/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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 
  addList : [o * o] --> o 
  cons : [o * o] --> o 
  false : [] --> o 
  if : [o * o * o * o * o] --> o 
  le : [o * o] --> o 
  len : [o] --> o 
  min : [o * o] --> o 
  nil : [] --> o 
  s : [o] --> o 
  sum : [o * o] --> o 
  take : [o * o] --> o 
  true : [] --> o 

  min(0, X) => 0 
  min(s(X), 0) => 0 
  min(s(X), s(Y)) => min(X, Y) 
  len(nil) => 0 
  len(cons(X, Y)) => s(len(Y)) 
  sum(X, 0) => X 
  sum(X, s(Y)) => s(sum(X, Y)) 
  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  take(0, cons(X, Y)) => X 
  take(s(X), cons(Y, Z)) => take(X, Z) 
  addList(X, Y) => if(le(0, min(len(X), len(Y))), 0, X, Y, nil) 
  if(false, X, Y, Z, U) => U 
  if(true, X, Y, Z, U) => if(le(s(X), min(len(Y), len(Z))), s(X), Y, Z, cons(sum(take(X, Y), take(X, Z)), U)) 

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 : [] --> mf 
  addList : [mf * mf] --> mf 
  cons : [mf * mf] --> mf 
  false : [] --> ye 
  if : [ye * mf * mf * mf * mf] --> mf 
  le : [mf * mf] --> ye 
  len : [mf] --> mf 
  min : [mf * mf] --> mf 
  nil : [] --> mf 
  s : [mf] --> mf 
  sum : [mf * mf] --> mf 
  take : [mf * mf] --> mf 
  true : [] --> ye 


+++ 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.