/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 
  cond1 : [o * o] --> o 
  cond2 : [o * o] --> o 
  div2 : [o] --> o 
  even : [o] --> o 
  false : [] --> o 
  neq : [o * o] --> o 
  p : [o] --> o 
  s : [o] --> o 
  true : [] --> o 
  y : [] --> o 

  cond1(true, X) => cond2(even(X), X) 
  cond2(true, X) => cond1(neq(X, 0), div2(X)) 
  cond2(false, X) => cond1(neq(X, 0), p(X)) 
  neq(0, 0) => false 
  neq(0, s(X)) => true 
  neq(s(X), 0) => true 
  neq(s(X), s(y)) => neq(X, y) 
  even(0) => true 
  even(s(0)) => false 
  even(s(s(X))) => even(X) 
  div2(0) => 0 
  div2(s(0)) => 0 
  div2(s(s(X))) => s(div2(X)) 
  p(0) => 0 
  p(s(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 : [] --> qd 
  cond1 : [rc * qd] --> za 
  cond2 : [rc * qd] --> za 
  div2 : [qd] --> qd 
  even : [qd] --> rc 
  false : [] --> rc 
  neq : [qd * qd] --> rc 
  p : [qd] --> qd 
  s : [qd] --> qd 
  true : [] --> rc 
  y : [] --> qd 


+++ 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.