/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 
  and : [o * o] --> o 
  double : [o] --> o 
  f : [o * o * o] --> o 
  false : [] --> o 
  gt : [o * o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  f(true, X, Y) => f(and(gt(X, Y), gt(Y, s(s(0)))), plus(s(0), X), double(Y)) 
  gt(0, X) => false 
  gt(s(X), 0) => true 
  gt(s(X), s(Y)) => gt(X, Y) 
  and(X, true) => X 
  and(X, false) => false 
  plus(X, 0) => X 
  plus(X, s(Y)) => s(plus(X, Y)) 
  double(0) => 0 
  double(s(X)) => s(s(double(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 : [] --> wc 
  and : [wc * wc] --> wc 
  double : [wc] --> wc 
  f : [wc * wc * wc] --> na 
  false : [] --> wc 
  gt : [wc * wc] --> wc 
  plus : [wc * wc] --> wc 
  s : [wc] --> wc 
  true : [] --> wc 


+++ 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.