/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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MAYBE
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  aver : [o * o] --> o 
  average : [o * o] --> o 
  double : [o] --> o 
  false : [] --> o 
  gt : [o * o] --> o 
  if : [o * o * o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  gt(0, X) => false 
  gt(s(X), 0) => true 
  gt(s(X), s(Y)) => gt(X, Y) 
  plus(0, X) => X 
  plus(s(X), Y) => s(plus(X, Y)) 
  double(0) => 0 
  double(s(X)) => s(s(double(X))) 
  average(X, Y) => aver(plus(X, Y), 0) 
  aver(X, Y) => if(gt(X, double(Y)), X, Y) 
  if(true, X, Y) => aver(X, s(Y)) 
  if(false, X, Y) => 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:

  0 : [] --> tc 
  aver : [tc * tc] --> tc 
  average : [tc * tc] --> tc 
  double : [tc] --> tc 
  false : [] --> ic 
  gt : [tc * tc] --> ic 
  if : [ic * tc * tc] --> tc 
  plus : [tc * tc] --> tc 
  s : [tc] --> tc 
  true : [] --> ic 


+++ 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.