/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 
  fac : [o] --> o 
  false : [] --> o 
  if : [o * o * o * o] --> o 
  loop : [o * o * o] --> o 
  lt : [o * o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  times : [o * o] --> o 
  true : [] --> o 

  lt(0, s(X)) => true 
  lt(X, 0) => false 
  lt(s(X), s(Y)) => lt(X, Y) 
  times(0, X) => 0 
  times(s(X), Y) => plus(Y, times(X, Y)) 
  plus(0, X) => X 
  plus(s(X), Y) => s(plus(X, Y)) 
  fac(X) => loop(X, s(0), s(0)) 
  loop(X, Y, Z) => if(lt(X, Y), X, Y, Z) 
  if(false, X, Y, Z) => loop(X, s(Y), times(Z, s(Y))) 
  if(true, X, Y, Z) => Z 

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 : [] --> bd 
  fac : [bd] --> bd 
  false : [] --> kc 
  if : [kc * bd * bd * bd] --> bd 
  loop : [bd * bd * bd] --> bd 
  lt : [bd * bd] --> kc 
  plus : [bd * bd] --> bd 
  s : [bd] --> bd 
  times : [bd * bd] --> bd 
  true : [] --> kc 


+++ 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.