/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 
  ack : [o * o] --> o 
  cons : [o * o] --> o 
  d : [o] --> o 
  false : [] --> o 
  if : [o * o] --> o 
  le : [o * o] --> o 
  nil : [] --> o 
  nr : [] --> o 
  numbers : [] --> o 
  s : [o] --> o 
  true : [] --> o 

  numbers => d(0) 
  d(X) => if(le(X, nr), X) 
  if(true, X) => cons(X, d(s(X))) 
  if(false, X) => nil 
  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  nr => ack(s(s(s(s(s(s(0)))))), 0) 
  ack(0, X) => s(X) 
  ack(s(X), 0) => ack(X, s(0)) 
  ack(s(X), s(Y)) => ack(X, ack(s(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 
  ack : [dd * dd] --> dd 
  cons : [dd * va] --> va 
  d : [dd] --> va 
  false : [] --> qb 
  if : [qb * dd] --> va 
  le : [dd * dd] --> qb 
  nil : [] --> va 
  nr : [] --> dd 
  numbers : [] --> va 
  s : [dd] --> dd 
  true : [] --> qb 


+++ 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.