/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 
  del : [o * o] --> o 
  eq : [o * o] --> o 
  false : [] --> o 
  if : [o * o * o * o] --> o 
  last : [o] --> o 
  nil : [] --> o 
  reverse : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  last(nil) => 0 
  last(cons(X, nil)) => X 
  last(cons(X, cons(Y, Z))) => last(cons(Y, Z)) 
  del(X, nil) => nil 
  del(X, cons(Y, Z)) => if(eq(X, Y), X, Y, Z) 
  if(true, X, Y, Z) => Z 
  if(false, X, Y, Z) => cons(Y, del(X, Z)) 
  eq(0, 0) => true 
  eq(0, s(X)) => false 
  eq(s(X), 0) => false 
  eq(s(X), s(Y)) => eq(X, Y) 
  reverse(nil) => nil 
  reverse(cons(X, Y)) => cons(last(cons(X, Y)), reverse(del(last(cons(X, Y)), cons(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 : [] --> vd 
  cons : [vd * be] --> be 
  del : [vd * be] --> be 
  eq : [vd * vd] --> dd 
  false : [] --> dd 
  if : [dd * vd * vd * be] --> be 
  last : [be] --> vd 
  nil : [] --> be 
  reverse : [be] --> be 
  s : [vd] --> vd 
  true : [] --> dd 


+++ 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.