/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.

  append : [o * o] --> o 
  cons : [o * o] --> o 
  false : [] --> o 
  hd : [o] --> o 
  ifappend : [o * o * o] --> o 
  is!6220empty : [o] --> o 
  nil : [] --> o 
  tl : [o] --> o 
  true : [] --> o 

  is!6220empty(nil) => true 
  is!6220empty(cons(X, Y)) => false 
  hd(cons(X, Y)) => X 
  tl(cons(X, Y)) => Y 
  append(X, Y) => ifappend(X, Y, is!6220empty(X)) 
  ifappend(X, Y, true) => Y 
  ifappend(X, Y, false) => cons(hd(X), append(tl(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:

  append : [pb * pb] --> pb 
  cons : [kb * pb] --> pb 
  false : [] --> wa 
  hd : [pb] --> kb 
  ifappend : [pb * pb * wa] --> pb 
  is!6220empty : [pb] --> wa 
  nil : [] --> pb 
  tl : [pb] --> pb 
  true : [] --> wa 


+++ 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.