/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 
  add : [o * o] --> o 
  app : [o * o] --> o 
  false : [] --> o 
  head : [o] --> o 
  high : [o * o] --> o 
  if!6220high : [o * o * o] --> o 
  if!6220low : [o * o * o] --> o 
  if!6220qs : [o * o * o * o] --> o 
  isempty : [o] --> o 
  le : [o * o] --> o 
  low : [o * o] --> o 
  nil : [] --> o 
  quicksort : [o] --> o 
  s : [o] --> o 
  tail : [o] --> o 
  true : [] --> o 

  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  app(nil, X) => X 
  app(add(X, Y), Z) => add(X, app(Y, Z)) 
  low(X, nil) => nil 
  low(X, add(Y, Z)) => if!6220low(le(Y, X), X, add(Y, Z)) 
  if!6220low(true, X, add(Y, Z)) => add(Y, low(X, Z)) 
  if!6220low(false, X, add(Y, Z)) => low(X, Z) 
  high(X, nil) => nil 
  high(X, add(Y, Z)) => if!6220high(le(Y, X), X, add(Y, Z)) 
  if!6220high(true, X, add(Y, Z)) => high(X, Z) 
  if!6220high(false, X, add(Y, Z)) => add(Y, high(X, Z)) 
  head(add(X, Y)) => X 
  tail(add(X, Y)) => Y 
  isempty(nil) => true 
  isempty(add(X, Y)) => false 
  quicksort(X) => if!6220qs(isempty(X), low(head(X), tail(X)), head(X), high(head(X), tail(X))) 
  if!6220qs(true, X, Y, Z) => nil 
  if!6220qs(false, X, Y, Z) => app(quicksort(X), add(Y, quicksort(Z))) 

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 : [] --> jh 
  add : [jh * jh] --> jh 
  app : [jh * jh] --> jh 
  false : [] --> cg 
  head : [jh] --> jh 
  high : [jh * jh] --> jh 
  if!6220high : [cg * jh * jh] --> jh 
  if!6220low : [cg * jh * jh] --> jh 
  if!6220qs : [cg * jh * jh * jh] --> jh 
  isempty : [jh] --> cg 
  le : [jh * jh] --> cg 
  low : [jh * jh] --> jh 
  nil : [] --> jh 
  quicksort : [jh] --> jh 
  s : [jh] --> jh 
  tail : [jh] --> jh 
  true : [] --> cg 


+++ 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.