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


--------------------------------------------------------------------------------


MAYBE
We consider the system theBenchmark.

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

  0 : [] --> o 
  false : [] --> o 
  if!6220quot : [o * o * o] --> o 
  le : [o * o] --> o 
  minus : [o * o] --> o 
  quot : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  minus(X, 0) => X 
  minus(s(X), s(Y)) => minus(X, Y) 
  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  quot(X, s(Y)) => if!6220quot(le(s(Y), X), X, s(Y)) 
  if!6220quot(true, X, Y) => s(quot(minus(X, Y), Y)) 
  if!6220quot(false, X, Y) => 0 

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 : [] --> bc 
  false : [] --> hb 
  if!6220quot : [hb * bc * bc] --> bc 
  le : [bc * bc] --> hb 
  minus : [bc * bc] --> bc 
  quot : [bc * bc] --> bc 
  s : [bc] --> bc 
  true : [] --> hb 


+++ 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.