/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 
  addLists : [o * o * o] --> o 
  append : [o * o] --> o 
  cons : [o * o] --> o 
  differentLengthError : [] --> o 
  false : [] --> o 
  head : [o] --> o 
  if : [o * o * o * o * o * o * o * o * o] --> o 
  inc : [o] --> o 
  isEmpty : [o] --> o 
  isZero : [o] --> o 
  nil : [] --> o 
  p : [o] --> o 
  s : [o] --> o 
  tail : [o] --> o 
  true : [] --> o 

  isEmpty(cons(X, Y)) => false 
  isEmpty(nil) => true 
  isZero(0) => true 
  isZero(s(X)) => false 
  head(cons(X, Y)) => X 
  tail(cons(X, Y)) => Y 
  tail(nil) => nil 
  append(nil, X) => cons(X, nil) 
  append(cons(X, Y), Z) => cons(X, append(Y, Z)) 
  p(s(s(X))) => s(p(s(X))) 
  p(s(0)) => 0 
  p(0) => 0 
  inc(s(X)) => s(inc(X)) 
  inc(0) => s(0) 
  addLists(X, Y, Z) => if(isEmpty(X), isEmpty(Y), isZero(head(X)), tail(X), tail(Y), cons(p(head(X)), tail(X)), cons(inc(head(Y)), tail(Y)), Z, append(Z, head(Y))) 
  if(true, true, X, Y, Z, U, V, W, Q) => W 
  if(true, false, X, Y, Z, U, V, W, Q) => differentLengthError 
  if(false, true, X, Y, Z, U, V, W, Q) => differentLengthError 
  if(false, false, false, X, Y, Z, U, V, W) => addLists(Z, U, V) 
  if(false, false, true, X, Y, Z, U, V, W) => addLists(X, Y, W) 
  addList(X, Y) => addLists(X, Y, 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 : [] --> ye 
  addList : [ci * ci] --> ci 
  addLists : [ci * ci * ci] --> ci 
  append : [ci * ye] --> ci 
  cons : [ye * ci] --> ci 
  differentLengthError : [] --> ci 
  false : [] --> ff 
  head : [ci] --> ye 
  if : [ff * ff * ff * ci * ci * ci * ci * ci * ci] --> ci 
  inc : [ye] --> ye 
  isEmpty : [ci] --> ff 
  isZero : [ye] --> ff 
  nil : [] --> ci 
  p : [ye] --> ye 
  s : [ye] --> ye 
  tail : [ci] --> ci 
  true : [] --> ff 


+++ 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.