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


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


MAYBE
We consider the system theBenchmark.

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

  0 : [] --> o 
  div : [o * o * o] --> o 
  divides : [o * o] --> o 
  false : [] --> o 
  gt : [o * o] --> o 
  if1 : [o * o * o] --> o 
  if2 : [o * o * o] --> o 
  prime : [o] --> o 
  s : [o] --> o 
  test : [o * o] --> o 
  true : [] --> o 

  gt(s(X), 0) => true 
  gt(0, X) => false 
  gt(s(X), s(Y)) => gt(X, Y) 
  divides(X, Y) => div(X, Y, Y) 
  div(0, 0, X) => true 
  div(0, s(X), Y) => false 
  div(s(X), 0, s(Y)) => div(s(X), s(Y), s(Y)) 
  div(s(X), s(Y), Z) => div(X, Y, Z) 
  prime(X) => test(X, s(s(0))) 
  test(X, Y) => if1(gt(X, Y), X, Y) 
  if1(true, X, Y) => if2(divides(X, Y), X, Y) 
  if1(false, X, Y) => true 
  if2(true, X, Y) => false 
  if2(false, X, Y) => test(X, s(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 : [] --> le 
  div : [le * le * le] --> me 
  divides : [le * le] --> me 
  false : [] --> me 
  gt : [le * le] --> me 
  if1 : [me * le * le] --> me 
  if2 : [me * le * le] --> me 
  prime : [le] --> me 
  s : [le] --> le 
  test : [le * le] --> me 
  true : [] --> me 


+++ 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.