/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 
  Cons : [o * o] --> o 
  cond : [o * o] --> o 
  downfrom : [o] --> o 
  f : [o] --> o 
  isList : [o] --> o 
  nil : [] --> o 
  s : [o] --> o 
  tt : [] --> o 

  isList(nil) => tt 
  isList(Cons(X, Y)) => isList(Y) 
  downfrom(0) => nil 
  downfrom(s(X)) => Cons(s(X), downfrom(X)) 
  f(X) => cond(isList(downfrom(X)), s(X)) 
  cond(tt, X) => f(X) 

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 : [] --> va 
  Cons : [va * sa] --> sa 
  cond : [ta * va] --> cb 
  downfrom : [va] --> sa 
  f : [va] --> cb 
  isList : [sa] --> ta 
  nil : [] --> sa 
  s : [va] --> va 
  tt : [] --> ta 


+++ 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.