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

  f(true, X, Y) => f(gt(X, Y), double(X), s(Y)) 
  gt(s(X), 0) => true 
  gt(0, X) => false 
  gt(s(X), s(Y)) => gt(X, Y) 
  double(X) => times(X, s(s(0))) 
  times(0, X) => 0 
  times(s(X), Y) => plus(Y, times(X, Y)) 
  plus(0, X) => X 
  plus(s(X), Y) => plus(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 : [] --> kc 
  double : [kc] --> kc 
  f : [xa * kc * kc] --> ba 
  false : [] --> xa 
  gt : [kc * kc] --> xa 
  plus : [kc * kc] --> kc 
  s : [kc] --> kc 
  times : [kc * kc] --> kc 
  true : [] --> xa 


+++ 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.