/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


MAYBE
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  cons : [o * o] --> o 
  conv : [o] --> o 
  conviter : [o * o] --> o 
  false : [] --> o 
  half : [o] --> o 
  if : [o * o * o] --> o 
  lastbit : [o] --> o 
  nil : [] --> o 
  s : [o] --> o 
  true : [] --> o 
  zero : [o] --> o 

  half(0) => 0 
  half(s(0)) => 0 
  half(s(s(X))) => s(half(X)) 
  lastbit(0) => 0 
  lastbit(s(0)) => s(0) 
  lastbit(s(s(X))) => lastbit(X) 
  zero(0) => true 
  zero(s(X)) => false 
  conv(X) => conviter(X, cons(0, nil)) 
  conviter(X, Y) => if(zero(X), X, Y) 
  if(true, X, Y) => Y 
  if(false, X, Y) => conviter(half(X), cons(lastbit(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 : [] --> vc 
  cons : [vc * yc] --> yc 
  conv : [vc] --> yc 
  conviter : [vc * yc] --> yc 
  false : [] --> dc 
  half : [vc] --> vc 
  if : [dc * vc * yc] --> yc 
  lastbit : [vc] --> vc 
  nil : [] --> yc 
  s : [vc] --> vc 
  true : [] --> dc 
  zero : [vc] --> dc 


+++ 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.