/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 
  cond1 : [o * o * o] --> o 
  cond2 : [o * o * o] --> o 
  cons : [o * o] --> o 
  equal : [o * o] --> o 
  false : [] --> o 
  gt : [o * o] --> o 
  if : [o * o * o] --> o 
  max : [o] --> o 
  member : [o * o] --> o 
  nil : [] --> o 
  or : [o * o] --> o 
  s : [o] --> o 
  sort : [o] --> o 
  st : [o * o] --> o 
  true : [] --> o 

  sort(X) => st(0, X) 
  st(X, Y) => cond1(member(X, Y), X, Y) 
  cond1(true, X, Y) => cons(X, st(s(X), Y)) 
  cond1(false, X, Y) => cond2(gt(X, max(Y)), X, Y) 
  cond2(true, X, Y) => nil 
  cond2(false, X, Y) => st(s(X), Y) 
  member(X, nil) => false 
  member(X, cons(Y, Z)) => or(equal(X, Y), member(X, Z)) 
  or(X, true) => true 
  or(X, false) => X 
  equal(0, 0) => true 
  equal(s(X), 0) => false 
  equal(0, s(X)) => false 
  equal(s(X), s(Y)) => equal(X, Y) 
  gt(0, X) => false 
  gt(s(X), 0) => true 
  gt(s(X), s(Y)) => gt(X, Y) 
  max(nil) => 0 
  max(cons(X, Y)) => if(gt(X, max(Y)), X, max(Y)) 
  if(true, X, Y) => X 
  if(false, 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 : [] --> qg 
  cond1 : [qg * qg * vf] --> vf 
  cond2 : [qg * qg * vf] --> vf 
  cons : [qg * vf] --> vf 
  equal : [qg * qg] --> qg 
  false : [] --> qg 
  gt : [qg * qg] --> qg 
  if : [qg * qg * qg] --> qg 
  max : [vf] --> qg 
  member : [qg * vf] --> qg 
  nil : [] --> vf 
  or : [qg * qg] --> qg 
  s : [qg] --> qg 
  sort : [vf] --> vf 
  st : [qg * vf] --> vf 
  true : [] --> qg 


+++ 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.