/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 
  baseError : [] --> o 
  false : [] --> o 
  if : [o * o * o * o * o] --> o 
  le : [o * o] --> o 
  log : [o * o] --> o 
  logZeroError : [] --> o 
  loop : [o * o * o * o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  times : [o * o] --> o 
  true : [] --> o 

  le(s(X), 0) => false 
  le(0, X) => true 
  le(s(X), s(Y)) => le(X, Y) 
  plus(0, X) => X 
  plus(s(X), Y) => s(plus(X, Y)) 
  times(0, X) => 0 
  times(s(X), Y) => plus(Y, times(X, Y)) 
  log(X, 0) => baseError 
  log(X, s(0)) => baseError 
  log(0, s(s(X))) => logZeroError 
  log(s(X), s(s(Y))) => loop(s(X), s(s(Y)), s(0), 0) 
  loop(X, s(s(Y)), s(Z), U) => if(le(X, s(Z)), X, s(s(Y)), s(Z), U) 
  if(true, X, Y, Z, U) => U 
  if(false, X, Y, Z, U) => loop(X, Y, times(Y, Z), s(U)) 

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 : [] --> ze 
  baseError : [] --> ze 
  false : [] --> ud 
  if : [ud * ze * ze * ze * ze] --> ze 
  le : [ze * ze] --> ud 
  log : [ze * ze] --> ze 
  logZeroError : [] --> ze 
  loop : [ze * ze * ze * ze] --> ze 
  plus : [ze * ze] --> ze 
  s : [ze] --> ze 
  times : [ze * ze] --> ze 
  true : [] --> ud 


+++ 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.