/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 
  false : [] --> o 
  if : [o * o * o * o * o * o] --> o 
  if2 : [o * o * o * o * o * o] --> o 
  le : [o * o] --> o 
  quot : [o * o] --> o 
  quotIter : [o * o * o * o * o] --> o 
  quotZeroErro : [] --> o 
  s : [o] --> o 
  true : [] --> o 

  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  quot(X, 0) => quotZeroErro 
  quot(X, s(Y)) => quotIter(X, s(Y), 0, 0, 0) 
  quotIter(X, s(Y), Z, U, V) => if(le(X, Z), X, s(Y), Z, U, V) 
  if(true, X, Y, Z, U, V) => V 
  if(false, X, Y, Z, U, V) => if2(le(Y, s(U)), X, Y, s(Z), s(U), V) 
  if2(false, X, Y, Z, U, V) => quotIter(X, Y, Z, U, V) 
  if2(true, X, Y, Z, U, V) => quotIter(X, Y, Z, 0, s(V)) 

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 : [] --> ge 
  false : [] --> vc 
  if : [vc * ge * ge * ge * ge * ge] --> ge 
  if2 : [vc * ge * ge * ge * ge * ge] --> ge 
  le : [ge * ge] --> vc 
  quot : [ge * ge] --> ge 
  quotIter : [ge * ge * ge * ge * ge] --> ge 
  quotZeroErro : [] --> ge 
  s : [ge] --> ge 
  true : [] --> vc 


+++ 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.