/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 
  append : [o * o] --> o 
  cons : [o * o] --> o 
  false : [] --> o 
  filterhigh : [o * o] --> o 
  filterlow : [o * o] --> o 
  ge : [o * o] --> o 
  if1 : [o * o * o * o] --> o 
  if2 : [o * o * o * o] --> o 
  nil : [] --> o 
  qsort : [o] --> o 
  s : [o] --> o 
  true : [] --> o 
  ys : [] --> o 

  qsort(nil) => nil 
  qsort(cons(X, Y)) => append(qsort(filterlow(X, cons(X, Y))), cons(X, qsort(filterhigh(X, cons(X, Y))))) 
  filterlow(X, nil) => nil 
  filterlow(X, cons(Y, Z)) => if1(ge(X, Y), X, Y, Z) 
  if1(true, X, Y, Z) => filterlow(X, Z) 
  if1(false, X, Y, Z) => cons(Y, filterlow(X, Z)) 
  filterhigh(X, nil) => nil 
  filterhigh(X, cons(Y, Z)) => if2(ge(Y, X), X, Y, Z) 
  if2(true, X, Y, Z) => filterhigh(X, Z) 
  if2(false, X, Y, Z) => cons(Y, filterhigh(X, Z)) 
  ge(X, 0) => true 
  ge(0, s(X)) => false 
  ge(s(X), s(Y)) => ge(X, Y) 
  append(nil, ys) => ys 
  append(cons(X, Y), ys) => cons(X, append(Y, ys)) 

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 : [] --> qe 
  append : [lf * lf] --> lf 
  cons : [qe * lf] --> lf 
  false : [] --> ue 
  filterhigh : [qe * lf] --> lf 
  filterlow : [qe * lf] --> lf 
  ge : [qe * qe] --> ue 
  if1 : [ue * qe * qe * lf] --> lf 
  if2 : [ue * qe * qe * lf] --> lf 
  nil : [] --> lf 
  qsort : [lf] --> lf 
  s : [qe] --> qe 
  true : [] --> ue 
  ys : [] --> lf 


+++ 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.