/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


MAYBE
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  decr : [o] --> o 
  f : [o * o] --> o 
  s : [o] --> o 
  swap : [o * o] --> o 
  tt : [] --> o 

  f(tt, X) => f(swap(X, 0), s(X)) 
  swap(s(X), Y) => swap(X, s(Y)) 
  swap(0, s(X)) => decr(s(X)) 
  decr(s(X)) => decr(X) 
  decr(0) => tt 

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 : [] --> pa 
  decr : [pa] --> va 
  f : [va * pa] --> t 
  s : [pa] --> pa 
  swap : [pa * pa] --> va 
  tt : [] --> va 


+++ 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.