/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 
  eq : [o * o] --> o 
  false : [] --> o 
  if1 : [o * o * o * o] --> o 
  if2 : [o * o * o * o] --> o 
  le : [o * o] --> o 
  min : [o] --> o 
  minsort : [o] --> o 
  nil : [] --> o 
  rm : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  eq(0, 0) => true 
  eq(0, s(X)) => false 
  eq(s(X), 0) => false 
  eq(s(X), s(Y)) => eq(X, Y) 
  minsort(nil) => nil 
  minsort(cons(X, Y)) => cons(min(cons(X, Y)), minsort(rm(min(cons(X, Y)), cons(X, Y)))) 
  min(nil) => 0 
  min(cons(X, nil)) => X 
  min(cons(X, cons(Y, Z))) => if1(le(X, Y), X, Y, Z) 
  if1(true, X, Y, Z) => min(cons(X, Z)) 
  if1(false, X, Y, Z) => min(cons(Y, Z)) 
  rm(X, nil) => nil 
  rm(X, cons(Y, Z)) => if2(eq(X, Y), X, Y, Z) 
  if2(true, X, Y, Z) => rm(X, Z) 
  if2(false, X, Y, Z) => cons(Y, rm(X, 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 : [] --> oe 
  cons : [oe * bg] --> bg 
  eq : [oe * oe] --> df 
  false : [] --> df 
  if1 : [df * oe * oe * bg] --> oe 
  if2 : [df * oe * oe * bg] --> bg 
  le : [oe * oe] --> df 
  min : [bg] --> oe 
  minsort : [bg] --> bg 
  nil : [] --> bg 
  rm : [oe * bg] --> bg 
  s : [oe] --> oe 
  true : [] --> df 


+++ 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.