/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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 
  cond1 : [o * o] --> o 
  cond2 : [o * o * o] --> o 
  false : [] --> o 
  gr : [o * o] --> o 
  neq : [o * o] --> o 
  p : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  cond1(s(X), Y) => cond2(gr(s(X), Y), s(X), Y) 
  cond2(true, X, Y) => cond1(Y, Y) 
  cond2(false, X, Y) => cond1(p(X), Y) 
  gr(0, X) => false 
  gr(s(X), 0) => true 
  gr(s(X), s(Y)) => gr(X, Y) 
  neq(0, 0) => false 
  neq(0, s(X)) => true 
  neq(s(X), 0) => true 
  neq(s(X), s(Y)) => neq(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 : [] --> bd 
  cond1 : [bd * bd] --> xa 
  cond2 : [tc * bd * bd] --> xa 
  false : [] --> tc 
  gr : [bd * bd] --> tc 
  neq : [bd * bd] --> tc 
  p : [bd] --> bd 
  s : [bd] --> bd 
  true : [] --> tc 


+++ 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.