/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 
  false : [] --> o 
  help : [o * o * o] --> o 
  if : [o * o * o * o] --> o 
  le : [o * o] --> o 
  minus : [o * o] --> o 
  mod : [o * o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  minus(X, 0) => X 
  minus(0, s(X)) => 0 
  minus(s(X), s(Y)) => minus(X, Y) 
  plus(X, 0) => X 
  plus(X, s(Y)) => s(plus(X, Y)) 
  mod(s(X), 0) => 0 
  mod(X, s(Y)) => help(X, s(Y), 0) 
  help(X, s(Y), Z) => if(le(Z, X), X, s(Y), Z) 
  if(true, X, s(Y), Z) => help(X, s(Y), plus(Z, s(Y))) 
  if(false, X, s(Y), Z) => minus(X, minus(Z, s(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 : [] --> ge 
  false : [] --> zc 
  help : [ge * ge * ge] --> ge 
  if : [zc * ge * ge * ge] --> ge 
  le : [ge * ge] --> zc 
  minus : [ge * ge] --> ge 
  mod : [ge * ge] --> ge 
  plus : [ge * ge] --> ge 
  s : [ge] --> ge 
  true : [] --> zc 


+++ 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.