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

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

As the system is orthogonal, it is terminating if it is innermost terminating by [Gra95].  Then, by [FuhGieParSchSwi11], it suffices to prove (innermost) termination of the typed system, with sort annotations chosen to respect the rules, as follows:

  f : [v] --> v 
  g : [v * i] --> v 
  h : [v * i] --> v 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

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

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

The following interpretation satisfies the requirements:

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

Using this interpretation, the requirements translate to:

  [[h(f(_x0), _x1)]] = 5 + x1 + 2x0 > 4 + x1 + 2x0 = [[f(g(_x0, _x1))]] 
  [[g(_x0, _x1)]] = 2 + x1 + 2x0 > 1 + x1 + 2x0 = [[h(_x0, _x1)]] 

We can thus remove the following rules:

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

[FuhGieParSchSwi11]  C. Fuhs, J. Giesl, M. Parting, P. Schneider-Kamp, and S. Swiderski.  Proving Termination by Dependency Pairs and Inductive Theorem Proving.  In volume 47(2) of Journal of Automated Reasoning.  133--160, 2011.
[Gra95]  B. Gramlich.  Abstract Relations Between Restricted Termination and Confluence Properties of Rewrite Systems.  In volume 24(1-2) of Fundamentae Informaticae.  3--23, 1995.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.