/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 
  append : [o * o] --> o 
  cons : [o * o] --> o 
  eq : [o * o] --> o 
  false : [] --> o 
  ge : [o * o] --> o 
  help : [o * o * o * o] --> o 
  if : [o * o * o * o * o] --> o 
  length : [o] --> o 
  nil : [] --> o 
  rev : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  ge(X, 0) => true 
  ge(0, s(X)) => false 
  ge(s(X), s(Y)) => ge(X, Y) 
  rev(X) => if(X, eq(0, length(X)), nil, 0, length(X)) 
  if(X, true, Y, Z, U) => Y 
  if(X, false, Y, Z, U) => help(s(Z), U, X, Y) 
  help(X, Y, cons(Z, U), V) => if(append(U, cons(Z, nil)), ge(X, Y), cons(Z, V), X, Y) 
  append(nil, X) => X 
  append(cons(X, Y), Z) => cons(X, append(Y, Z)) 
  length(nil) => 0 
  length(cons(X, Y)) => s(length(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 : [] --> be 
  append : [be * be] --> be 
  cons : [be * be] --> be 
  eq : [be * be] --> tc 
  false : [] --> tc 
  ge : [be * be] --> tc 
  help : [be * be * be * be] --> be 
  if : [be * tc * be * be * be] --> be 
  length : [be] --> be 
  nil : [] --> be 
  rev : [be] --> be 
  s : [be] --> be 
  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.