/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 
  cons : [o * o] --> o 
  nil : [] --> o 
  s : [o] --> o 
  sub : [o * o] --> o 
  zero : [o] --> o 
  zero2 : [o * o] --> o 

  sub(0, 0) => 0 
  sub(s(X), 0) => s(X) 
  sub(0, s(X)) => 0 
  sub(s(X), s(Y)) => sub(X, Y) 
  zero(nil) => zero2(0, nil) 
  zero(cons(X, Y)) => zero2(sub(X, X), cons(X, Y)) 
  zero2(0, nil) => nil 
  zero2(0, cons(X, Y)) => cons(sub(X, X), zero(Y)) 
  zero2(s(X), nil) => zero(nil) 
  zero2(s(X), cons(Y, Z)) => zero(cons(Y, 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 : [] --> mc 
  cons : [mc * uc] --> uc 
  nil : [] --> uc 
  s : [mc] --> mc 
  sub : [mc * mc] --> mc 
  zero : [uc] --> uc 
  zero2 : [mc * uc] --> uc 


+++ 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.