/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 
  cond : [o * o * o] --> o 
  false : [] --> o 
  gt : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 
  while : [o * o * o] --> o 

  while(true, X, Y) => cond(gt(X, 0), X, Y) 
  cond(true, s(X), Y) => while(gt(Y, 0), X, Y) 
  cond(false, X, Y) => while(gt(s(Y), 0), s(Y), s(Y)) 
  gt(s(X), 0) => true 
  gt(0, X) => false 
  gt(s(X), s(Y)) => gt(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:

  0 : [] --> ob 
  cond : [sb * ob * ob] --> ya 
  false : [] --> sb 
  gt : [ob * ob] --> sb 
  s : [ob] --> ob 
  true : [] --> sb 
  while : [sb * ob * ob] --> ya 


+++ 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.