/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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 
  cons : [o * o] --> o 
  del : [o * o] --> o 
  empty : [o] --> o 
  eq : [o * o] --> o 
  false : [] --> o 
  ge : [o * o] --> o 
  if1 : [o * o * o * o] --> o 
  if2 : [o * o * o * o] --> o 
  if3 : [o * o] --> o 
  max : [o] --> o 
  nil : [] --> o 
  s : [o] --> o 
  sort : [o] --> o 
  true : [] --> o 

  max(nil) => 0 
  max(cons(X, nil)) => X 
  max(cons(X, cons(Y, Z))) => if1(ge(X, Y), X, Y, Z) 
  if1(true, X, Y, Z) => max(cons(X, Z)) 
  if1(false, X, Y, Z) => max(cons(Y, Z)) 
  del(X, nil) => nil 
  del(X, cons(Y, Z)) => if2(eq(X, Y), X, Y, Z) 
  if2(true, X, Y, Z) => Z 
  if2(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) 
  sort(X) => if3(empty(X), X) 
  if3(true, X) => nil 
  if3(false, X) => sort(del(max(X), X)) 
  empty(nil) => true 
  empty(cons(X, Y)) => false 
  ge(X, 0) => true 
  ge(0, s(X)) => false 
  ge(s(X), s(Y)) => ge(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 : [] --> ig 
  cons : [ig * qf] --> qf 
  del : [ig * qf] --> qf 
  empty : [qf] --> mg 
  eq : [ig * ig] --> mg 
  false : [] --> mg 
  ge : [ig * ig] --> mg 
  if1 : [mg * ig * ig * qf] --> ig 
  if2 : [mg * ig * ig * qf] --> qf 
  if3 : [mg * qf] --> qf 
  max : [qf] --> ig 
  nil : [] --> qf 
  s : [ig] --> ig 
  sort : [qf] --> qf 
  true : [] --> mg 


+++ 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.