/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 
  add : [o * o] --> o 
  app : [o * o] --> o 
  eq : [o * o] --> o 
  false : [] --> o 
  head : [o] --> o 
  if : [o * o * o * o] --> o 
  if!6220min : [o * o] --> o 
  if!6220rm : [o * o * o] --> o 
  if2 : [o * o * o * o] --> o 
  le : [o * o] --> o 
  min : [o] --> o 
  mins : [o * o * o] --> o 
  minsort : [o] --> o 
  nil : [] --> o 
  null : [o] --> o 
  rm : [o * o] --> o 
  s : [o] --> o 
  tail : [o] --> o 
  true : [] --> o 

  eq(0, 0) => true 
  eq(0, s(X)) => false 
  eq(s(X), 0) => false 
  eq(s(X), s(Y)) => eq(X, Y) 
  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)) 
  min(add(X, nil)) => X 
  min(add(X, add(Y, Z))) => if!6220min(le(X, Y), add(X, add(Y, Z))) 
  if!6220min(true, add(X, add(Y, Z))) => min(add(X, Z)) 
  if!6220min(false, add(X, add(Y, Z))) => min(add(Y, Z)) 
  head(add(X, Y)) => X 
  tail(add(X, Y)) => Y 
  tail(nil) => nil 
  null(nil) => true 
  null(add(X, Y)) => false 
  rm(X, nil) => nil 
  rm(X, add(Y, Z)) => if!6220rm(eq(X, Y), X, add(Y, Z)) 
  if!6220rm(true, X, add(Y, Z)) => rm(X, Z) 
  if!6220rm(false, X, add(Y, Z)) => add(Y, rm(X, Z)) 
  minsort(X) => mins(X, nil, nil) 
  mins(X, Y, Z) => if(null(X), X, Y, Z) 
  if(true, X, Y, Z) => Z 
  if(false, X, Y, Z) => if2(eq(head(X), min(X)), X, Y, Z) 
  if2(true, X, Y, Z) => mins(app(rm(head(X), tail(X)), Y), nil, app(Z, add(head(X), nil))) 
  if2(false, X, Y, Z) => mins(tail(X), add(head(X), Y), 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 : [] --> pk 
  add : [pk * pk] --> pk 
  app : [pk * pk] --> pk 
  eq : [pk * pk] --> cj 
  false : [] --> cj 
  head : [pk] --> pk 
  if : [cj * pk * pk * pk] --> pk 
  if!6220min : [cj * pk] --> pk 
  if!6220rm : [cj * pk * pk] --> pk 
  if2 : [cj * pk * pk * pk] --> pk 
  le : [pk * pk] --> cj 
  min : [pk] --> pk 
  mins : [pk * pk * pk] --> pk 
  minsort : [pk] --> pk 
  nil : [] --> pk 
  null : [pk] --> cj 
  rm : [pk * pk] --> pk 
  s : [pk] --> pk 
  tail : [pk] --> pk 
  true : [] --> cj 


+++ 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.