/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 
  false : [] --> o 
  ge : [o * o] --> o 
  half : [o] --> o 
  if : [o * o] --> o 
  log : [o] --> o 
  p : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  half(X) => if(ge(X, s(s(0))), X) 
  if(false, X) => 0 
  if(true, X) => s(half(p(p(X)))) 
  p(0) => 0 
  p(s(X)) => X 
  ge(X, 0) => true 
  ge(0, s(X)) => false 
  ge(s(X), s(Y)) => ge(X, Y) 
  log(0) => 0 
  log(s(X)) => s(log(half(s(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:

  0 : [] --> hc 
  false : [] --> tb 
  ge : [hc * hc] --> tb 
  half : [hc] --> hc 
  if : [tb * hc] --> hc 
  log : [hc] --> hc 
  p : [hc] --> hc 
  s : [hc] --> hc 
  true : [] --> tb 


+++ 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.