/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.

  a : [] --> o 
  f : [o] --> o 
  g : [o * o] --> o 
  h : [o * o] --> o 

  f(g(f(a), h(a, f(a)))) => f(h(g(f(a), a), g(f(a), f(a)))) 

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(f(a), h(a, f(a)))) >? f(h(g(f(a), a), g(f(a), f(a)))) 

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

The following interpretation satisfies the requirements:

  a = 0 
  f = \y0.y0 
  g = \y0y1.y0 + 3y1 
  h = \y0y1.3 + y0 + y1 

Using this interpretation, the requirements translate to:

  [[f(g(f(a), h(a, f(a))))]] = 9 > 3 = [[f(h(g(f(a), a), g(f(a), f(a))))]] 

We can thus remove the following rules:

  f(g(f(a), h(a, f(a)))) => f(h(g(f(a), a), g(f(a), f(a)))) 

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.