/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 
  double : [o] --> o 
  false : [] --> o 
  first : [] --> o 
  greater : [o * o] --> o 
  if : [o * o * o * o] --> o 
  le : [o * o * o] --> o 
  s : [o] --> o 
  second : [] --> o 
  triple : [o] --> o 
  true : [] --> o 

  le(0, X, Y) => greater(X, Y) 
  le(s(X), 0, Y) => false 
  le(s(X), s(Y), 0) => false 
  le(s(X), s(Y), s(Z)) => le(X, Y, Z) 
  greater(X, 0) => first 
  greater(0, s(X)) => second 
  greater(s(X), s(Y)) => greater(X, Y) 
  double(0) => 0 
  double(s(X)) => s(s(double(X))) 
  triple(X) => if(le(X, X, double(X)), X, 0, 0) 
  if(false, X, Y, Z) => true 
  if(first, X, Y, Z) => if(le(s(X), Y, s(Z)), s(X), Y, s(Z)) 
  if(second, X, Y, Z) => if(le(s(X), s(Y), Z), s(X), s(Y), 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 : [] --> pe 
  double : [pe] --> pe 
  false : [] --> le 
  first : [] --> le 
  greater : [pe * pe] --> le 
  if : [le * pe * pe * pe] --> re 
  le : [pe * pe * pe] --> le 
  s : [pe] --> pe 
  second : [] --> le 
  triple : [pe] --> re 
  true : [] --> re 


+++ 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.