/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 
  cond : [o * o * o] --> o 
  double : [o] --> o 
  false : [] --> o 
  le : [o * o] --> o 
  log : [o * o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  square : [o] --> o 
  true : [] --> o 

  log(X, s(s(Y))) => cond(le(X, s(s(Y))), X, Y) 
  cond(true, X, Y) => s(0) 
  cond(false, X, Y) => double(log(X, square(s(s(Y))))) 
  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  double(0) => 0 
  double(s(X)) => s(s(double(X))) 
  square(0) => 0 
  square(s(X)) => s(plus(square(X), double(X))) 
  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 : [] --> pd 
  cond : [yb * pd * pd] --> pd 
  double : [pd] --> pd 
  false : [] --> yb 
  le : [pd * pd] --> yb 
  log : [pd * pd] --> pd 
  plus : [pd * pd] --> pd 
  s : [pd] --> pd 
  square : [pd] --> pd 
  true : [] --> yb 


+++ 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.