/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 
  exp : [o * o] --> o 
  false : [] --> o 
  ge : [o * o] --> o 
  help : [o * o * o * o * o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  times : [o * o] --> o 
  tower : [o * o] --> o 
  towerIter : [o * o * o * o] --> o 
  true : [] --> o 

  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)) 
  exp(X, 0) => s(0) 
  exp(X, s(Y)) => times(X, exp(X, Y)) 
  ge(X, 0) => true 
  ge(0, s(X)) => false 
  ge(s(X), s(Y)) => ge(X, Y) 
  tower(X, Y) => towerIter(0, X, Y, s(0)) 
  towerIter(X, Y, Z, U) => help(ge(X, Y), X, Y, Z, U) 
  help(true, X, Y, Z, U) => U 
  help(false, X, Y, Z, U) => towerIter(s(X), Y, Z, exp(Z, 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 : [] --> ie 
  exp : [ie * ie] --> ie 
  false : [] --> gd 
  ge : [ie * ie] --> gd 
  help : [gd * ie * ie * ie * ie] --> ie 
  plus : [ie * ie] --> ie 
  s : [ie] --> ie 
  times : [ie * ie] --> ie 
  tower : [ie * ie] --> ie 
  towerIter : [ie * ie * ie * ie] --> ie 
  true : [] --> gd 


+++ 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.