/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 
  gcd : [o * o] --> o 
  if!6220gcd : [o * o * o] --> o 
  le : [o * o] --> o 
  minus : [o * o] --> o 
  pred : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  pred(s(X)) => X 
  minus(X, 0) => X 
  minus(X, s(Y)) => pred(minus(X, Y)) 
  gcd(0, X) => X 
  gcd(s(X), 0) => s(X) 
  gcd(s(X), s(Y)) => if!6220gcd(le(Y, X), s(X), s(Y)) 
  if!6220gcd(true, X, Y) => gcd(minus(X, Y), Y) 
  if!6220gcd(false, X, Y) => gcd(minus(Y, X), 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 : [] --> bd 
  false : [] --> cc 
  gcd : [bd * bd] --> bd 
  if!6220gcd : [cc * bd * bd] --> bd 
  le : [bd * bd] --> cc 
  minus : [bd * bd] --> bd 
  pred : [bd] --> bd 
  s : [bd] --> bd 
  true : [] --> cc 


+++ 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.