/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 
  1 : [] --> o 
  false : [] --> o 
  half : [o] --> o 
  help : [o * o] --> o 
  ifa : [o * o] --> o 
  ifb : [o * o * o] --> o 
  logZeroError : [] --> o 
  logarithm : [o] --> o 
  lt : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  lt(0, s(X)) => true 
  lt(X, 0) => false 
  lt(s(X), s(Y)) => lt(X, Y) 
  logarithm(X) => ifa(lt(0, X), X) 
  ifa(true, X) => help(X, 1) 
  ifa(false, X) => logZeroError 
  help(X, Y) => ifb(lt(Y, X), X, Y) 
  ifb(true, X, Y) => help(half(X), s(Y)) 
  ifb(false, X, Y) => Y 
  half(0) => 0 
  half(s(0)) => 0 
  half(s(s(X))) => s(half(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 : [] --> dd 
  1 : [] --> dd 
  false : [] --> sb 
  half : [dd] --> dd 
  help : [dd * dd] --> dd 
  ifa : [sb * dd] --> dd 
  ifb : [sb * dd * dd] --> dd 
  logZeroError : [] --> dd 
  logarithm : [dd] --> dd 
  lt : [dd * dd] --> sb 
  s : [dd] --> dd 
  true : [] --> sb 


+++ 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.