/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.

  f : [o] --> o 
  g : [o] --> o 

  f(g(X)) => g(g(f(X))) 
  f(g(X)) => g(g(g(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]):

  f(g(X)) >? g(g(f(X))) 
  f(g(X)) >? g(g(g(X))) 

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

The following interpretation satisfies the requirements:

  f = \y0.3y0 
  g = \y0.1 + y0 

Using this interpretation, the requirements translate to:

  [[f(g(_x0))]] = 3 + 3x0 > 2 + 3x0 = [[g(g(f(_x0)))]] 
  [[f(g(_x0))]] = 3 + 3x0 >= 3 + x0 = [[g(g(g(_x0)))]] 

We can thus remove the following rules:

  f(g(X)) => g(g(f(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]):

  f(g(X)) >? g(g(g(X))) 

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

The following interpretation satisfies the requirements:

  f = \y0.3 + 3y0 
  g = \y0.y0 

Using this interpretation, the requirements translate to:

  [[f(g(_x0))]] = 3 + 3x0 > x0 = [[g(g(g(_x0)))]] 

We can thus remove the following rules:

  f(g(X)) => g(g(g(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.