/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 
  empty : [o] --> o 
  eq : [o * o] --> o 
  false : [] --> o 
  head : [o] --> o 
  if : [o * o * o * o] --> o 
  if!6220min : [o * o] --> o 
  if!6220replace : [o * o * o * o] --> o 
  le : [o * o] --> o 
  min : [o] --> o 
  nil : [] --> o 
  replace : [o * o * o] --> o 
  s : [o] --> o 
  sort : [o] --> o 
  sortIter : [o * o] --> o 
  tail : [o] --> o 
  true : [] --> o 

  eq(0, 0) => true 
  eq(0, s(X)) => false 
  eq(s(X), 0) => false 
  eq(s(X), s(Y)) => eq(X, Y) 
  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  min(cons(X, nil)) => X 
  min(cons(X, cons(Y, Z))) => if!6220min(le(X, Y), cons(X, cons(Y, Z))) 
  if!6220min(true, cons(X, cons(Y, Z))) => min(cons(X, Z)) 
  if!6220min(false, cons(X, cons(Y, Z))) => min(cons(Y, Z)) 
  replace(X, Y, nil) => nil 
  replace(X, Y, cons(Z, U)) => if!6220replace(eq(X, Z), X, Y, cons(Z, U)) 
  if!6220replace(true, X, Y, cons(Z, U)) => cons(Y, U) 
  if!6220replace(false, X, Y, cons(Z, U)) => cons(Z, replace(X, Y, U)) 
  empty(nil) => true 
  empty(cons(X, Y)) => false 
  head(cons(X, Y)) => X 
  tail(nil) => nil 
  tail(cons(X, Y)) => Y 
  sort(X) => sortIter(X, nil) 
  sortIter(X, Y) => if(empty(X), X, Y, append(Y, cons(min(X), nil))) 
  if(true, X, Y, Z) => Y 
  if(false, X, Y, Z) => sortIter(replace(min(X), head(X), tail(X)), 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 : [] --> vi 
  append : [vi * vi] --> vi 
  cons : [vi * vi] --> vi 
  empty : [vi] --> oh 
  eq : [vi * vi] --> oh 
  false : [] --> oh 
  head : [vi] --> vi 
  if : [oh * vi * vi * vi] --> vi 
  if!6220min : [oh * vi] --> vi 
  if!6220replace : [oh * vi * vi * vi] --> vi 
  le : [vi * vi] --> oh 
  min : [vi] --> vi 
  nil : [] --> vi 
  replace : [vi * vi * vi] --> vi 
  s : [vi] --> vi 
  sort : [vi] --> vi 
  sortIter : [vi * vi] --> vi 
  tail : [vi] --> vi 
  true : [] --> oh 


+++ 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.