/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 
  false : [] --> o 
  if : [o * o * o * o] --> o 
  inc : [o] --> o 
  le : [o * o] --> o 
  log : [o] --> o 
  logIter : [o * o] --> o 
  logZeroError : [] --> o 
  minus : [o * o] --> o 
  quot : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  minus(X, 0) => X 
  minus(s(X), s(Y)) => minus(X, Y) 
  quot(0, s(X)) => 0 
  quot(s(X), s(Y)) => s(quot(minus(X, Y), s(Y))) 
  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  inc(s(X)) => s(inc(X)) 
  inc(0) => s(0) 
  log(X) => logIter(X, 0) 
  logIter(X, Y) => if(le(s(0), X), le(s(s(0)), X), quot(X, s(s(0))), inc(Y)) 
  if(false, X, Y, Z) => logZeroError 
  if(true, false, X, s(Y)) => Y 
  if(true, true, X, Y) => logIter(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 : [] --> ne 
  false : [] --> rd 
  if : [rd * rd * ne * ne] --> ne 
  inc : [ne] --> ne 
  le : [ne * ne] --> rd 
  log : [ne] --> ne 
  logIter : [ne * ne] --> ne 
  logZeroError : [] --> ne 
  minus : [ne * ne] --> ne 
  quot : [ne * ne] --> ne 
  s : [ne] --> ne 
  true : [] --> rd 


+++ 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.