/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 
  adx : [o] --> o 
  cons : [o * o] --> o 
  hd : [o] --> o 
  incr : [o] --> o 
  nats : [] --> o 
  s : [o] --> o 
  tl : [o] --> o 
  zeros : [] --> o 

  nats => adx(zeros) 
  zeros => cons(0, zeros) 
  incr(cons(X, Y)) => cons(s(X), incr(Y)) 
  adx(cons(X, Y)) => incr(cons(X, adx(Y))) 
  hd(cons(X, Y)) => X 
  tl(cons(X, Y)) => 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 : [] --> da 
  adx : [cb] --> cb 
  cons : [da * cb] --> cb 
  hd : [cb] --> da 
  incr : [cb] --> cb 
  nats : [] --> cb 
  s : [da] --> da 
  tl : [cb] --> cb 
  zeros : [] --> cb 


+++ 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.