/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


MAYBE
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  cond1 : [o * o * o] --> o 
  cond2 : [o * o * o] --> o 
  diff : [o * o] --> o 
  equal : [o * o] --> o 
  false : [] --> o 
  gt : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  diff(X, Y) => cond1(equal(X, Y), X, Y) 
  cond1(true, X, Y) => 0 
  cond1(false, X, Y) => cond2(gt(X, Y), X, Y) 
  cond2(true, X, Y) => s(diff(X, s(Y))) 
  cond2(false, X, Y) => s(diff(s(X), Y)) 
  gt(0, X) => false 
  gt(s(X), 0) => true 
  gt(s(X), s(Y)) => gt(X, Y) 
  equal(0, 0) => true 
  equal(s(X), 0) => false 
  equal(0, s(X)) => false 
  equal(s(X), s(Y)) => equal(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 : [] --> id 
  cond1 : [md * id * id] --> id 
  cond2 : [md * id * id] --> id 
  diff : [id * id] --> id 
  equal : [id * id] --> md 
  false : [] --> md 
  gt : [id * id] --> md 
  s : [id] --> id 
  true : [] --> md 


+++ 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.