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

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

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

The following interpretation satisfies the requirements:

  f = \y0y1.3 + 2y0 + 3y1 
  g = \y0.3 + y0 
  h = \y0.3 + y0 

Using this interpretation, the requirements translate to:

  [[f(_x0, g(_x0))]] = 12 + 5x0 > x0 = [[_x0]] 
  [[f(_x0, h(_x1))]] = 12 + 2x0 + 3x1 > 9 + 2x0 + 3x1 = [[f(h(_x0), _x1)]] 

We can thus remove the following rules:

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