/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 
  bitIter : [o * o] --> o 
  bits : [o] --> o 
  false : [] --> o 
  half : [o] --> o 
  if : [o * o * o] --> o 
  inc : [o] --> o 
  p : [o] --> o 
  s : [o] --> o 
  true : [] --> o 
  zero : [o] --> o 

  half(0) => 0 
  half(s(0)) => 0 
  half(s(s(X))) => s(half(X)) 
  inc(0) => 0 
  inc(s(X)) => s(inc(X)) 
  zero(0) => true 
  zero(s(X)) => false 
  p(0) => 0 
  p(s(X)) => X 
  bits(X) => bitIter(X, 0) 
  bitIter(X, Y) => if(zero(X), X, inc(Y)) 
  if(true, X, Y) => p(Y) 
  if(false, X, Y) => bitIter(half(X), 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 : [] --> yc 
  bitIter : [yc * yc] --> yc 
  bits : [yc] --> yc 
  false : [] --> ec 
  half : [yc] --> yc 
  if : [ec * yc * yc] --> yc 
  inc : [yc] --> yc 
  p : [yc] --> yc 
  s : [yc] --> yc 
  true : [] --> ec 
  zero : [yc] --> ec 


+++ 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.