/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.

  c : [] --> o 
  d : [] --> o 
  f : [o] --> o 
  h : [o] --> o 

  f(f(X)) => c 
  c => d 
  h(X) => c 

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(f(X)) >? c 
  c >? d 
  h(X) >? c 

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

The following interpretation satisfies the requirements:

  c = 1 
  d = 0 
  f = \y0.3 + 3y0 
  h = \y0.3 + y0 

Using this interpretation, the requirements translate to:

  [[f(f(_x0))]] = 12 + 9x0 > 1 = [[c]] 
  [[c]] = 1 > 0 = [[d]] 
  [[h(_x0)]] = 3 + x0 > 1 = [[c]] 

We can thus remove the following rules:

  f(f(X)) => c 
  c => d 
  h(X) => c 

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.