/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 
  cond : [o * o * o * o] --> o 
  d : [o * o * o] --> o 
  div : [o * o] --> o 
  false : [] --> o 
  ge : [o * o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  div(X, s(Y)) => d(X, s(Y), 0) 
  d(X, s(Y), Z) => cond(ge(X, Z), X, Y, Z) 
  cond(true, X, Y, Z) => s(d(X, s(Y), plus(s(Y), Z))) 
  cond(false, X, Y, Z) => 0 
  ge(X, 0) => true 
  ge(0, s(X)) => false 
  ge(s(X), s(Y)) => ge(X, Y) 
  plus(X, 0) => X 
  plus(X, s(Y)) => s(plus(X, 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 : [] --> yc 
  cond : [ic * yc * yc * yc] --> yc 
  d : [yc * yc * yc] --> yc 
  div : [yc * yc] --> yc 
  false : [] --> ic 
  ge : [yc * yc] --> ic 
  plus : [yc * yc] --> yc 
  s : [yc] --> yc 
  true : [] --> ic 


+++ 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.