/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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 
  d : [o * o] --> o 
  g : [o * o] --> o 
  h : [o * o] --> o 

  h(g(X, Y), Z) => h(d(g(X, Y), Z), g(Z, 0)) 
  d(0, X) => 0 
  d(g(X, 0), Y) => X 
  d(g(X, g(Y, 0)), 0) => X 
  d(g(X, g(Y, 0)), g(Z, U)) => g(d(g(X, g(Y, 0)), Z), Y) 
  d(g(X, g(Y, g(Z, U))), V) => g(X, d(g(Y, g(Z, U)), V)) 

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 : [] --> jc 
  d : [jc * jc] --> jc 
  g : [jc * jc] --> jc 
  h : [jc * jc] --> aa 


+++ 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.