/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 
  a : [] --> o 
  append : [o * o] --> o 
  appendAkk : [o * o] --> o 
  cons : [o * o] --> o 
  eq : [o * o] --> o 
  f : [o * o] --> o 
  false : [] --> o 
  length : [o] --> o 
  nil : [] --> o 
  reverse : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  f(true, X) => f(eq(s(length(X)), length(cons(a, X))), append(cons(a, nil), X)) 
  length(nil) => 0 
  length(cons(X, Y)) => s(length(Y)) 
  eq(0, 0) => true 
  eq(s(X), 0) => false 
  eq(0, s(X)) => false 
  eq(s(X), s(Y)) => eq(X, Y) 
  append(X, Y) => appendAkk(reverse(X), Y) 
  appendAkk(nil, X) => X 
  appendAkk(cons(X, Y), Z) => appendAkk(Y, cons(X, Z)) 
  reverse(nil) => nil 
  reverse(cons(X, Y)) => append(reverse(Y), cons(X, nil)) 

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 : [] --> yb 
  a : [] --> yb 
  append : [sd * sd] --> sd 
  appendAkk : [sd * sd] --> sd 
  cons : [yb * sd] --> sd 
  eq : [yb * yb] --> cc 
  f : [cc * sd] --> ma 
  false : [] --> cc 
  length : [sd] --> yb 
  nil : [] --> sd 
  reverse : [sd] --> sd 
  s : [yb] --> yb 
  true : [] --> cc 


+++ 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.