/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 
  cons : [o * o] --> o 
  filter : [o * o * o] --> o 
  nats : [o] --> o 
  s : [o] --> o 
  sieve : [o] --> o 
  zprimes : [] --> o 

  filter(cons(X, Y), 0, Z) => cons(0, filter(Y, Z, Z)) 
  filter(cons(X, Y), s(Z), U) => cons(X, filter(Y, Z, U)) 
  sieve(cons(0, X)) => cons(0, sieve(X)) 
  sieve(cons(s(X), Y)) => cons(s(X), sieve(filter(Y, X, X))) 
  nats(X) => cons(X, nats(s(X))) 
  zprimes => sieve(nats(s(s(0)))) 

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 : [] --> ac 
  cons : [ac * cc] --> cc 
  filter : [cc * ac * ac] --> cc 
  nats : [ac] --> cc 
  s : [ac] --> ac 
  sieve : [cc] --> cc 
  zprimes : [] --> cc 


+++ 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.