/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 
  add : [o * o] --> o 
  cons : [o * o] --> o 
  fib : [o] --> o 
  fib1 : [o * o] --> o 
  s : [o] --> o 
  sel : [o * o] --> o 

  fib(X) => sel(X, fib1(s(0), s(0))) 
  fib1(X, Y) => cons(X, fib1(Y, add(X, Y))) 
  add(0, X) => X 
  add(s(X), Y) => s(add(X, Y)) 
  sel(0, cons(X, Y)) => X 
  sel(s(X), cons(Y, Z)) => sel(X, Z) 

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 : [] --> pb 
  add : [pb * pb] --> pb 
  cons : [pb * lb] --> lb 
  fib : [pb] --> pb 
  fib1 : [pb * pb] --> lb 
  s : [pb] --> pb 
  sel : [pb * lb] --> pb 


+++ 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.