/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 
  average : [o * o] --> o 
  false : [] --> o 
  if : [o * o * o * o * o * o] --> o 
  if2 : [o * o * o * o * o] --> o 
  if3 : [o * o * o * o] --> o 
  if4 : [o * o * o] --> o 
  le : [o * o] --> o 
  p : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  p(s(X)) => X 
  p(0) => 0 
  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  average(X, Y) => if(le(X, 0), le(Y, 0), le(Y, s(0)), le(Y, s(s(0))), X, Y) 
  if(true, X, Y, Z, U, V) => if2(X, Y, Z, U, V) 
  if(false, X, Y, Z, U, V) => average(p(U), s(V)) 
  if2(true, X, Y, Z, U) => 0 
  if2(false, X, Y, Z, U) => if3(X, Y, Z, U) 
  if3(true, X, Y, Z) => 0 
  if3(false, X, Y, Z) => if4(X, Y, Z) 
  if4(true, X, Y) => s(0) 
  if4(false, X, Y) => average(s(X), p(p(Y))) 

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 : [] --> ff 
  average : [ff * ff] --> ff 
  false : [] --> dc 
  if : [dc * dc * dc * dc * ff * ff] --> ff 
  if2 : [dc * dc * dc * ff * ff] --> ff 
  if3 : [dc * dc * ff * ff] --> ff 
  if4 : [dc * ff * ff] --> ff 
  le : [ff * ff] --> dc 
  p : [ff] --> ff 
  s : [ff] --> ff 
  true : [] --> dc 


+++ 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.