/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.

  !minus : [o * o] --> o 
  0 : [] --> o 
  false : [] --> o 
  if : [o * o * o] --> o 
  leq : [o * o] --> o 
  mod : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  leq(0, X) => true 
  leq(s(X), 0) => false 
  leq(s(X), s(Y)) => leq(X, Y) 
  if(true, X, Y) => X 
  if(false, X, Y) => Y 
  !minus(X, 0) => X 
  !minus(s(X), s(Y)) => !minus(X, Y) 
  mod(0, X) => 0 
  mod(s(X), 0) => 0 
  mod(s(X), s(Y)) => if(leq(Y, X), mod(!minus(s(X), s(Y)), s(Y)), s(X)) 

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:

  !minus : [rc * rc] --> rc 
  0 : [] --> rc 
  false : [] --> gc 
  if : [gc * rc * rc] --> rc 
  leq : [rc * rc] --> gc 
  mod : [rc * rc] --> rc 
  s : [rc] --> rc 
  true : [] --> gc 


+++ 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.