/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 
  double : [o] --> o 
  eq : [o * o] --> o 
  f : [o * o] --> o 
  half : [o] --> o 
  s : [o] --> o 
  tt : [] --> o 

  f(tt, X) => f(eq(X, half(double(X))), s(X)) 
  eq(s(X), s(Y)) => eq(X, Y) 
  eq(0, 0) => tt 
  double(s(X)) => s(s(double(X))) 
  double(0) => 0 
  half(s(s(X))) => s(half(X)) 
  half(0) => 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 : [] --> lb 
  double : [lb] --> lb 
  eq : [lb * lb] --> na 
  f : [na * lb] --> y 
  half : [lb] --> lb 
  s : [lb] --> lb 
  tt : [] --> na 


+++ 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.