/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 
  false : [] --> o 
  id : [o] --> o 
  if!6220mod : [o * o * o * o * o] --> o 
  if2 : [o * o * o * o] --> o 
  if3 : [o * o * o] --> o 
  le : [o * o] --> o 
  minus : [o * o] --> o 
  mod : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 
  zero : [o] --> o 

  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  zero(0) => true 
  zero(s(X)) => false 
  id(0) => 0 
  id(s(X)) => s(id(X)) 
  minus(X, 0) => X 
  minus(s(X), s(Y)) => minus(X, Y) 
  mod(X, Y) => if!6220mod(zero(X), zero(Y), le(Y, X), id(X), id(Y)) 
  if!6220mod(true, X, Y, Z, U) => 0 
  if!6220mod(false, X, Y, Z, U) => if2(X, Y, Z, U) 
  if2(true, X, Y, Z) => 0 
  if2(false, X, Y, Z) => if3(X, Y, Z) 
  if3(true, X, Y) => mod(minus(X, Y), s(Y)) 
  if3(false, X, Y) => 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 : [] --> xe 
  false : [] --> yc 
  id : [xe] --> xe 
  if!6220mod : [yc * yc * yc * xe * xe] --> xe 
  if2 : [yc * yc * xe * xe] --> xe 
  if3 : [yc * xe * xe] --> xe 
  le : [xe * xe] --> yc 
  minus : [xe * xe] --> xe 
  mod : [xe * xe] --> xe 
  s : [xe] --> xe 
  true : [] --> yc 
  zero : [xe] --> yc 


+++ 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.