/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.

  append : [o * o] --> o 
  cons : [o * o] --> o 
  elem : [o] --> o 
  empty : [] --> o 
  false : [] --> o 
  if : [o * o * o * o * o * o] --> o 
  isEmpty : [o] --> o 
  left : [o] --> o 
  listify : [o * o] --> o 
  nil : [] --> o 
  node : [o * o * o] --> o 
  right : [o] --> o 
  toList : [o] --> o 
  true : [] --> o 
  y : [] --> o 

  isEmpty(empty) => true 
  isEmpty(node(X, Y, Z)) => false 
  left(empty) => empty 
  left(node(X, Y, Z)) => X 
  right(empty) => empty 
  right(node(X, Y, Z)) => Z 
  elem(node(X, Y, Z)) => Y 
  append(nil, X) => cons(X, nil) 
  append(cons(y, X), Y) => cons(y, append(X, Y)) 
  listify(X, Y) => if(isEmpty(X), isEmpty(left(X)), right(X), node(left(left(X)), elem(left(X)), node(right(left(X)), elem(X), right(X))), Y, append(Y, X)) 
  if(true, X, Y, Z, U, V) => U 
  if(false, false, X, Y, Z, U) => listify(Y, Z) 
  if(false, true, X, Y, Z, U) => listify(X, U) 
  toList(X) => listify(X, nil) 

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:

  append : [ff * yd] --> ff 
  cons : [yd * ff] --> ff 
  elem : [yd] --> yd 
  empty : [] --> yd 
  false : [] --> vd 
  if : [vd * vd * yd * yd * ff * ff] --> ff 
  isEmpty : [yd] --> vd 
  left : [yd] --> yd 
  listify : [yd * ff] --> ff 
  nil : [] --> ff 
  node : [yd * yd * yd] --> yd 
  right : [yd] --> yd 
  toList : [yd] --> ff 
  true : [] --> vd 
  y : [] --> yd 


+++ 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.