/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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 
  false : [] --> o 
  fib : [o] --> o 
  fibo : [o] --> o 
  if : [o * o * o * o * o] --> o 
  lt : [o * o] --> o 
  s : [o] --> o 
  sum : [o * o] --> o 
  true : [] --> o 

  lt(0, s(X)) => true 
  lt(X, 0) => false 
  lt(s(X), s(Y)) => lt(X, Y) 
  fibo(0) => fib(0) 
  fibo(s(0)) => fib(s(0)) 
  fibo(s(s(X))) => sum(fibo(s(X)), fibo(X)) 
  fib(0) => s(0) 
  fib(s(0)) => s(0) 
  fib(s(s(X))) => if(true, 0, s(s(X)), 0, 0) 
  if(true, X, s(s(Y)), Z, U) => if(lt(s(X), s(s(Y))), s(X), s(s(Y)), U, X) 
  if(false, X, s(s(Y)), Z, U) => sum(fibo(Z), fibo(U)) 
  sum(X, 0) => X 
  sum(X, s(Y)) => s(sum(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 : [] --> ne 
  false : [] --> dd 
  fib : [ne] --> ne 
  fibo : [ne] --> ne 
  if : [dd * ne * ne * ne * ne] --> ne 
  lt : [ne * ne] --> dd 
  s : [ne] --> ne 
  sum : [ne * ne] --> ne 
  true : [] --> dd 


+++ 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.