/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 
  div : [o * o * o] --> o 
  division : [o * o] --> o 
  false : [] --> o 
  if : [o * o * o * o] --> o 
  inc : [o] --> o 
  lt : [o * o] --> o 
  minus : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  division(X, Y) => div(X, Y, 0) 
  div(X, Y, Z) => if(lt(X, Y), X, Y, inc(Z)) 
  if(true, X, Y, Z) => Z 
  if(false, X, s(Y), Z) => div(minus(X, s(Y)), s(Y), Z) 
  minus(X, 0) => X 
  minus(s(X), s(Y)) => minus(X, Y) 
  lt(X, 0) => false 
  lt(0, s(X)) => true 
  lt(s(X), s(Y)) => lt(X, Y) 
  inc(0) => s(0) 
  inc(s(X)) => s(inc(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:

  0 : [] --> hd 
  div : [hd * hd * hd] --> hd 
  division : [hd * hd] --> hd 
  false : [] --> vc 
  if : [vc * hd * hd * hd] --> hd 
  inc : [hd] --> hd 
  lt : [hd * hd] --> vc 
  minus : [hd * hd] --> hd 
  s : [hd] --> hd 
  true : [] --> vc 


+++ 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.