/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 
  add : [o * o] --> o 
  cond1 : [o * o * o] --> o 
  cond2 : [o * o * o] --> o 
  cond3 : [o * o * o] --> o 
  false : [] --> o 
  gr : [o * o] --> o 
  p : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  cond1(true, X, Y) => cond2(gr(X, 0), X, Y) 
  cond2(true, X, Y) => cond1(gr(add(X, Y), 0), p(X), Y) 
  cond2(false, X, Y) => cond3(gr(Y, 0), X, Y) 
  cond3(true, X, Y) => cond1(gr(add(X, Y), 0), X, p(Y)) 
  cond3(false, X, Y) => cond1(gr(add(X, Y), 0), X, Y) 
  gr(0, X) => false 
  gr(s(X), 0) => true 
  gr(s(X), s(Y)) => gr(X, Y) 
  add(0, X) => X 
  add(s(X), Y) => s(add(X, Y)) 
  p(0) => 0 
  p(s(X)) => 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 : [] --> xd 
  add : [xd * xd] --> xd 
  cond1 : [bd * xd * xd] --> gc 
  cond2 : [bd * xd * xd] --> gc 
  cond3 : [bd * xd * xd] --> gc 
  false : [] --> bd 
  gr : [xd * xd] --> bd 
  p : [xd] --> xd 
  s : [xd] --> xd 
  true : [] --> bd 


+++ 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.