/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] --> o 
  g : [o] --> o 
  h : [o * o] --> o 

  f(g(h(X, Y)), f(a, a)) => f(h(X, X), g(f(Y, 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(h(X, Y)), f(a, a)) >? f(h(X, X), g(f(Y, a))) 

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

The following interpretation satisfies the requirements:

  a = 0 
  f = \y0y1.y1 + 2y0 
  g = \y0.2 + 2y0 
  h = \y0y1.1 + y0 + y1 

Using this interpretation, the requirements translate to:

  [[f(g(h(_x0, _x1)), f(a, a))]] = 8 + 4x0 + 4x1 > 4 + 4x0 + 4x1 = [[f(h(_x0, _x0), g(f(_x1, a)))]] 

We can thus remove the following rules:

  f(g(h(X, Y)), f(a, a)) => f(h(X, X), g(f(Y, 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.