/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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 
  10 : [] --> o 
  1024 : [] --> o 
  1024!62201 : [o] --> o 
  double : [o] --> o 
  false : [] --> o 
  if : [o * o] --> o 
  lt : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  1024 => 1024!62201(0) 
  1024!62201(X) => if(lt(X, 10), X) 
  if(true, X) => double(1024!62201(s(X))) 
  if(false, X) => s(0) 
  lt(0, s(X)) => true 
  lt(X, 0) => false 
  lt(s(X), s(Y)) => lt(X, Y) 
  double(0) => 0 
  double(s(X)) => s(s(double(X))) 
  10 => double(s(double(s(s(0))))) 

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 : [] --> ic 
  10 : [] --> ic 
  1024 : [] --> ic 
  1024!62201 : [ic] --> ic 
  double : [ic] --> ic 
  false : [] --> ob 
  if : [ob * ic] --> ic 
  lt : [ic * ic] --> ob 
  s : [ic] --> ic 
  true : [] --> ob 


+++ 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.