/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


MAYBE
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  count : [o * o] --> o 
  empty : [] --> o 
  false : [] --> o 
  if : [o * o * o * o * o * o] --> o 
  inc : [o] --> o 
  isEmpty : [o] --> o 
  left : [o] --> o 
  node : [o * o] --> o 
  nrOfNodes : [o] --> o 
  right : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  isEmpty(empty) => true 
  isEmpty(node(X, Y)) => false 
  left(empty) => empty 
  left(node(X, Y)) => X 
  right(empty) => empty 
  right(node(X, Y)) => Y 
  inc(0) => s(0) 
  inc(s(X)) => s(inc(X)) 
  count(X, Y) => if(isEmpty(X), isEmpty(left(X)), right(X), node(left(left(X)), node(right(left(X)), right(X))), Y, inc(Y)) 
  if(true, X, Y, Z, U, V) => U 
  if(false, false, X, Y, Z, U) => count(Y, Z) 
  if(false, true, X, Y, Z, U) => count(X, U) 
  nrOfNodes(X) => count(X, 0) 

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 : [] --> fe 
  count : [xc * fe] --> fe 
  empty : [] --> xc 
  false : [] --> uc 
  if : [uc * uc * xc * xc * fe * fe] --> fe 
  inc : [fe] --> fe 
  isEmpty : [xc] --> uc 
  left : [xc] --> xc 
  node : [xc * xc] --> xc 
  nrOfNodes : [xc] --> fe 
  right : [xc] --> xc 
  s : [fe] --> fe 
  true : [] --> uc 


+++ 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.