) => X 
  f(empty, cons(X, Y)) => f(cons(X, Y), 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]):

  f(cons(X, Y), Z) >? f(Z, Y) 

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

The following interpretation satisfies the requirements:

  cons = \y0y1.1 + y0 + 3y1 
  f = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[f(cons(_x0, _x1), _x2)]] = 1 + x0 + x2 + 3x1 > x1 + x2 = [[f(_x2, _x1)]] 

We can thus remove the following rules:

  f(cons(X, Y), Z) => f(Z, Y) 

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.