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


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


MAYBE
We consider the system theBenchmark.

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

  0 : [] --> o 
  even : [o] --> o 
  false : [] --> o 
  half : [o] --> o 
  if!6220times : [o * o * o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  times : [o * o] --> o 
  true : [] --> o 

  even(0) => true 
  even(s(0)) => false 
  even(s(s(X))) => even(X) 
  half(0) => 0 
  half(s(s(X))) => s(half(X)) 
  plus(0, X) => X 
  plus(s(X), Y) => s(plus(X, Y)) 
  times(0, X) => 0 
  times(s(X), Y) => if!6220times(even(s(X)), s(X), Y) 
  if!6220times(true, s(X), Y) => plus(times(half(s(X)), Y), times(half(s(X)), Y)) 
  if!6220times(false, s(X), Y) => plus(Y, times(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 : [] --> dd 
  even : [dd] --> xb 
  false : [] --> xb 
  half : [dd] --> dd 
  if!6220times : [xb * dd * dd] --> dd 
  plus : [dd * dd] --> dd 
  s : [dd] --> dd 
  times : [dd * dd] --> dd 
  true : [] --> xb 


+++ 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.