/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.

  and : [o * o * o] --> o 
  band : [o * o] --> o 
  not : [o] --> o 

  and(not(not(X)), Y, not(Z)) => and(Y, band(X, Z), X) 

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:

  and : [r * r * r] --> t 
  band : [r * r] --> r 
  not : [r] --> r 

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

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

  and(not(not(X)), Y, not(Z)) >? and(Y, band(X, Z), X) 

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

The following interpretation satisfies the requirements:

  and = \y0y1y2.y0 + 2y1 + 2y2 
  band = \y0y1.y0 + y1 
  not = \y0.3 + 3y0 

Using this interpretation, the requirements translate to:

  [[and(not(not(_x0)), _x1, not(_x2))]] = 18 + 2x1 + 6x2 + 9x0 > x1 + 2x2 + 4x0 = [[and(_x1, band(_x0, _x2), _x0)]] 

We can thus remove the following rules:

  and(not(not(X)), Y, not(Z)) => and(Y, band(X, Z), X) 

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.