/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 
  divides : [o * o] --> o 
  false : [] --> o 
  filter : [o * o] --> o 
  from : [o] --> o 
  head : [o] --> o 
  if : [o * o * o] --> o 
  primes : [] --> o 
  s : [o] --> o 
  sieve : [o] --> o 
  tail : [o] --> o 
  true : [] --> o 

  primes => sieve(from(s(s(0)))) 
  from(X) => cons(X, from(s(X))) 
  head(cons(X, Y)) => X 
  tail(cons(X, Y)) => Y 
  if(true, X, Y) => X 
  if(false, X, Y) => Y 
  filter(s(s(X)), cons(Y, Z)) => if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) 
  sieve(cons(X, Y)) => cons(X, filter(X, sieve(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 : [] --> tc 
  cons : [tc * tc] --> tc 
  divides : [tc * tc] --> wb 
  false : [] --> wb 
  filter : [tc * tc] --> tc 
  from : [tc] --> tc 
  head : [tc] --> tc 
  if : [wb * tc * tc] --> tc 
  primes : [] --> tc 
  s : [tc] --> tc 
  sieve : [tc] --> tc 
  tail : [tc] --> tc 
  true : [] --> wb 


+++ 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.